Adaptive Runge-Kutta methodsby Peter Stone, Dept. of Applied and Environmental Sciences, RMITpeter.stone@rmit.edu.au . . or . . peterstone@optusnet.com.auVersion: 10.12.2003restart;;

RMIT file path to read Maple m-file of differential equation solving procedures with the interface desolve. read "J:\\Class_Notes/Peter Stone/MapleMath/procdrs/DEsol.m";;Another file path. read "D:\\Maple 9/extensions/DEsol.m";;

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RMIT file path to read Maple m-file of procedures for numerical root-findingread "J:\\Class_Notes/Peter Stone/MapleMath/procdrs/roots.m";;Another file path. read "D:\\MapleMath/procdrs/roots.m";;

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;The Maple procedure desolveRK given in this worksheet uses a combined 4th and 5th order Runge-Kutta method, with adaptive step-size control, to provide a numerical solution for a differential equation of the form NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiZkc2JCUieEclInlH.The difference between the 4th and 5th order values gives an estimate for the error of the 4th order value.Since the error in the 4th order value is proportional to NiMqJCUiaEciIiY=, we have the relation NiMvKiYmJSJoRzYjIiIhIiIiJkYmNiNGKSEiIikqJiYlKGVwc2lsb25HRidGKSZGMEYrRiwqJkYpRikiIiZGLA==between step sizes NiMmJSJoRzYjIiIh and NiMmJSJoRzYjIiIi, and the corresponding errors NiMmJShlcHNpbG9uRzYjIiIh and NiMmJShlcHNpbG9uRzYjIiIi.This relation can be used to decrease the step-size, if the error is too large and repeat the failed step, or increase the step-size for the next step, if the error is small enough for the current step to suggest that this should be possible.The actual value given is the 5th order value, even though the error control really applies to the 4th order value. (Sneaky or what!)A combined 7th and 8th order Runge-Kutta method is also available. The error in the 7th order method is estimated by an 8th order method, and the step-size is controlled using the relation NiMvKiYmJSJoRzYjIiIhIiIiJkYmNiNGKSEiIikqJiYlKGVwc2lsb25HRidGKSZGMEYrRiwqJkYpRikiIilGLA== between step sizes NiMmJSJoRzYjIiIh and NiMmJSJoRzYjIiIi, and the corresponding errors NiMmJShlcHNpbG9uRzYjIiIh and NiMmJShlcHNpbG9uRzYjIiIi.For more details of the method see: "Numerical Recipes in C ", Cambridge University Press, pp 714 -722.The procedure below uses constants from: "Numerical Methods for Differential Equations", by John R. Dormand, CRC Press, pages 84 & 110.

This utility routine is required by examples in a later section.;

Calling Sequence: comparewithfcn( pts, f ,options ) comparewithfcn( pts, fx, x , options ) Parameters: pts - a list of points NiM3JDckJiUieEc2IyIiIiYlInlHRic7NyQmRiY2IyIiIyZGKkYuNyQmRiY2IyUibkcmRipGMw==. f or fx - a function of one variable or an expression fx defining a function of one variable. x - the independent variable is required when the 2nd argument is an expression fx. Description:The procedure comparewithfcn tabulates the list of points vertically as the first two columns of a matrix.The values of the 2nd components are compared with the values obtained by applying the given function to the corresponding 1st components, and listing these "exact values" in the 3rd column of the matrix. The absolute or relative error is given in the 4th column.Alternatively, the same information can be printed out line by line.The maximum of the absolute or relative errors is given. When the absolute error is tabulated, the mean absolute error is given.Options:mode=linebyline/matrixWith the option "mode=linebyline" the information is printed out line by line. This is the default option.With the option "mode=matrix" the tabulated information is given in a matrix as a return value of the procedure.errtype=relative/absoluteThis option determines whether the absolute or relative error is given. The default is "errtype=relative" in which case the relative error is tabulated."errtype=RELATIVE" and "errtype=REL" are equivalent to "errtype=relative""errtype=ABSOLUTE" and "errtype=ABS" are equivalent to "errtype=absolute"."errtype" can also be typed as "errortype".Note: If the true value of the function is zero, the relative error is infinite. The maximum relative error is computed for the remaining relative errors, ignoring any infinite relative errors.

How to activate: To make the procedure active open the subsection, place the cursor anywhere after the prompt [ > and press [Enter]. You can then close up the subsection.

comparewithfcn:= proc(pts,f) local xx,y1,y2,r,i,j,n,rows,fn,x,prec,prcsn,saveDigits, startoptions,Options,md,t,maxerr,format1,format2,g,e,zero, proctype,ertyp,xmax,prec1,prec2,vars,averr; proctype := false; if nargs<2 then error "invalid arguments; the basic syntax is 'comparewithfcn([[x1,y1],..,[xn,yn]],f(x),x)' or 'comparewithfcn([[x1,y1],..,[xn,yn]],f)'" end if; if not type(pts,listlist) then error "the 1st argument, %1, is invalid .. it should be a list of points",pts; end if; if nargs>2 and not type(args[3],equation) then x := args[3]; if not type(x,name) then error "the 3rd optional argument must be the name of the independent variable" end if; startoptions := 4; vars := indets(f,name) minus indets(f,realcons); if vars<>{x} then if not type(f,algebraic) or not has(indets(f),{Int,Sum}) then error "the 2nd argument, %1, is invalid .. it should be an expression which depends only on the single variable %2",f,x; end if; end if; else if type(f,procedure) or (type(f,`@`) and type({op(f)},set(procedure))) then proctype := true; startoptions := 3; else error "the 2nd argument, %1, is invalid .. it should be a function of one variable or an expression in the one variable given as a 3rd argument",f; end if; end if; # Get the options. md := 0; ertyp := 1; if nargs>=startoptions then Options :=[args[startoptions..nargs]]; if not type(Options,list(equation)) then error "each optional argument must be an equation" end if; if hasoption(Options,'mode','md','Options') then if not (md='linebyline' or md='matrix') then error "\"mode\" must be 'matrix' or 'linebyline'" end if; if md='linebyline' then md := 0 else md := 1 end if; end if; if hasoption(Options,'errtype','ertyp','Options') or hasoption(Options,'errortype','ertyp','Options') then if not member(ertyp,{'absolute','ABSOLUTE','ABS','relative','RELATIVE','REL'}) then error "\"errtype\" option must be 'absolute' <-> 'ABSOLUTE' <-> 'ABS' or 'relative' <-> 'RELATIVE' <-> 'REL'" end if; if member(ertyp,{'absolute','ABSOLUTE','ABS'}) then ertyp := 0 else ertyp := 1 end if; end if; if nops(Options)>0 then error "%1 is not a valid option for %2",op(1,Options), procname; end if; end if; n := nops(pts); prcsn := 0; # Check the data and find its maximum precision. for i to n do if nops(pts[i])<>2 then error "the 1st argument must be a list of points, where each point is itself a list with two members" end if; t := pts[i,1]; if type(t,float) and type(t,numeric) then prec1 := length(convert(op(1,t),string)); elif type(t,realcons) then prec1 := Digits; else error "the 1st argument must be a list of points, where each point is itself a list of two real numbers" end if; t := pts[i,2]; if type(t,float) and type(t,numeric) then prec2 := length(convert(op(1,t),string)); elif type(t,realcons) then prec2 := Digits; else error "the 1st argument must be a list of points, where each point is itself a list of two real numbers" end if; prec := max(prec1,prec2); if prec>prcsn then prcsn := prec end if; end do; saveDigits := Digits; Digits := prcsn; prec := trunc(prcsn/2); if proctype then fn := f; else fn := unapply(evalf(f),x); end if; rows := NULL; maxerr := 0; averr := 0; zero := false; format1 := cat("%",convert(prcsn+3,string),".",convert(prcsn-1,string),g); format2 := cat("%",convert(prec+3,string),".",convert(prec-1,string),e); for i from 1 to n do xx := evalf(pts[i,1]); y1 := evalf(pts[i,2]); y2 := traperror(evalf(fn(xx))); if y2=lasterror or not type(y2,numeric) then error "function failed to evaluate to a real floating point number at %1",xx; end if; if ertyp=0 then r := evalf(abs(y1-y2)); averr := averr+r; if r>maxerr then maxerr := r; xmax := xx; end if; else if y2<>0 then r := evalf(abs(y1-y2)/abs(y2)); if r>maxerr then maxerr := r; xmax := xx; end if; else zero := true; r := infinity; end if; end if; if md=0 then printf(format1,xx); printf(` `); printf(format1,y1); printf(` function val: `); printf(format1,y2); if ertyp=0 then printf(` abs err: `); printf(format2,r); else printf(` rel err: `); if y2<>0 then printf(format2,r) else printf(infinity); end if; end if; printf(`\n`); else rows := rows,[xx,y1,y2,r]; end if; end do; print(``); if ertyp=0 then printf(` Maximum absolute error: `); else printf(` Maximum relative error: `); end if; printf(format2,maxerr); if maxerr<>0 then printf(`\n obtained for the input value: `); printf(format1,xmax); end if; if ertyp=1 and zero then printf(`\n excluding any cases where the function value is zero.`); end if; if ertyp=0 then averr := averr/n; print(``); printf(` Mean absolute error: `); printf(format2,averr); end if; Digits := saveDigits; if md=1 then print(``); if ertyp=0 then return array([[x,"discrete value","function value","absolute err"],rows]); else return array([[x,"discrete value","function value","relative err"],rows]); end if; else return NULL; end if; end proc:;

Examples appear in a later section

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Calling Sequence: desolveRK( {de,ic}, rng ) desolveRK( {de,ic}, y(x), rng )Parameters: de - a first order differential equation with the derivative given in the form diff(y(x),x), (if x and y are the independent and dependent variables respectively). ic - an initial condition in the form y(x0) = y0. rng - a range of values x=a..b containing the initial value x0 of the independent variable. Description:The procedure desolveRK uses a prescribed Runge-Kutta method to find a numerical solution to a first order differential equation of the form NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiZkc2JCUieEclInlH.The numerical solution is given as a list of points along the solution curve with equally spaced values of the first coordinate.Options: The numerical solution can be given in two ways:As a list of points along the solution curve.As a numerical procedure of a real variable, defined throughout the specified range, which uses an appropriate method to interpolate between pre-computed points along the solution curve.In both cases there is a choice of using a fixed or variable step-size. In the case of variable step-size a method of adaptive step-size is used in an attempt to control the error.Options for both fixed and variable step-size: stepsize=fixed/variableWith the option "stepsize=fixed", equally spaced x values are used in construction of the numerical solution.With the option "stepsize=variable", the stepsize is controlled by estmating the error at each step.The default option is "stepsize=variable".Note: The stepsize option can be omitted when the method is specified. output=points/discrete/rkstep/rkinterp/continuous/procedureWith the options "output=points" or "output=discrete" desolveRK returns the numerical solution of the differential equation as a discrete set of points along the approximate solution curve.With the option "output=continuous" , "output=procedure" , "output=rkstep" or "output=rkinterp" desolveRK returns the numerical solution of the differential equation as a numerical procedure. With the option "output=rkstep", the numerical function requires a number of evaluations of the direction field for each new function value calculated. With the option "output=rkinterp", the numerical function uses extensive pre-computed data to interpolate between the points of the discrete solution instead of using the direction field. Note: "output=continuous" , "output=procedure" are equivalent to "output=rkstep" and the option "output=rkinterp" is only available for variable step-size when the combined 4th and 5th Runge-Kutta method is used via the option "method=rk45".info=true/false The option info=true allows the progress of the procedure to be monitored by printing the result of each step as it is computed. Options for fixed step-size only: method=euler/impeuler/rk2,rk3,rk4,rk5,rk6,rk7,rk8METHODS USED:method=euler, the Euler methodmethod=impeuler, the improved Euler methodmethod=rk2, the "classical" mid-point order 2 methodmethod=rk3, the "classical" order 3 methodmethod=rk4, the "classical" order 4 methodmethod=rk5, the Fehlberg order 5 methodmethod=rk6, a Prince-Dormand order 6 methodmethod=rk7, a Butcher order 7 methodmethod=rk8, a Butcher order 8 methodThe default is "method=euler".alpha=NiMlJmFscGhhRw== If this option is used, it overrides the method option to provide a custom order 2 Runge-Kutta method.The formula used has form: NiMvJiUiZkc2IyIiIi1GJTYkJiUieEc2IyUia0cmJSJ5R0Ys NiMvJiUiZkc2IyIiIy1GJTYkLCYmJSJ4RzYjJSJrRyIiIiomJSZhbHBoYUdGLyUiaEdGL0YvLCYmJSJ5R0YtRi8qKCUlYmV0YUdGLyZGJTYjRi9GL0YyRi9GLw== NiMvJiUieUc2IywmJSJrRyIiIkYpRiksKCZGJTYjRihGKSomJSJzR0YpJiUiZkc2I0YpRilGKSomJSJ0R0YpJkYwNiMiIiNGKUYp When a numerical value NiMlJmFscGhhRw== is chosen for alpha, where NiMyIiIhJSZhbHBoYUc= and NiMxJSZhbHBoYUciIiI=, s and t are calculated from the relations: NiMvLCYlInNHIiIiJSJ0R0YmRiY=,and NiMlJmFscGhhRw== = NiMvJSViZXRhRyomIiIiRiYqJiIiI0YmJSJ0R0YmISIi.This means that the second function evaluation used by the method occurs at a fraction NiMlJmFscGhhRw== of the distance across the step.For example, "alpha=0.5" is equivalent to choosing "method=rk2" and "alpha=1" is equivalent to choosing "method=impeuler".Options for variable step-size only:method=rk45/rk78This option gives a choice between either an adaptive combined 4th and 5th order Runge-Kutta method via "method=rk45", or an adaptive combined 7th and 8th order Runge-Kutta method via "method=rk78". The default option is "method=rk45".hstart=hThe step-size for the first trial step. The default is "hstart=0.1*NiMpIiM1LSUlY2VpbEc2IywkKiYlJ0RpZ2l0c0ciIiJGJCEiIkYs".hmax=hThe maximum step-size. The default is "hmax=0.25" when method=rk45, and "hmax=0.5" when method=rk78.hmin=h The minimum step-size. The default is "hmin=min(0.5*NiMpIiM1LCQiIiYhIiI=, NiMqJiUnaHN0YXJ0RyIiIiIlKz8hIiI=)".Note: hstart must lie between hmin and hmax.maxsteps=nThe maximum number of steps to be used. An error message results if the maximum number of steps is reached before reaching the end of the solution interval. The default is "maxsteps=2000".tolerance=NiMlKGVwc2lsb25H When the error control option is set to "relativeerror", the step size is chosen adaptively to try to ensure that the relative error for each step is no greater than NiMlKGVwc2lsb25H. Other error control options use NiMlKGVwc2lsb25H in an appropriate way. See "errorcontrol" for more information. The default is "tolerance=10^(-Digits)". errorcontrol=auto/absolute/relative/accumulativeFor the current setting of tolerance=NiMlKGVwc2lsb25H, the step size is chosen to try to ensure that when "errorcontrol" is:auto - NiMxLSUkYWJzRzYjLCYlInlHIiIiJkYoNiMlImtHISIiKiYlKGVwc2lsb25HRiktJSRtYXhHNiUtRiU2I0YqLUYlNiMqJi0lImZHNiQmJSJ4R0YrRipGKSUiaEdGKSUldGlueUdGKQ== absolute - NiMxLSUkYWJzRzYjLCYlInlHIiIiJkYoNiMlImtHISIiKiYlKGVwc2lsb25HRiktRiU2Iy0lJG1heEc2JCZGKDYjRjMlJXRpbnlHRik= , where NiMmJSJ5RzYjJSRtYXhH is updated as the computation progresses, with an initial estimate provided by the option "maxvalue". This error control option would be an appropriate choice if the solution is oscillatory in nature. relative - NiMxLSUkYWJzRzYjLCYlInlHIiIiJkYoNiMlImtHISIiKiYlKGVwc2lsb25HRiktJSRtYXhHNiQtRiU2I0YqJSV0aW55R0Yp This is a general purpose choice except that problems would arise if y is close to 0. If this is likely to happen it would be better to use the default option "auto". accumulative - NiMxLSUkYWJzRzYjLCYlInlHIiIiJkYoNiMlImtHISIiKiYlKGVwc2lsb25HRiktJSRtYXhHNiQtRiU2IyomLSUiZkc2JCYlInhHRitGKkYpJSJoR0YpJSV0aW55R0Yp This is the most stringent error control option, which attempts to control the global error."tiny" is NiMpIiM1LCQqJiIiJCIiIiUnRGlnaXRzR0YoISIi, and is included to avoid the possibility of division by 0.The default is "errorcontrol=auto".maxvalue=aThis is only used in the case where the "errorcontrol" option has been set to "absolute". If no value is provided, the maximum value is updated as the computation procedes.

How to activate: To make the procedures active open the subsections, place the cursor anywhere after the prompt [ > and press [Enter]. You can then close up the subsections.Note: All four procedures must be activated for desolveRK to work in all appropriate situations.

desolveRK := proc(ff) local vars,derivs,x,yx,y,df,dff,Options,mthd,rg, de,ic,lsic,rsic,x0,y0,tt,rng,z,outpt,stps,lftstps, rghtstps,drv,lftprt,rghtprt,reverse,saveDigits,lft,rght, startopts,xx,yy,ee,arg2OK,stpsz,dblrk,nvars; reverse := proc(lst::list) local n,i; n := nops(lst); [seq(lst[n-i],i=0..n-2)]; end proc: dblrk := proc(lft::realcons,rght::realcons,x0::realcons, lftprc::procedure,rghtprc::procedure) proc(x::realcons) local left,right,leftproc,rightproc,xx0,xx,saveDigits; options `Copyright 2002 by Peter Stone`; leftproc := lftprc; rightproc := rghtprc; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); xx0 := evalf(x0); left := evalf(lft); right := evalf(rght); xx := evalf(x); if xx<=right and xx>=xx0 then Digits := saveDigits; return rightproc(xx); elif xx>=left and xx<=xx0 then Digits := saveDigits; return leftproc(xx); else error "argument must be between %1 and %2",left,right; end if; end proc; end proc: # of dblrk # start of main procedure desolveRK if type(ff,set(equation)) and nops(ff)=2 then de := op(1,ff); ic := op(2,ff); if not has(de,diff) then de := op(2,ff); ic := op(1,ff); end if; else error "the 1st argument, %1, is invalid .. it should be an equation or a set of 2 equations" end if; startopts := 3; if nargs>1 then ee := args[2]; if type(ee,range) or type(ee,name=range) then rng := ee; else arg2OK := true; if type(ee,function) and nops(ee)=1 then yy := op(0,ee); xx := op(1,ee); if type(xx,name) and type(yy,name) then startopts := 4; if nargs>2 then rng := args[3]; else error "expecting a 3rd argument" end if; else arg2OK := false; end if; else arg2OK := false; end if; if not arg2OK then error "2nd argument, %1, has incorrect form for the dependent variable",ee; end if; end if; else error "expecting a 2nd argument" end if; stps := 20; stpsz := 'variable'; outpt := 'rkstep'; mthd := 'rk45'; Options :=[]; if nargs>=startopts then Options:=[args[startopts..nargs]]; if not type(Options,list(equation)) then error "each optional argument must be an equation" end if; if hasoption(Options,'stepsize','stpsz','Options') then if not(stpsz='fixed' or stpsz='variable') then error "\"stepsize\" must be 'fixed' or 'variable'" end if; else if hasoption(Options,'method','mthd') then if mthd='rk45' or mthd='rk78' then stpsz := variable else stpsz := 'fixed' end if; end if; end if; if stpsz='fixed' then outpt := 'points' end if; if hasoption(Options,'output','outpt') then if not member(outpt,{'points','discrete','rkstep','rkinterp','continuous','procedure'}) then error "\"output\" must be one of 'points','discrete','continuous','rkstep','rkinterp' or 'procedure'" end if; if outpt='discrete' then outpt := 'points' end if; if outpt='continuous' or outpt='procedure' then outpt := 'rkstep' end if; if mthd<>'rk45' and outpt='rkinterp' then error "when \"method\" is %1, \"output\" cannot be 'rkinterp', but must be one of 'points','discrete','continuous','rkstep' or 'procedure'",mthd; end if; end if; if stpsz='fixed' then if hasoption(Options,'steps','stps') then if not type(stps,'posint') then error"\"steps\" must be a positive integer" end if; end if; else # the step-size is variable if hasoption(Options,'steps') then error"\"steps\" is not a valid option when the step-size is variable" end if; if hasoption(Options,'method','mthd') then if not (mthd='rk45' or mthd='rk78') then error "\"method\" must be 'rk45' or 'rk78'" end if; Options := convert({op(Options)} minus {method=mthd},list); end if; end if; end if; if has(de,'D') then de := convert(de,'diff') end if; derivs := indets(de,'specfunc(anything,diff)'); if derivs={} then error "the equation, %1, is not an ordinary differential equation",de; end if; nvars := nops(indets(derivs,'name')); if nvars<>1 then if nvars=0 then error "there is a problem with the independent variable occurring in the derivative(s)"; else error "there should only be one independent variable in the differential equation" end if; end if; nvars := nops(indets(derivs,'anyfunc(name)')); if nvars<>1 then if nvars=0 then error "there is a problem with the dependent variable occurring in the derivative(s)" else error "there should only be one dependent variable in the differential equation" end if; end if; if nops(derivs)<>1 then error "there are too many derivatives in the differential equation .. note that the differential equation must be of order 1" end if; df := op(1,derivs); if type(df,function) and op(0,df)=diff and nops(df)=2 then yx := op(1,df); if not type(yx,anyfunc(name)) then error "the 1st argument %1, in the derivative, %2, is invalid .. it should be the 'unknown' dependent variable",yx,df; end if; x := op(2,df); if not type(x,name) then error "the 2nd argument %1, in the derivative, %2, is invalid .. it should be the dependent variable",x,df; end if; else error "the derivative, %1, does not make sense",df; end if; y := op(0,yx); vars := indets(de,name); if member(y,vars) then error "%1 and %2 cannot both appear in the differential equation",yx,y; end if; if op(1,yx)<>x then error "the derivative, %1, does not make sense",df; end if; if startopts=4 then if x<>xx or y<>yy then error "cannot solve the differential equation for %1",ee; end if; end if; lsic := lhs(ic); if type(lsic,function) and op(0,lsic)=y and nops(lsic)=1 and type(op(1,lsic),algebraic) then x0 := op(1,lsic); else error "the initial condition is not decipherable" end if; rsic := rhs(ic); if type(rsic,realcons) then y0 := rsic; else error "the initial condition is not decipherable" end if; if type(rng,equation) then z := op(1,rng); if not type(z,name) or z<>x then error "left side, %1, of equation for solution range must be the independent variable",x; end if; rg := op(2,rng); else rg := rng; end if; if not type(rg,realcons..realcons) then error "the range for the solution must have real end values" end if; lft := op(1,rg); rght := op(2,rg); if signum(rght-lft)=0 then error "the range for the solution must have distinct end points" end if; if signum(rght-lft)<0 then # swap over tt := lft; lft := rght; rght := tt; end if; if signum(x0-rght)>0 or signum(x0-lft)<0 then error "the range for the solution must contain the initial value of the independent variable" end if; drv := rhs(isolate(de,df)); if nops([drv])<>1 then error "cannot obtain a unique expression for the derivative" end if; drv := subs(yx=y,drv); if stpsz=fixed then if signum(x0-lft)=0 then return rungk(drv,x=x0..rght,y=y0,op(Options)); elif signum(x0-rght)=0 then return rungk(drv,x=x0..lft,y=y0,op(Options)); else lftstps := trunc(evalf((x0-lft)/(rght-lft)*stps,Digits+5)); rghtstps := stps-lftstps; Options := convert({op(Options)} minus {steps=stps},list); rghtprt := rungk(drv,x=x0..rght,y=y0, steps=rghtstps,op(Options)); lftprt := rungk(drv,x=x0..lft,y=y0, steps=lftstps,op(Options)); if outpt=rkstep then return dblrk(lft,rght,x0,lftprt,rghtprt); else return [op(reverse(lftprt)),op(rghtprt)]; end if; end if; else # step-size is variable if mthd='rk45' then if signum(x0-lft)=0 then return rungk45(drv,x=x0..rght,y=y0,op(Options)); elif signum(x0-rght)=0 then return rungk45(drv,x=x0..lft,y=y0,op(Options)); else rghtprt := rungk45(drv,x=x0..rght,y=y0,op(Options)); lftprt := rungk45(drv,x=x0..lft,y=y0,op(Options)); if outpt=rkstep or outpt=rkinterp then return dblrk(lft,rght,x0,lftprt,rghtprt); else return [op(reverse(lftprt)),op(rghtprt)]; end if; end if; else # mthd='rk78' if signum(x0-lft)=0 then return rungk78(drv,x=x0..rght,y=y0,op(Options)) elif signum(x0-rght)=0 then return rungk78(drv,x=x0..lft,y=y0,op(Options)) else rghtprt := rungk78(drv,x=x0..rght,y=y0,op(Options)); lftprt := rungk78(drv,x=x0..lft,y=y0,op(Options)); if outpt=rkstep then return dblrk(lft,rght,x0,lftprt,rghtprt); else return [op(reverse(lftprt)),op(rghtprt)]; end if; end if; end if; end if; end proc:

rungk := proc(fxy::algebraic,xrng::(name=realcons..realcons), iy::(name=realcons)) local x0,xn,y0,soln,xk,yk,k,xmax,h,x,y,fn,gn,xrg, c1,c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a42,a43, a51,a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81, a82,a83,a84,a85,a86,a87,a91,a94,a95,a96,a97,a98,aA1, aA4,aA5,aA6,aA7,aA8,aA9,aB1,aB4,aB5,aB6,aB7,aB8,aB9, aBA,aC1,aC6,aC7,aC8,aC9,aCA,aD1,aD4, aD5,aD6,aD7,aD8,aD9,aDA,aDC,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA, fB,fC,fD,b1,b2,b3,b4,b5,b6,b7,b8,b9,bA,bB,bC,bD, k1,k2,k3,hh,hk1,hk2,hk4,saveDigits,Options,stps,mthd,j,i, prntflg,s,t,c,eulerstep,impeulerstep,rk2step,rk3step,rk4step, rk5step,rk6step,rk7step,rk8step,customrk2step,outpt; eulerstep := proc(x_eulerstep::realcons) local xk,yk,jF,jM,jS,n,h, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_eulerstep); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; ys := yk + h*fn(xk,yk); Digits := saveDigits; evalf(ys); end proc: # of eulerstep impeulerstep := proc(x_impeulerstep::realcons) local xk,yk,jF,jM,jS,n,h,k1,k2, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_impeulerstep); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; k1 := h*fn(xk,yk); k2 := h*fn(xk + h,yk + k1); ys := yk + (k1 + k2)/2; Digits := saveDigits; evalf(ys); end proc: # of impeulerstep rk2step := proc(x_rk2step::realcons) local xk,yk,jF,jM,jS,n,h,hh,k1, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk2step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; hh := h/2; k1 := hh*fn(xk,yk); ys := yk + h*fn(xk + hh,yk + k1); Digits := saveDigits; evalf(ys); end proc: # of rk2step customrk2step := proc(x_customrk2step::realcons) local xk,yk,jF,jM,jS,n,h,f1,f2,s,t,c, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; c := _ALPH; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_customrk2step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; t := 1/(2*c); s := 1 - t; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); f2 := fn(xk + c*h,yk + c*f1*h); ys := yk + h*(s*f1 + t*f2); Digits := saveDigits; evalf(ys); end proc: # of customrk2step rk3step := proc(x_rk3step::realcons) local xk,yk,jF,jM,jS,n,h,k1,k2,k3, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk3step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; k1 := h*fn(xk,yk); k2 := h*fn(xk+h/2,yk + k1/2); k3 := h*fn(xk+h,yk - k1 + 2*k2); ys := yk + (k1 + 4*k2 + k3)/6; Digits := saveDigits; evalf(ys); end proc: # of rk3step rk4step := proc(x_rk4step::realcons) local xk,yk,jF,jM,jS,n,h,hh,hk1,hk2,k3,hk4, data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk4step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; hh := h/2; hk1 := hh*fn(xk,yk); hk2 := hh*fn(xk + hh,yk + hk1); k3 := h*fn(xk + hh,yk + hk2); hk4 := hh*fn(xk + h,yk + k3); ys := yk + (hk1 + hk2 + hk2 + k3 + hk4)/3; Digits := saveDigits; evalf(ys); end proc: # of rk4step rk5step := proc(x_rk5step::realcons) local c2,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51, a52,a53,a54,a61,a62,a63,a64,a65,a71,a73,a74,a75,a76, f1,f2,f3,f4,f5,f6,b1,b3,b4,b5,b6, xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk5step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := 0.2; c3 := 0.3; c4 := 0.8; c5 := evalf(8/9); a21 := c2; a31 := 0.075; a32 := 0.225; a41 := evalf(44/45); a42 := evalf(-56/15); a43 := evalf(32/9); a51 := evalf(19372/6561); a52 := evalf(-25360/2187); a53 := evalf(64448/6561); a54 := evalf(-212/729); a61 := evalf(9017/3168); a62 := evalf(-355/33); a63 := evalf(46732/5247); a64 := evalf(49/176); a65 := evalf(-5103/18656); a71 := evalf(35/384); a73 := evalf(500/1113); a74 := evalf(125/192); a75 := evalf(-2187/6784); a76 := evalf(11/84); b1 := a71; b3 := a73; b4 := a74; b5 := a75; b6 := a76; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + h,yk + t*h); ys := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h; Digits := saveDigits; evalf(ys); end proc: # of rk5step rk6step := proc(x_rk6step::realcons) local c2,c3,c4,c5,c6,a21,a31,a32,a41,a42,a43,a51, a52,a53,a54,a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76, a81,a82,a83,a84,a85,a86,f1,f2,f3,f4,f5,f6,f7,f8,b1,b3,b4,b5, b6,b7,b8,xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk6step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := 0.1; c3 := evalf(2/9); c4 := evalf(3/7); c5 := 0.6; c6 := 0.8; # interior weights a21 := 0.1; a31 := evalf(-2/81); a32 := evalf(20/81); a41 := evalf(615/1372); a42 := evalf(-270/343); a43 := evalf(1053/1372); a51 := evalf(3243/5500); a52 := evalf(-54/55); a53 := evalf(50949/71500); a54 := evalf(4998/17875); a61 := evalf(-26492/37125); a62 := evalf(72/55); a63 := evalf(2808/23375); a64 := evalf(-24206/37125); a65 := evalf(338/459); a71 := evalf(5561/2376); a72 := evalf(-35/11); a73 := evalf(-24117/31603); a74 := evalf(899983/200772); a75 := evalf(-5225/1836); a76 := evalf(3925/4056); a81 := evalf(465467/266112); a82 := evalf(-2945/1232); a83 := evalf(-5610201/14158144); a84 := evalf(10513573/3212352); a85 := evalf(-424325/205632); a86 := evalf(376225/454272); # exterior weights b1 := evalf(61/864); b3 := evalf(98415/321776); b4 := evalf(16807/146016); b5 := evalf(1375/7344); b6 := evalf(1375/5408); b7 := evalf(-37/1120); b8 := 0.1; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + h,yk + t*h); t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6; f8 := fn(xk + h,yk + t*h); ys := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h; Digits := saveDigits; evalf(ys); end proc: # of rk6step rk7step := proc(x_rk7step::realcons) local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a43, a51,a53,a54,a61,a64,a65,a71,a74,a75,a76,a81,a85,a86,a87, a91,a94,a95,a96,a97,a98,aA1,aA4,aA5,aA6,aA7,aA8,aA9,aB1, aB4,aB5,aB6,aB7,aB8,aB9,aBA,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA, fB,b1,b6,b7,b8,b9,bA,bB,xk,yk,t,jF,jM,jS,n,h,data,fn,xx, ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk7step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := evalf(2/27); c3 := evalf(1/9); c4 := evalf(1/6); c5 := evalf(5/12); c6 := 0.5; c7 := evalf(5/6); c8 := c4; c9 := evalf(2/3); cA := evalf(1/3); # interior weights a21 := c2; a31 := evalf(1/36); a32 := evalf(1/12); a41 := evalf(1/24); a43 := 0.125; a51 := c5; a53 := -1.5625; a54 := 1.5625; a61 := 0.05; a64 := 0.25; a65 := 0.2; a71 := evalf(-25/108); a74 := evalf(125/108); a75 := evalf(-65/27); a76 := evalf(125/54); a81 := evalf(31/300); a85 := evalf(61/225); a86 := evalf(-2/9); a87 := evalf(13/900); a91 := 2; a94 := evalf(-53/6); a95 := evalf(704/45); a96 := evalf(-107/9); a97 := evalf(67/90); a98 := 3; aA1 := evalf(-91/108); aA4 := evalf(23/108); aA5 := evalf(-976/135); aA6 := evalf(311/54); aA7 := evalf(-19/60); aA8 := evalf(17/6); aA9 := -a32; aB1 := evalf(2383/4100); aB4 := evalf(-341/164); aB5 := evalf(4496/1025); aB6 := evalf(-301/82); aB7 := evalf(2133/4100); aB8 := evalf(45/82); aB9 := evalf(45/164); aBA := evalf(18/41); # exterior weights b1 := evalf(41/840); b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bB := b1; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aB1*f1 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9 + aBA*fA; fB := fn(xk + h,yk + t*h); ys := yk + (b1*f1 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA + bB*fB)*h; Digits := saveDigits; evalf(ys); end proc: # of rk7step rk8step := proc(x_rk8step::realcons) local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a43, a51,a53,a54,a61,a64,a65,a71,a74,a75,a76,a81,a85,a86,a87, a91,a94,a95,a96,a97,a98,aA1,aA4,aA5,aA6,aA7,aA8,aA9,aC1, aC6,aC7,aC8,aC9,aCA,aD1,aD4,aD5,aD6,aD7,aD8,aD9,aDA,f1, f2,f3,f4,f5,f6,f7,f8,f9,fA,fC,fD,b6,b7,b8,b9,bA,bC,bD, xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk8step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := evalf(2/27); c3 := evalf(1/9); c4 := evalf(1/6); c5 := evalf(5/12); c6 := 0.5; c7 := evalf(5/6); c8 := c4; c9 := evalf(2/3); cA := evalf(1/3); # interior weights a21 := c2; a31 := evalf(1/36); a32 := evalf(1/12); a41 := evalf(1/24); a43 := 0.125; a51 := c5; a53 := -1.5625; a54 := 1.5625; a61 := 0.05; a64 := 0.25; a65 := 0.2; a71 := evalf(-25/108); a74 := evalf(125/108); a75 := evalf(-65/27); a76 := evalf(125/54); a81 := evalf(31/300); a85 := evalf(61/225); a86 := evalf(-2/9); a87 := evalf(13/900); a91 := 2; a94 := evalf(-53/6); a95 := evalf(704/45); a96 := evalf(-107/9); a97 := evalf(67/90); a98 := 3; aA1 := evalf(-91/108); aA4 := evalf(23/108); aA5 := evalf(-976/135); aA6 := evalf(311/54); aA7 := evalf(-19/60); aA8 := evalf(17/6); aA9 := -a32; aC1 := evalf(3/205); aC6 := evalf(-6/41); aC7 := evalf(-3/205); aC8 := evalf(-3/41); aC9 := -aC8; aCA := evalf(6/41); aD1 := evalf(-1777/4100); aD4 := evalf(-341/164); aD5 := evalf(4496/1025); aD6 := evalf(-289/82); aD7 := evalf(2193/4100); aD8 := evalf(51/82); aD9 := evalf(33/164); aDA := evalf(12/41); #aDC := 1; # exterior weights b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bC := evalf(41/840); bD := bC; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aC1*f1 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9 + aCA*fA; fC := fn(xk,yk + t*h); t := aD1*f1 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9 + aDA*fA + fC; fD := fn(xk + h,yk + t*h); ys := yk + (b6*f6 +b7*f7 + b8*f8 + b9*f9 + bA*fA + bC*fC + bD*fD)*h; Digits := saveDigits; evalf(ys); end proc: # of rk8step # start of main procedure rungk x := op(1,xrng); y := op(1,iy); if not type(indets(fxy,name) minus {x,y},set(realcons)) then error "the 1st argument, %1, must depend only on the variables %2 and %3",fxy,x,y; end if; xrg := op(2,xrng); x0 := op(1,xrg); xn := op(2,xrg); y0 := op(2,iy); # Get the options. # Set the default values to start with. stps := 20; mthd := 'rk4'; outpt := 'points'; prntflg := false; if nargs > 3 then Options:=[args[4..nargs]]; if not type(Options,list(equation)) then error "each optional argument must be an equation" end if; if hasoption(Options,'steps','stps','Options') then if not type(stps,posint) then error"\"steps\" must be a positive integer" end if; end if; if hasoption(Options,'method','mthd','Options') then if not member(mthd,{'euler','impeuler','rk2','rk3','rk4','rk5','rk6','rk7','rk8'}) then error "\"method\" must be 'euler','impeuler','rk2','rk3','rk4','rk5','rk6','rk7' or 'rk8'" end if; end if; if hasoption(Options,'alpha','alph','Options') then c := evalf(alph); if not type(alph,realcons) or c<=0 or c>1 then error "\"alpha\" must be a positive real constant <= 1" else mthd := 'customrk2'; end if; end if; if hasoption(Options,'output','outpt','Options') then if not member(outpt,{'points','discrete','rkstep','continuous','procedure'}) then error "\"output\" must be one of 'points','discrete','continuous','rkstep' or 'procedure'" end if; if outpt='discrete' then outpt := 'points' end if; if outpt='continuous' or outpt='procedure' then outpt := 'rkstep' end if; end if; if hasoption(Options,'info','prntflg','Options') then if prntflg<>true then prntflg := false end if; end if; if nops(Options)>0 then error "%1 is not a valid option for %2",op(1,Options), procname; end if; end if; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); xk := evalf(x0); xmax := evalf(xn); yk := evalf(y0); h := (xmax-xk)/stps; # Convert any real constants in the expression to floats. fn := subs({_FXY=evalf(fxy),_X=x,_Y=y}, proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc); soln := [xk,yk]; if mthd='euler' then if prntflg then print(`method: euler`)end if; for k from 1 to stps do yk := yk + h*fn(xk,yk); xk := xk + h; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(eulerstep)); else return evalf([soln]); end if; elif mthd='impeuler' then if prntflg then print(`method: improved euler`) end if; for k from 1 to stps do k1 := h*fn(xk,yk); xk := xk + h; k2 := h*fn(xk,yk + k1); yk := yk + (k1 + k2)/2; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(impeulerstep)); else return evalf([soln]); end if; elif mthd='rk2' then if prntflg then print(`method: classical mid-point Runge-Kutta order 2`) end if; hh := h/2; for k from 1 to stps do k1 := hh*fn(xk,yk); yk := yk + h*fn(xk + hh,yk + k1); xk := xk + h; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk2step)); else return evalf([soln]); end if; elif mthd='customrk2' then if prntflg then print(`method: custom Runge-Kutta order 2`); print(`evaluation at fraction `,c,` of distance across step`); end if; t := 1/(2*c); s := 1 - t; for k from 1 to stps do f1 := fn(xk,yk); f2 := fn(xk + c*h,yk + c*f1*h); yk := yk + h*(s*f1 + t*f2); xk := xk + h; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y,_ALPH=c}, eval(customrk2step)); else return evalf([soln]); end if; elif mthd='rk3' then if prntflg then print(`method: classical Runge-Kutta order 3`) end if; for k from 1 to stps do k1 := h*fn(xk,yk); k2 := h*fn(xk+h/2,yk + k1/2); k3 := h*fn(xk+h,yk - k1 + 2*k2); yk := yk + (k1 + 4*k2 + k3)/6; xk := xk + h; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk3step)); else return evalf([soln]); end if; elif mthd='rk4' then if prntflg then print(`method: classical Runge-Kutta order 4`) end if; hh := h/2; for k from 1 to stps do hk1 := hh*fn(xk,yk); hk2 := hh*fn(xk + hh,yk + hk1); k3 := h*fn(xk + hh,yk + hk2); hk4 := hh*fn(xk + h,yk + k3); yk := yk + (hk1 + hk2 + hk2 + k3 + hk4)/3; xk := xk + h; soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk4step)); else return evalf([soln]); end if; elif mthd='rk5' then if prntflg then print(`method: Runge-Kutta order 5`) end if; c2 := 0.25; c3 := 0.375; c4 := evalf(12/13); c5 := 1.0; c6 := 0.5; a21 := c2; a31 := 0.09375; a32 := 0.28125; a41 := evalf(1932/2197); a42 := evalf(-7200/2197); a43 := evalf(7296/2197); a51 := evalf(439/216); a52 := -8.0; a53 := evalf(3680/513); a54 := evalf(-845/4104); a61 := evalf(-8/27); a62 := 2.0; a63 := evalf(-3544/2565); a64 := evalf(1859/4104); a65 := evalf(-11/40); b1 := evalf(16/135); b3 := evalf(6656/12825); b4 := evalf(28561/56430); b5 := -0.18; b6 := evalf(2/55); for k from 1 to stps do f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); yk := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h; xk := xk + h: soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk5step)); else return evalf([soln]); end if; elif mthd='rk6' then if prntflg then print(`method: Runge-Kutta order 6`) end if; c2 := 0.1; c3 := evalf(2/9); c4 := evalf(3/7); c5 := 0.6; c6 := 0.8; # The interior weights a21 := 0.1; a31 := evalf(-2/81); a32 := evalf(20/81); a41 := evalf(615/1372); a42 := evalf(-270/343); a43 := evalf(1053/1372); a51 := evalf(3243/5500); a52 := evalf(-54/55); a53 := evalf(50949/71500); a54 := evalf(4998/17875); a61 := evalf(-26492/37125); a62 := evalf(72/55); a63 := evalf(2808/23375); a64 := evalf(-24206/37125); a65 := evalf(338/459); a71 := evalf(5561/2376); a72 := evalf(-35/11); a73 := evalf(-24117/31603); a74 := evalf(899983/200772); a75 := evalf(-5225/1836); a76 := evalf(3925/4056); a81 := evalf(465467/266112); a82 := evalf(-2945/1232); a83 := evalf(-5610201/14158144); a84 := evalf(10513573/3212352); a85 := evalf(-424325/205632); a86 := evalf(376225/454272); # The exterior weights for the higher order formula b1 := evalf(61/864); b3 := evalf(98415/321776); b4 := evalf(16807/146016); b5 := evalf(1375/7344); b6 := evalf(1375/5408); b7 := evalf(-37/1120); b8 := 0.1; for k from 1 to stps do f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a72*f2 + a73*f3 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + h,yk + t*h); t := a81*f1 + a82*f2 + a83*f3 + a84*f4 + a85*f5 + a86*f6; f8 := fn(xk + h,yk + t*h); yk := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8)*h; xk := xk + h: soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk6step)); else return evalf([soln]); end if; elif mthd='rk7' or mthd='rk8' then c2 := evalf(2/27); c3 := evalf(1/9); c4 := evalf(1/6); c5 := evalf(5/12); c6 := 0.5; c7 := evalf(5/6); c8 := c4; c9 := evalf(2/3); cA := evalf(1/3); #cB := 1; #cD := 1; a21 := c2; a31 := evalf(1/36); a32 := evalf(1/12); a41 := evalf(1/24); a43 := 0.125; a51 := c5; a53 := -1.5625; a54 := 1.5625; a61 := 0.05; a64 := 0.25; a65 := 0.2; a71 := evalf(-25/108); a74 := evalf(125/108); a75 := evalf(-65/27); a76 := evalf(125/54); a81 := evalf(31/300); a85 := evalf(61/225); a86 := evalf(-2/9); a87 := evalf(13/900); a91 := 2; a94 := evalf(-53/6); a95 := evalf(704/45); a96 := evalf(-107/9); a97 := evalf(67/90); a98 := 3; aA1 := evalf(-91/108); aA4 := evalf(23/108); aA5 := evalf(-976/135); aA6 := evalf(311/54); aA7 := evalf(-19/60); aA8 := evalf(17/6); aA9 := -a32; aB1 := evalf(2383/4100); aB4 := evalf(-341/164); aB5 := evalf(4496/1025); aB6 := evalf(-301/82); aB7 := evalf(2133/4100); aB8 := evalf(45/82); aB9 := evalf(45/164); aBA := evalf(18/41); aC1 := evalf(3/205); aC6 := evalf(-6/41); aC7 := evalf(-3/205); aC8 := evalf(-3/41); aC9 := -aC8; aCA := evalf(6/41); aD1 := evalf(-1777/4100); aD4 := evalf(-341/164); aD5 := evalf(4496/1025); aD6 := evalf(-289/82); aD7 := evalf(2193/4100); aD8 := evalf(51/82); aD9 := evalf(33/164); aDA := evalf(12/41); aDC := 1; if mthd='rk7' then if prntflg then print(`method: Runge-Kutta order 7`) end if; b1 := evalf(41/840); b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bB := b1; for k from 1 to stps do f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aB1*f1 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9 + aBA*fA; fB := fn(xk + h,yk + t*h); yk := yk + (b1*f1 + b6*f6 + b7*f7 + b8*f8 + b9*f9 + bA*fA + bB*fB)*h; xk := xk + h: soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk7step)); else return evalf([soln]); end if; else # mthd='rk8' if prntflg then print(`method: Runge-Kutta order 8`) end if; b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bC := evalf(41/840); bD := bC; for k from 1 to stps do f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aC1*f1 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9 + aCA*fA; fC := fn(xk,yk + t*h); t := aD1*f1 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9 + aDA*fA + fC; fD := fn(xk + h,yk + t*h); yk := yk + (b6*f6 +b7*f7 + b8*f8 + b9*f9 + bA*fA + bC*fC + bD*fD)*h; xk := xk + h: soln := soln,[xk,yk]; if prntflg then print(`step`,k,` `,[xk,yk]) end if; end do; Digits := saveDigits; if outpt=rkstep then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk8step)); else return evalf([soln]); end if; end if; end if; end proc:

rungk45 := proc(fxy,eqn1,eqn2) local ytemp,fn,soln,xk,yk,k,xend,xstart,h,safety,errcontrol, c1,c2,c3,c4,c5,c8,c9,a21,a31,a32,a41,a42,a43,a51,a52,a53,a54, a61,a62,a63,a64,a65,a71,a72,a73,a74,a75,a76,a81,a83,a84, a85,a86,a87,a91,a93,a94,a95,a96,a97,f1,f2,f3,f4,f5, f6,f7,f8,f9,b1,b3,b4,b5,b6,e1,e3,e4,e5,e6,e7,yout,yerr,rs, maxstps,tiny,yscale,err,htemp,xnew,hnext,saveDigits,eps, pgrow,pshrink,Options,j,prntflg,t,tt,errcntl,ymax,ymaxtemp, x,y,errc,x0,y0,xn,f,outpt,hmx,hmn,hstrt,temp,sgn,finished, maxstepsize,laststep,minstepsize,ftd,inc,rk45step,rk45interp; rk45interp := proc(x_rk45interp::realcons) local e2,e3,e4,e5,e6,e7,s,t1,t2,t3,sm,sms,ss, b1,b3,b4,b5,b6,b7,b8,b9,xF,xS,yF,f1,f3,f4,f5,f6, f7,f8,f9,t,jF,jM,jS,n,h,data,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); xx := evalf(x_rk45interp); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; e2 := evalf(-500/1113); e3 := evalf(-125/192); e4 := evalf(2187/6784); e5 := evalf(-11/84); e6 := evalf(125/24); e7 := evalf(16/3); # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xF := data[jF,1]; xS := data[jS,1]; yF := data[jF,2]; f1 := data[jF,3]; f3 := data[jF,5]; f4 := data[jF,6]; f5 := data[jF,7]; f6 := data[jF,8]; f7 := data[jF,9]; f8 := data[jF,10]; f9 := data[jF,11]; # Calculate the parameters. h := xx-xF; s := h/(xS-xF); t1 := (-1710+(3104+(-2439+696*s)*s)*s)*s; t2 := (-6+(32+(-51+24*s)*s)*s)*s; t3 := 7+(-31+32*s)*s; sm := s-1; sms := sm*s; ss := sms*sm; b1:= t1/384+1; b3 := t2*e2; b4 := t2*e3; b5 := t2*e4; b6 := t2*e5; b7 := sms*t3/8; b8 := ss*e6; b9 := ss*(3*s-1)*e7; # Calculate the interpolated y value. t := b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6 + b7*f7 + b8*f8 + b9*f9; ys := yF + t*h; Digits := saveDigits; evalf(ys); end proc: # of rk45interp rk45step := proc(x_rk45step::realcons) local c2,c3,c4,c5,a21,a31,a32,a41,a42,a43,a51,a52,a53,a54, a61,a62,a63,a64,a65,a71,a73,a74,a75,a76,f1,f2,f3,f4,f5, f6,b1,b3,b4,b5,b6, xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk45step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := 0.2; c3 := 0.3; c4 := 0.8; c5 := evalf(8/9); a21 := c2; a31 := 0.075; a32 := 0.225; a41 := evalf(44/45); a42 := evalf(-56/15); a43 := evalf(32/9); a51 := evalf(19372/6561); a52 := evalf(-25360/2187); a53 := evalf(64448/6561); a54 := evalf(-212/729); a61 := evalf(9017/3168); a62 := evalf(-355/33); a63 := evalf(46732/5247); a64 := evalf(49/176); a65 := evalf(-5103/18656); a71 := evalf(35/384); a73 := evalf(500/1113); a74 := evalf(125/192); a75 := evalf(-2187/6784); a76 := evalf(11/84); b1 := a71; b3 := a73; b4 := a74; b5 := a75; b6 := a76; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF> 1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + h,yk + t*h); ys := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h; Digits := saveDigits; evalf(ys); end proc: # of rk45step # start of main procedure rungk45 x := lhs(eqn1); y := lhs(eqn2); rs := rhs(eqn1); x0 := lhs(rs); xn := rhs(rs); y0 := rhs(eqn2); # Get the options. # Set the default values to start with. maxstps := 2000; t := Float(1,-Digits); hmx := 0.25; hstrt := evalf(0.1*10^(-ceil(Digits/10))); hmn := min(0.000005,hstrt/2000); errcntl := 0; ymaxtemp := 0; outpt := 'rkstep'; prntflg := false; Options := []; if nargs > 3 then Options:=[args[4..nargs]]; if not type(Options,list(equation)) then error "each optional argument must be an equation" end if; if hasoption(Options,'maxsteps','maxstps','Options') then if not type(maxstps,posint) then error "\"maxsteps\" must be a positive integer" end if; end if; if hasoption(Options,'tolerance','t','Options') then tt := evalf(t); if not type(tt,float) or tt>Float(1,-iquo(Digits,2)) or tt<Float(1,-Digits*2) then error "\"tolerance\" must evaluate to a float between %1 and %2",Float(1,-Digits*2),Float(1,-iquo(Digits,2)); end if; end if; if hasoption(Options,'hmax','hmx','Options') then if not type(hmx,realcons) or evalf(hmx)<=0 or has(hmx,infinity) then error "\"hmax\" must be a finite positive real constant" end if; end if; if hasoption(Options,'hstart','hstrt','Options') then if not type(hstrt,realcons) or evalf(hstrt)<=0 or has(hstrt,infinity) then error "\"hstart\" must be a finite positive real constant" end if; hmn := min(0.0005,hstrt/2000); end if; if hasoption(Options,'hmin','hmn','Options') then if not type(hmn,realcons) or evalf(hmn)<=0 or has(hmn,infinity) then error "\"hmin\" must be a finite positive real constant" end if; end if; if evalf(hstrt>hmx) or evalf(hstrt<hmn) then error "\"hstart\" must be between \"hmin\" and \"hmax\"" end if; if hasoption(Options,'errorcontrol','errc','Options') then if errc='auto' then errcntl := 0 elif errc='relative' then errcntl := 1 elif errc='absolute' then errcntl := 2 elif errc='accumulative' then errcntl := 3 else error "\"errorcontrol\" must be 'auto','absolute','relative' or 'accumulative'" end if; end if; if hasoption(Options,'maxvalue','ymaxtemp','Options') then if not type(ymaxtemp,realcons) then error "\"maxvalue\" must be a real constant" end if; end if; if hasoption(Options,'output','outpt','Options') then if not member(outpt,{'points','discrete','rkstep','rkinterp', 'continuous','procedure'}) then error "\"output\" must be one of 'points','discrete','continuous','rkstep','rkinterp' or 'procedure'" end if; if outpt='discrete' then outpt := 'points' end if; if outpt='continuous' or outpt='procedure' then outpt := 'rkstep' end if; end if; if hasoption(Options,'info','prntflg','Options') then if prntflg<>true then prntflg := false end if; end if; if nops(Options)>0 then error "%1 is not a valid option for %2",op(1,Options), procname; end if; end if; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := subs({_FXY=evalf(fxy),_X=x,_Y=y}, proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc); c2 := 0.2; c3 := 0.3; c4 := 0.8; c5 := evalf(8/9); c8 := c2; c9 := 0.5; a21 := c2; a31 := 0.075; a32 := 0.225; a41 := evalf(44/45); a42 := evalf(-56/15); a43 := evalf(32/9); a51 := evalf(19372/6561); a52 := evalf(-25360/2187); a53 := evalf(64448/6561); a54 := evalf(-212/729); a61 := evalf(9017/3168); a62 := evalf(-355/33); a63 := evalf(46732/5247); a64 := evalf(49/176); a65 := evalf(-5103/18656); a71 := evalf(35/384); a73 := evalf(500/1113); a74 := evalf(125/192); a75 := evalf(-2187/6784); a76 := evalf(11/84); a81 := evalf(5207/48000); a83 := evalf(92/795); a84 := evalf(-79/960); a85 := evalf(53217/848000); a86 := evalf(-11/300); a87 := evalf(4/125); a91 := evalf(613/6144); a93 := evalf(125/318); a94 := evalf(-125/3072); a95 := evalf(8019/108544); a96 := evalf(-11/192); a97 := evalf(1/32); b1 := a71; b3 := a73; b4 := a74; b5 := a75; b6 := a76; e1 := evalf(71/57600); e3 := evalf(-71/16695); e4 := evalf(71/1920); e5 := evalf(-17253/339200); e6 := evalf(22/525); e7 := -0.025; xstart := evalf(x0); xend:= evalf(xn); sgn := sign(xend-xstart); h := sgn*hstrt; eps := evalf(t); safety := 0.9; pgrow := -0.2; pshrink := -0.25; errcontrol := 0.000189; #(5/safety)^(1/pgrow) tiny := Float(1,-3*saveDigits); xk := evalf(x0); yk := evalf(y0); if ymaxtemp<>0 then ymax := abs(evalf(ymaxtemp)) else ymax := max(abs(yk),tiny) end if; soln := NULL; finished := false; f1 := fn(xk,yk); for k from 1 to maxstps do if errcntl=0 then yscale := max(abs(yk),abs(f1*h),tiny) elif errcntl=1 then yscale := max(abs(yk),tiny) elif errcntl=2 then yscale := abs(ymax) else yscale := max(abs(f1*h),tiny) end if; if abs(h)>=hmx then h := sgn*hmx; maxstepsize := true; else maxstepsize := false; end if; if abs(h)<=hmn then h := sgn*hmn; minstepsize := true; else minstepsize := false; end if; if (xk+h-xend)*(xk+h-xstart) > 0 then h := xend-xk; laststep := true; else laststep := false; end if; # Do step. do f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a42*f2 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a52*f2 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a62*f2 + a63*f3 + a64*f4 + a65*f5; f6 := fn(xk + h,yk + t*h); yout := yk + (b1*f1 + b3*f3 + b4*f4 + b5*f5 + b6*f6)*h; t := a71*f1 + a73*f3 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + h,yk + t*h); # error estimate yerr := (e1*f1 + e3*f3 + e4*f4 + e5*f5 + e6*f6 + e7*f7)*h; err := abs(yerr/yscale)/eps; if err<=1.0 or minstepsize then break end if; # Shrink, but not too much. if prntflg then print(`reducing step-size and repeating step`); end if; htemp := safety*h*err^pshrink; if h>=0 then h := max(htemp,0.1*h) else h := min(htemp,0.1*h) end if; xnew := xk + h; if xnew=xk then error "stepsize underflow" end if; end do; if outpt='points' or outpt='rkstep' then soln := soln,[xk,yk]; else # extra evaluations needed for interpolation t := a81*f1 + a83*f3 + a84*f4 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a93*f3 + a94*f4 + a95*f5 + a96*f6 + a97*f7; f9 := fn(xk + c9*h,yk + t*h); soln := soln,[xk,yk,f1,f2,f3,f4,f5,f6,f7,f8,f9]; end if; if err>errcontrol then hnext := safety*h*err^pgrow; inc := false; else if abs(h)<hmx then hnext := 5*h; inc := true; else h := sgn*hmx; maxstepsize := true; inc := false; end if; end if; xk := xk + h; yk := yout; f1 := fn(xk,yk); if prntflg then print(`abs err estimate -> `,evalf(abs(yerr),5),`abs err bound -> `,evalf(abs(yscale)*eps,5)); print(`step`,k,` `,h,` `,[xk,yk]); if laststep then print(`last step`); elif inc then print(`increasing step-size by a factor of 5`) elif maxstepsize then print(`used maximum step-size`) elif not minstepsize then print(`using error to adjust step-size`) else print(`used minimum step-size`) end if; print(``); end if; if (xk-xend)*(xend-xstart)>=0 then finished := true; break; end if; if abs(yk)>ymax then ymax := abs(yk) end if; h := hnext; end do; if not finished and k>=maxstps then error "reached maximum number of steps before reaching end of interval" end if; soln := soln,[xk,yk]; Digits := saveDigits; if outpt='rkstep' then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk45step)); elif outpt='rkinterp' then return subs(_SOLN=[soln],eval(rk45interp)); else return evalf([soln]); end if; end proc: # of rungk45

rungk78 := proc(fxy,eqn1,eqn2) local ak1,ak2,ak3,ak4,ak5,ak6,ytemp,fn,soln,xk,yk,k, xend,xstart,h,c1,c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31, a32,a41,a42,a43,a51,a52,a53,a54,a61,a62,a63,a64,a65, a71,a72,a73,a74,a75,a76,a81,a82,a83,a84,a85,a86,a87, a91,a94,a95,a96,a97,a98,aA1,aA4,aA5,aA6,aA7,aA8,aA9, aB1,aB4,aB5,aB6,aB7,aB8,aB9,aBA,aC1,aC6,aC7,aC8,aC9, aCA,aD1,aD4,aD5,aD6,aD7,aD8,aD9,aDA,f1,f2,f3,f4,f5, f6,f7,f8,f9,fA,fB,fC,fD,b6,b7,b8,b9,bA,bB,bC,bD, e1,eB,eC,eD,yout,yerr,rs,hmx,hmn,hstrt,temp,sgn, maxstps,tiny,yscale,err,htemp,xnew,hnext,saveDigits,eps, Options,j,prntflg,t,tt,errcntl,ymax,ymaxtemp,x,y,errc, x0,y0,xn,f,outpt,minstepsize,ftd,inc,rk78step,finished, maxstepsize,laststep,safety,pgrow,pshrink,errcontrol; rk78step := proc(x_rk78step::realcons) local c2,c3,c4,c5,c6,c7,c8,c9,cA,a21,a31,a32,a41,a43, a51,a53,a54,a61,a64,a65,a71,a74,a75,a76,a81,a85,a86, a87,a91,a94,a95,a96,a97,a98,aA1,aA4,aA5,aA6,aA7,aA8, aA9,aC1,aC6,aC7,aC8,aC9,aCA,aD1,aD4,aD5,aD6,aD7,aD8, aD9,aDA,f1,f2,f3,f4,f5,f6,f7,f8,f9,fA,fC,fD,b6,b7,b8, b9,bA,bC,bD,xk,yk,t,jF,jM,jS,n,h,data,fn,xx,ys,saveDigits; options `Copyright 2002 by Peter Stone`; data := _SOLN; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc; xx := evalf(x_rk78step); n := nops(data); if (data[1,1]<data[n,1] and (xx>data[n,1] or xx<data[1,1])) or (data[n,1]<data[1,1] and (xx<data[n,1] or xx>data[1,1])) then error "independent variable is outside the interpolation interval: %1",evalf(data[1,1])..evalf(data[n,1]); end if; c2 := evalf(2/27); c3 := evalf(1/9); c4 := evalf(1/6); c5 := evalf(5/12); c6 := 0.5; c7 := evalf(5/6); c8 := c4; c9 := evalf(2/3); cA := evalf(1/3); a21 := c2; a31 := evalf(1/36); a32 := evalf(1/12); a41 := evalf(1/24); a43 := 0.125; a51 := c5; a53 := -1.5625; a54 := 1.5625; a61 := 0.05; a64 := 0.25; a65 := 0.2; a71 := evalf(-25/108); a74 := evalf(125/108); a75 := evalf(-65/27); a76 := evalf(125/54); a81 := evalf(31/300); a85 := evalf(61/225); a86 := evalf(-2/9); a87 := evalf(13/900); a91 := 2; a94 := evalf(-53/6); a95 := evalf(704/45); a96 := evalf(-107/9); a97 := evalf(67/90); a98 := 3; aA1 := evalf(-91/108); aA4 := evalf(23/108); aA5 := evalf(-976/135); aA6 := evalf(311/54); aA7 := evalf(-19/60); aA8 := evalf(17/6); aA9 := -a32; aC1 := evalf(3/205); aC6 := evalf(-6/41); aC7 := evalf(-3/205); aC8 := evalf(-3/41); aC9 := -aC8; aCA := evalf(6/41); aD1 := evalf(-1777/4100); aD4 := evalf(-341/164); aD5 := evalf(4496/1025); aD6 := evalf(-289/82); aD7 := evalf(2193/4100); aD8 := evalf(51/82); aD9 := evalf(33/164); aDA := evalf(12/41); b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bC := evalf(41/840); bD := bC; # Peform a binary search for the interval containing x. n := nops(data); jF := 0; jS := n+1; if data[1,1]<data[n,1] then while jS-jF>1 do jM := trunc((jF+jS)/2); if xx>=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; else while jS-jF>1 do jM := trunc((jF+jS)/2); if xx<=data[jM,1] then jF := jM else jS := jM end if; end do; if jM = n then jF := n-1; jS := n end if; end if; # Get the data needed from the list. xk := data[jF,1]; yk := data[jF,2]; # Do one step with step-size .. h := xx-xk; f1 := fn(xk,yk); t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aC1*f1 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9 + aCA*fA; fC := fn(xk,yk + t*h); t := aD1*f1 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9 + aDA*fA + fC; fD := fn(xk + h,yk + t*h); # Calculate the intermediate y value. t := b6*f6 +b7*f7 + b8*f8 + b9*f9 + bA*fA + bC*fC + bD*fD; ys := yk + t*h; Digits := saveDigits; evalf(ys); end proc: # of rk78step # start of main procedure rungk78 x := lhs(eqn1); y := lhs(eqn2); rs := rhs(eqn1); x0 := lhs(rs); xn := rhs(rs); y0 := rhs(eqn2); # Get the options. # Set the default values to start with. maxstps := 2000; t := Float(1,-Digits); hmx := 0.5; hstrt := evalf(0.1*10^(-ceil(Digits/10))); hmn := min(0.000005,hstrt/2000); errcntl := 0; ymaxtemp := 0; outpt := 'rkstep'; prntflg := false; Options := []; if nargs>3 then Options:=[args[4..nargs]]; if not type(Options,list(equation)) then error "each optional argument must be an equation" end if; if hasoption(Options,'maxsteps','maxstps','Options') then if not type(maxstps,posint) then error "\"maxsteps\" must be a positive integer" end if; end if; if hasoption(Options,'tolerance','t','Options') then tt := evalf(t); if not type(tt,float) or tt>Float(1,-iquo(Digits,2)) or tt<Float(1,-Digits*2) then error "\"tolerance\" must evaluate to a float between %1 and %2",Float(1,-Digits*2),Float(1,-iquo(Digits,2)); end if; end if; if hasoption(Options,'hmax','hmx','Options') then if not type(hmx,realcons) or evalf(hmx)<=0 or has(hmx,infinity) then error "\"hmax\" must be a finite positive real constant" end if; end if; if hasoption(Options,'hstart','hstrt','Options') then if not type(hstrt,realcons) or evalf(hstrt)<=0 or has(hstrt,infinity) then error "\"hstart\" must be a finite positive real constant" end if; hmn := min(0.000005,hstrt/2000); end if; if hasoption(Options,'hmin','hmn','Options') then if not type(hmn,realcons) or evalf(hmn)<=0 or has(hmn,infinity) then error "\"hmin\" must be a finite positive real constant" end if; end if; if evalf(hstrt>hmx) or evalf(hstrt<hmn) then error "\"hstart\" must be between \"hmin\" and \"hmax\"" end if; if hasoption(Options,errorcontrol,'errc','Options') then if errc='auto' then errcntl := 0 elif errc='relative' then errcntl := 1 elif errc='absolute' then errcntl := 2 elif errc='accumulative' then errcntl := 3 else error "\"errorcontrol\" must be 'auto','absolute','relative' or 'accumulative'" end if; end if; if hasoption(Options,'maxvalue','ymaxtemp','Options') then if not type(ymaxtemp,realcons) then error "\"maxvalue\" must be a real constant" end if; end if; if hasoption(Options,'output','outpt','Options') then if not member(outpt,{'points','discrete','rkstep', 'continuous','procedure'}) then error "\"output\" must be one of 'points','discrete','continuous','rkstep' or 'procedure'" end if; if outpt='discrete' then outpt := 'points' end if; if outpt='continuous' or outpt='procedure' then outpt := 'rkstep' end if; end if; if hasoption(Options,'info','prntflg','Options') then if prntflg<>true then prntflg := false end if; end if; if nops(Options)>0 then error "%1 is not a valid option for %2",op(1,Options), procname; end if; end if; saveDigits := Digits; Digits := max(trunc(evalhf(Digits)),Digits+5); # procedure to evaluate the gradient field fn := subs({_FXY=evalf(fxy),_X=x,_Y=y}, proc(_X,_Y) local val; val := traperror(evalf(_FXY)); if val=lasterror or not type(val,numeric) then error "evaluation of gradient field failed at %1",evalf([_X,_Y],saveDigits); end if; val; end proc); c2 := evalf(2/27); c3 := evalf(1/9); c4 := evalf(1/6); c5 := evalf(5/12); c6 := 0.5; c7 := evalf(5/6); c8 := c4; c9 := evalf(2/3); cA := evalf(1/3); #cB := 1; #cD := 1; a21 := c2; a31 := evalf(1/36); a32 := evalf(1/12); a41 := evalf(1/24); a43 := 0.125; a51 := c5; a53 := -1.5625; a54 := 1.5625; a61 := 0.05; a64 := 0.25; a65 := 0.2; a71 := evalf(-25/108); a74 := evalf(125/108); a75 := evalf(-65/27); a76 := evalf(125/54); a81 := evalf(31/300); a85 := evalf(61/225); a86 := evalf(-2/9); a87 := evalf(13/900); a91 := 2; a94 := evalf(-53/6); a95 := evalf(704/45); a96 := evalf(-107/9); a97 := evalf(67/90); a98 := 3; aA1 := evalf(-91/108); aA4 := evalf(23/108); aA5 := evalf(-976/135); aA6 := evalf(311/54); aA7 := evalf(-19/60); aA8 := evalf(17/6); aA9 := -a32; aB1 := evalf(2383/4100); aB4 := evalf(-341/164); aB5 := evalf(4496/1025); aB6 := evalf(-301/82); aB7 := evalf(2133/4100); aB8 := evalf(45/82); aB9 := evalf(45/164); aBA := evalf(18/41); aC1 := evalf(3/205); aC6 := evalf(-6/41); aC7 := evalf(-3/205); aC8 := evalf(-3/41); aC9 := -aC8; aCA := evalf(6/41); aD1 := evalf(-1777/4100); aD4 := evalf(-341/164); aD5 := evalf(4496/1025); aD6 := evalf(-289/82); aD7 := evalf(2193/4100); aD8 := evalf(51/82); aD9 := evalf(33/164); aDA := evalf(12/41); b6 := evalf(34/105); b7 := evalf(9/35); b8 := b7; b9 := evalf(9/280); bA := b9; bC := evalf(41/840); bD := bC; e1 := -bC; eB := e1; eC := bC; eD := bC; xstart := evalf(x0); xend:= evalf(xn); sgn := sign(xend-xstart); h := sgn*hstrt; eps := evalf(t); safety := 0.9; pgrow := -0.125; pshrink := -0.14285714285714285714; errcontrol := 0.000001101996057; #(5/safety)^(1/pgrow) tiny := Float(1,-3*saveDigits); xk := evalf(x0); yk := evalf(y0); if ymaxtemp<>0 then ymax := abs(evalf(ymaxtemp)) else ymax := max(abs(yk),tiny) end if; soln := NULL; finished := false; f1 := fn(xk,yk); for k from 1 to maxstps do if errcntl=0 then yscale := max(abs(yk),abs(f1*h),tiny) elif errcntl=1 then yscale := max(abs(yk),tiny) elif errcntl=2 then yscale := abs(ymax) else yscale := max(abs(f1*h),tiny) end if; if abs(h)>=hmx then h := sgn*hmx; maxstepsize := true; else maxstepsize := false; end if; if abs(h)<=hmn then h := sgn*hmn; minstepsize := true; else minstepsize := false; end if; if (xk+h-xend)*(xk+h-xstart)>0 then h := xend-xk; laststep := true; else laststep := false; end if; # Do step. do t := a21*f1; f2 := fn(xk + c2*h,yk + t*h); t := a31*f1 + a32*f2; f3 := fn(xk + c3*h,yk + t*h); t := a41*f1 + a43*f3; f4 := fn(xk + c4*h,yk + t*h); t := a51*f1 + a53*f3 + a54*f4; f5 := fn(xk + c5*h,yk + t*h); t := a61*f1 + a64*f4 + a65*f5; f6 := fn(xk + c6*h,yk + t*h); t := a71*f1 + a74*f4 + a75*f5 + a76*f6; f7 := fn(xk + c7*h,yk + t*h); t := a81*f1 + a85*f5 + a86*f6 + a87*f7; f8 := fn(xk + c8*h,yk + t*h); t := a91*f1 + a94*f4 + a95*f5 + a96*f6 + a97*f7 + a98*f8; f9 := fn(xk + c9*h,yk + t*h); t := aA1*f1 + aA4*f4 + aA5*f5 + aA6*f6 + aA7*f7 + aA8*f8 + aA9*f9; fA := fn(xk + cA*h,yk + t*h); t := aB1*f1 + aB4*f4 + aB5*f5 + aB6*f6 + aB7*f7 + aB8*f8 + aB9*f9 + aBA*fA; fB := fn(xk + h,yk + t*h); t := aC1*f1 + aC6*f6 + aC7*f7 + aC8*f8 + aC9*f9 + aCA*fA; fC := fn(xk,yk + t*h); t := aD1*f1 + aD4*f4 + aD5*f5 + aD6*f6 + aD7*f7 + aD8*f8 + aD9*f9 + aDA*fA + fC; fD := fn(xk + h,yk + t*h); yout := yk + (b6*f6 +b7*f7 + b8*f8 + b9*f9 + bA*fA + bC*fC + bD*fD)*h; # error estimate yerr := (e1*f1 + eB*fB + eC*fC + eD*fD)*h; err := abs(yerr/yscale)/eps; if err<=1.0 or minstepsize then break end if; # Shrink, but not too much. if prntflg then print(`reducing step-size and repeating step`); end if; htemp := safety*h*err^pshrink; if h>=0 then h := max(htemp,0.1*h) else h := min(htemp,0.1*h) end if; xnew := xk + h; if xnew=xk then error "stepsize underflow" end if; end do; soln := soln,[xk,yk]; if err>errcontrol then hnext := safety*h*err^pgrow; inc := false; else if abs(h)<hmx then hnext := 5*h; inc := true; else h := sgn*hmx; maxstepsize := true; inc := false; end if; end if; xk := xk + h; yk := yout; f1 := fn(xk,yk); if prntflg then print(`abs err estimate -> `,evalf(abs(yerr),5),`abs err bound -> `,evalf(abs(yscale)*eps,5)); print(`step`,k,` `,h,` `,[xk,yk]); if laststep then print(`last step`); elif inc then print(`increasing step-size by a factor of 5`) elif maxstepsize then print(`used maximum step-size`) elif not minstepsize then print(`using error to adjust step-size`) else print(`used minimum step-size`) end if; print(``); end if; if (xk-xend)*(xend-xstart)>=0 then finished := true; break; end if; if abs(yk)>ymax then ymax := abs(yk) end if; h := hnext; end do; if not finished and k>=maxstps then error "reached maximum number of steps before reaching end of interval" end if; soln := soln,[xk,yk]; Digits := saveDigits; if outpt='rkstep' then return subs({_SOLN=[soln],_FXY=fxy,_X=x,_Y=y},eval(rk78step)); else return evalf([soln]); end if; end proc: # of rungk78;

Examples are given in the next section.;

Note: The utility routine comparewithfcn . . comparewithfcn is needed for these examples.;

First we find the analytical solution to the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEciIiMlInlHRiZGKA==, with the initial condition NiMvLSUieUc2IyIiISIiIw==.de := diff(y(x),x)=-x^2*y(x); ic := y(0)=2; dsolve({de,ic},y(x)); g := unapply(rhs(%),x); plot(g(x),x=-1..2);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComKUYsIiIjIiIiRilGMSEiIg==NiM+JSNpY0cvLSUieUc2IyIiISIiIw==NiMvLSUieUc2IyUieEcsJComIiIjIiIiLSUkZXhwRzYjLCQqJiIiJCEiIkYnRjFGMkYrRis=NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkKiYiIiMiIiItJSRleHBHNiMsJComI0YvIiIkRi8qJCk5JEY2Ri9GLyEiIkYvRi9GKEYoRig=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

;Now we use the procedure desolveRK to find a discrete set of points along the solution curve between NiMvJSJ4RyIiIQ== and NiMvJSJ4RyIiIw==, and plot these points along with the curve for the analytical solution NiMvJSJ5RyomIiIjIiIiLSUkZXhwRzYjLCQqJiUieEciIiRGLiEiIkYvRic=. The width of each step is controlled by the procedure in an attempt to ensure that the values obtained are correct to about 10 digits (with Maple running in the default mode).The method used is the combined 4th and 5th order Runge-Kutta method in which the error in the 4th order values is estimated by computing 5th order values as well. The values returned by the procedure are those obtained by the 5th order Runge-Kutta method. This method is obtained with the default options ''stepsize=variable'' and ''method=rk45'' (which could be omitted). Note: The option ''output=points'' must be included in order to obtain a discrete solution. de := diff(y(x),x)=-x^2*y(x); ic := y(0)=2; pts1 := desolveRK({de,ic},y(x),x=0..2,stepsize=variable,method=rk45,output=points): plot([g(x),pts1],x=0..2,y=0..2,style=[line,point], symbol=circle,color=[red,blue]);

NiM+SSNkZUc2Ii8tSSVkaWZmR0kqcHJvdGVjdGVkR0YpNiQtSSJ5R0YlNiNJInhHRiVGLiwkKiZGLiIiI0YrIiIiISIiNiM+SSNpY0c2Ii8tSSJ5R0YlNiMiIiEiIiM=

Warning, unable to evaluate 1 of the 2 functions to numeric values in the region; see the plotting command's help page to ensure the calling sequence is correct

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

We can compare these points with points along the analytical solution curve.comparewithfcn(pts1,g);

0 2 function val: 2 rel err: 0.0000e-01 .01 1.999999333 function val: 1.999999333 rel err: 0.0000e-01 .06 1.999856005 function val: 1.999856005 rel err: 0.0000e-01 .124256127 1.998721433 function val: 1.998721433 rel err: 0.0000e-01 .191363094 1.995333662 function val: 1.995333662 rel err: 0.0000e-01 .251848403 1.989378873 function val: 1.989378873 rel err: 0.0000e-01 .308585186 1.980505651 function val: 1.980505651 rel err: 0.0000e-01 .36299773 1.968365358 function val: 1.968365358 rel err: 0.0000e-01 .415900492 1.952610701 function val: 1.952610701 rel err: 0.0000e-01 .467904699 1.9328591 function val: 1.9328591 rel err: 0.0000e-01 .519587659 1.908636773 function val: 1.908636773 rel err: 0.0000e-01 .571635467 1.879269789 function val: 1.879269789 rel err: 0.0000e-01 .62509834 1.843615432 function val: 1.843615432 rel err: 0.0000e-01 .682220391 1.799135819 function val: 1.799135819 rel err: 0.0000e-01 .742588475 1.744817736 function val: 1.744817736 rel err: 0.0000e-01 .788592815 1.698384505 function val: 1.698384505 rel err: 0.0000e-01 .836098304 1.645952838 function val: 1.645952838 rel err: 0.0000e-01 .87894878 1.594887084 function val: 1.594887084 rel err: 0.0000e-01 .918456646 1.544790989 function val: 1.544790989 rel err: 0.0000e-01 .955670367 1.495121391 function val: 1.495121391 rel err: 0.0000e-01 .991176745 1.445650627 function val: 1.445650627 rel err: 0.0000e-01 1.025355468 1.396277889 function val: 1.39627789 rel err: 7.1619e-10 1.058486377 1.34694643 function val: 1.34694643 rel err: 0.0000e-01 1.090797074 1.297608754 function val: 1.297608754 rel err: 0.0000e-01 1.12249002 1.24820612 function val: 1.24820612 rel err: 0.0000e-01 1.15376305 1.198649985 function val: 1.198649985 rel err: 0.0000e-01 1.18483112 1.148797125 function val: 1.148797126 rel err: 8.7048e-10 1.215958645 1.098405698 function val: 1.098405698 rel err: 0.0000e-01 1.247523605 1.047040088 function val: 1.047040089 rel err: 9.5507e-10 1.280184575 .993813279 function val: .99381328 rel err: 2.0125e-10 1.315523155 .936379168 function val: .936379168 rel err: 4.2718e-10 1.348344548 .88341124 function val: .88341124 rel err: 2.2640e-10 1.376497018 .838431277 function val: .838431277 rel err: 1.1927e-10 1.402417804 .797505481 function val: .797505482 rel err: 7.5235e-10 1.428386837 .757074627 function val: .757074626 rel err: 9.2461e-10 1.452387076 .720299453 function val: .720299453 rel err: 2.7766e-10 1.47482964 .686489386 function val: .686489386 rel err: 5.8267e-10 1.496057664 .655074796 function val: .655074797 rel err: 6.1062e-10 1.516291088 .625685577 function val: .625685576 rel err: 1.1188e-09 1.535681425 .598061962 function val: .598061962 rel err: 1.3377e-09 1.554340098 .572008862 function val: .572008862 rel err: 6.9929e-10 1.572352848 .547372685 function val: .547372685 rel err: 5.4807e-10 1.589787894 .524028259 function val: .52402826 rel err: 5.7249e-10 1.606700886 .5018709 function val: .5018709 rel err: 3.9851e-10 1.623138144 .480811235 function val: .480811235 rel err: 4.1596e-10 1.639138854 .460771711 function val: .460771711 rel err: 6.5108e-10 1.654736607 .441684133 function val: .441684132 rel err: 6.7922e-10 1.669960532 .423487866 function val: .423487866 rel err: 1.1807e-09 1.68483613 .40612851 function val: .40612851 rel err: 9.8491e-10 1.699385913 .389556871 function val: .38955687 rel err: 7.7011e-10 1.7136299 .373728172 function val: .373728172 rel err: 8.0272e-10 1.727586012 .358601427 function val: .358601427 rel err: 5.5772e-10 1.741270376 .344138936 function val: .344138936 rel err: 1.7435e-09 1.754697586 .330305874 function val: .330305874 rel err: 3.0275e-10 1.767880906 .317069954 function val: .317069954 rel err: 2.2077e-09 1.780832443 .304401143 function val: .304401143 rel err: 3.2851e-10 1.793563288 .292271428 function val: .292271428 rel err: 3.4215e-10 1.806083638 .280654613 function val: .280654613 rel err: 7.1262e-10 1.818402898 .269526143 function val: .269526143 rel err: 1.4841e-09 1.830529769 .258862958 function val: .258862958 rel err: 0.0000e-01 1.842472317 .248643364 function val: .248643364 rel err: 0.0000e-01 1.854238047 .238846915 function val: .238846915 rel err: 8.3736e-10 1.865833949 .229454318 function val: .229454318 rel err: 1.3074e-09 1.877266557 .220447337 function val: .220447337 rel err: 0.0000e-01 1.888541982 .211808726 function val: .211808725 rel err: 4.7212e-10 1.899665956 .203522143 function val: .203522144 rel err: 2.4567e-09 1.910643865 .195572101 function val: .195572102 rel err: 1.0226e-09 1.921480776 .1879439 function val: .1879439 rel err: 1.0641e-09 1.932181465 .180623581 function val: .18062358 rel err: 5.5364e-10 1.94275044 .173597874 function val: .173597874 rel err: 1.1521e-09 1.953191962 .166854161 function val: .166854161 rel err: 1.1987e-09 1.963510065 .160380433 function val: .160380433 rel err: 1.2470e-09 1.973708574 .15416525 function val: .15416525 rel err: 1.2973e-09 1.983791113 .148197716 function val: .148197717 rel err: 6.7477e-10 1.993761131 .142467443 function val: .142467443 rel err: 1.4038e-09 2 .138966902 function val: .138966902 rel err: 0.0000e-01NiMlIUc= Maximum relative error: 2.4567e-09 obtained for the input value: 1.899665956

Using the 7-8 Runge-Kutta method we can obtain more widely spaced points, and still have comparable errors to those obtained with the default 4-5 Runge-Kutta method.de := diff(y(x),x)=-x^2*y(x); ic := y(0)=2; pts2 := desolveRK({de,ic},x=0..2,method=rk78,output=points): plot([g(x),pts2],x=0..2,y=0..2,style=[line,point], symbol=circle,color=[red,blue]); comparewithfcn(pts2,g);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComKUYsIiIjIiIiRilGMSEiIg==NiM+JSNpY0cvLSUieUc2IyIiISIiIw==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 0 2 function val: 2 rel err: 0.0000e-01 .01 1.999999333 function val: 1.999999333 rel err: 0.0000e-01 .06 1.999856005 function val: 1.999856005 rel err: 0.0000e-01 .31 1.980237619 function val: 1.980237619 rel err: 0.0000e-01 .513871979 1.911551878 function val: 1.911551878 rel err: 0.0000e-01 .703731724 1.780644785 function val: 1.780644785 rel err: 0.0000e-01 .897715966 1.571440252 function val: 1.571440252 rel err: 0.0000e-01 1.080209671 1.313898369 function val: 1.31389837 rel err: 7.6109e-10 1.217577172 1.095776774 function val: 1.095776774 rel err: 0.0000e-01 1.34176905 .893983296 function val: .893983297 rel err: 6.7115e-10 1.452525622 .720088954 function val: .720088954 rel err: 5.5549e-10 1.552700101 .574277365 function val: .574277365 rel err: 3.4826e-10 1.653974988 .442605769 function val: .442605769 rel err: 4.5187e-10 1.745462118 .33978239 function val: .33978239 rel err: 1.4715e-09 1.828878398 .260298042 function val: .260298042 rel err: 7.6835e-10 1.905698342 .199125714 function val: .199125714 rel err: 1.0044e-09 1.977019351 .152186419 function val: .15218642 rel err: 6.5709e-10 2 .138966902 function val: .138966902 rel err: 0.0000e-01NiMlIUc= Maximum relative error: 1.4715e-09 obtained for the input value: 1.745462118

;Now we find a continuous solution for differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGJkYo, with the initial condition NiMvLSUieUc2IyIiISIiIw== using the procedure desolveRK in the default mode. In this case the output is a numerical procedure gn, which can be used in a similar way to an ordinary function, provided that it is given input values which lie within the interval used for its construction.This numerical procedure incorporates the data of a discrete solution constructed using adaptive step-size control.Interpolation between the points of the discrete solution is achieved by performing a Runge-Kutta step of the appropriate width. This width will be less the width of the interval given by the x coordinates of the two points of the underlying discrete solution in which a given input value x lies. Warning: When using the numerical procedure gn in a plot command, it must be enclosed in quotes as'gn(x)' or 'gn'(x), in order to delay evaluation until a numerical value for the input is given. de := diff(y(x),x)=-x^2*y(x); ic := y(0)=2; gn := desolveRK({de,ic},x=0..2); plot([g(x),'gn'(x)],x=0..2,y=0..2,color=[red,green],thickness=[1,2]);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComKUYsIiIjIiIiRilGMSEiIg==NiM+JSNpY0cvLSUieUc2IyIiISIiIw==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

The solution can be compared with the analytical solution using numerical examples.xx := evalf(6*sqrt(2)/5); gn(xx); evalf(evalf(g(xx),15));

NiM+JSN4eEckIit1aTAocCIhIio=NiMkIisjXEg9I1IhIzU=NiMkIisjXEg9I1IhIzU=

The code for the procedure gn can be viewed as follows.interface(verboseproc=2): eval(gn); interface(verboseproc=1):We are not restricted to solutions which go to the right. In fact we can construct a solution which goes both to the left and to the right. We also have the option of using the 7-8 Runge-Kutta adaptive method instead of the 4-5 Runge-Kutta method. de := diff(y(x),x)=-x^2*y(x); ic := y(0)=2; fn := desolveRK({de,ic},x=-2..2,method=rk78); plot([g(x),'fn(x)'],x=-2..2,color=[red,green],thickness=[1,2]);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComKUYsIiIjIiIiRilGMSEiIg==NiM+JSNpY0cvLSUieUc2IyIiISIiIw==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

xx := evalf(-6*sqrt(2)/5); fn(xx); evalf(evalf(g(xx),15));

NiM+JSN4eEckISt1aTAocCIhIio=NiMkIithQCQqPjUhIik=NiMkIithQCQqPjUhIik=

;

First we find the analytical solution to the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYlInhHIiIjJSJ5R0Yo, with the initial condition NiMvLSUieUc2IyIiISIiJQ==.The solution is plotted along with the gradient field. de := diff(y(x),x)=x^2/y(x); ic := y(0)=4; dsolve({de,ic},y(x)); g := unapply(rhs(%),x); p1 := DEtools[DEplot](de,y(x),x=-3..3,y=0..6,arrows=medium,color=blue, dirgrid=[25,25]): p2 := plot(g(x),x=-3..3,y=0..3.2,color=brown,thickness=2): plots[display]([p1,p2]);

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Now we find a continuous solution in the form of a numerical procedure gn using the procedure desolveRK in the default mode.The continuous solution is defined on the interval from NiMvJSJ4RywkKSIjQyomIiIiRikiIiQhIiJGKw== to NiMvJSJ4RyIiJA==. de := diff(y(x),x)=x^2/y(x); ic := y(0)=4; gn := desolveRK({de,ic},y(x),x=-(24)^(1/3)..3); plot([g(x),'gn'(x)],x=-(24)^(1/3)..3,thickness=[1,2],color=[red,green]);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJkYsIiIjRikhIiI=NiM+JSNpY0cvLSUieUc2IyIiISIiJQ==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

Values given by the numerical procedure gn are correct to 10 digits over the greater part of the interval from NiMvJSJ4RywkKSIjQyomIiIiRikiIiQhIiJGKw== to NiMvJSJ4RyIiJA==. xx := evalf(sqrt(8)); gn(xx); evalf(evalf(g(xx),15));

NiM+JSN4eEckIitDclVHRyEiKg==NiMkIisqUihRdmIhIio=NiMkIisqUihRdmIhIio=

The values given by the numerical procedure gn are not very accurate near the left hand end of the interval.xx := -2.88; gn(xx); evalf(evalf(g(xx),15));

NiM+JSN4eEckISQpRyEiIw==NiMkIit0NzNNRiEjNQ==NiMkIisqPiIzTUYhIzU=

;

;In this example we construct a continuous numerical solution to the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYtJSFHNiMqJiUieUdGJiIiJUYoRiYsJkYmRiYqJkYuRiYiIz9GKEYoRiY=, subject to the initial condition NiMvLSUieUc2IyIiISIiIg==. The continuous solution is constructed over the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIiIg== using the procedure desolveRK with the options ''method=rk45'' and ''output=rkinterp''.The application of this last option means that, when the numerical procedure for the continuous solution is evaluated, rather performing a Runge-Kutta step, an alternative method of interpolation is used which does not require any new evaluations of the direction field NiMvLSUiZkc2JCUieEclInlHKiYtJSFHNiMqJkYoIiIiIiIlISIiRi4sJkYuRi4qJkYoRi4iIz9GMEYwRi4=.This should make the numerical procedure for the solution faster to evaluate. The numerical solution is compared with the analytical solution NiMvJSJ5RyomIiM/IiIiLCZGJ0YnKiYiIz5GJy0lJGV4cEc2IywkKiYlInhHRiciIiUhIiJGMkYnRidGMg==. de := diff(y(x),x)= y(x)*(1-y(x)/20)/4; ic := y(0)=1; dsolve({de,ic},y(x)): g := unapply(rhs(%),x);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComIyIiIiIiJUYwKiZGKUYwLCZGMEYwKiYjRjAiIz9GMEYpRjAhIiJGMEYwRjA=NiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKCwkKiYiIz8iIiIsJkYvRi8qJiIjPkYvLSUkZXhwRzYjLCQqJiNGLyIiJUYvOSRGLyEiIkYvRi9GO0YvRihGKEYo

de := diff(y(x),x)= y(x)*(1-y(x)/20)/4; ic := y(0)=1; gn := desolveRK({de,ic},x=0..4,output=rkinterp); plot(['gn'(x),g(x)],x=0..4,color=[red,green],thickness=[1,2]);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComIyIiIiIiJUYwKiZGKUYwLCZGMEYwKiYjRjAiIz9GMEYpRjAhIiJGMEYwRjA=NiM+JSNpY0cvLSUieUc2IyIiISIiIg==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

xx := evalf(Pi); gn(xx); evalf(evalf(g(xx),15));

NiM+JSN4eEckIithRWZUSiEiKg==NiMkIistIyl5cD8hIio=NiMkIistIyl5cD8hIio=

;

;Question: (a) Construct a continuous numerical solution to the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLChGJkYmKiQlInhHIiIjRigqJCUieUdGLEYo over the interval NiMvJSJ4RywkIiIjISIi to NiMvJSJ4RyIiIw==, with the initial condition NiMvLSUieUc2IyIiIUYn.(b) Plot a graph of the solution obtained in (a) along with the associated gradient field for the differential equation. (c) Find the first positive zero of the solution. (d) Find the coordinates of the the maximum pointNiMtJSFHNiQmJSJ4RzYjJSRtYXhHJiUieUdGKA== on the graph of the solution. Solution: (a) We use the combined 7th and 8th order Runge-Kutta method with adaptive step-size control to construct a continuous numerical solution for the differential equation over the interval from NiMvJSJ4RywkIiIjISIi to NiMvJSJ4RyIiIw==. Note that the solution goes both to the left and right of the initial pointNiMtJSFHNiQiIiFGJg==. de := diff(y(x),x)=1-x^2-y(x)^2; ic := y(0)=0; fn := desolveRK({de,ic},x=-2..2,method=rk78); plot(fn,-2..2,color=brown,thickness=2);

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

(b) In this case Maple's dsolve cannot obtain an analytical solution, but we can obtain a check on the general shape of the solution curve by plotting it together with the direction field.p1 := DEtools[DEplot](de,y(x),x=-2..2,y=-1..1,arrows=medium, color=blue,dirgrid=[25,25]): p2 := plot('fn'(x),x=-2..2,y=-1..1,color=brown,thickness=2): plots[display]([p1,p2]);

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'7$Fjgl$"39P+'HgU$y&*F/7$F_hl$"3!**Ht.t!*\v)F/7$Fdhl$"31-D?Dv<<*)F/7$Fihl$"3u**3]adK\*)F/F_h\m7'7$F_il$"3o")4IYj$fd*F/7$Fdil$"3ObB.()pRd()F/7$Fiil$"31wPSH_2:*)F/7$F^jl$"31oHA%ocT&*)F/F\i\m7'7$Fdjl$"3uaTW]7!>d*F/7$Fijl$"3I#=*)G3K9w)F/7$F^[m$"3c(3Uv]qE"*)F/7$Fc[m$"3#QfzA5P6'*)F/Fii\m7'7$Fi[m$"3Ed9'Q')fZc*F/7$F^\m$"3wz=ZpMdo()F/7$Fc\m$"3#)Q\\**))>5*)F/7$Fh\m$"3)[F$Hhoqr*)F/Ffj\m7'7$F^]m$"3O))Hd0yM^&*F/7$Fc]m$"3o[.wFb)>y)F/7$Fh]m$"3<2d"*>F`3*)F/7$F]^m$"3s%faU(>e))*)F/Fc[]m7'7$Fc^m$"3Cr^@KjtC&*F/7$Fh^m$"3zl"=6+(f3))F/7$F]_m$"38Lek_"z."*)F/7$Fb_m$"31*4!GwA"p,*F/F`\]m7'7$$!36xL[N\\-*)F/$"3!>!>PB+,r%*F/7$$!3i$G$=J<<kxF/$"39N9'*4LKi))F/7$$!3_'>'p7wzi")F/$"3vfD))[E'H#*)F/7$$!39j4*fDa%eyF/$"3;r+U\IDl!*F/F_]]m7'7$F]am$"3!oueWU\GP*F/7$Fbam$"3A!fu)3R[g*)F/7$Fgam$"3?`7R-WCf*)F/7$$!3y`=sC"*)H:'F/$"34,#G9#HGS"*F/Fb^]m7'7$Fbbm$"3XK^Oa5qS#*F/7$Fgbm$"3e/#o*yAj#4*F/7$F\cm$"3?!e?q&4=A!*F/7$Facm$"3_CjTh#*>F#*F/Fa_]m7'7$Fgcm$"3%Q133^Zj7*F/7$F\dm$"3?t__Ae)p?*F/7$Fadm$"3KE#elG\e3*F/7$Ffdm$"3EB'f_%\?$H*F/F^`]m7'7$$!3eI4HlpTsCF/$"3CWP:"R_.1*F/7$$!3x$*RU!oj"4')Fdz$"3y#fz@%4)HF*F/7$$!3G$pVTMfm:"F/$"3C%*>;BnAE"*F/7$F[fm$"3Gh!=yHkwK*F/F]a]m7'7$$!3$\It\-Z"QzFdz$"3XA6N=o()R!*F/7$$"3CQL(\-Z"QzFdz$"3e9A)\^cMH*F/7$F[gm$"3**G%yB1G$R"*F/7$$"3#)GU4#Hzo'QFdz$"3ohF+Q<yP$*F/F`b]m7'7$$"33FSU!oj"4')Fdz$"3OXP:"R_.1*F/7$$"3#R$4HlpTsCF/$"3o"fz@%4)HF*F/7$F\hm$"38$*>;BnAE"*F/7$$"3G;nn8(f.2#F/$"3<g!=yHkwK*F/Fcc]m7'7$FchmF\`]m7$FfhmF_`]m7$Fihm$"3WF#elG\e3*F/7$F\imFe`]mFbd]m7'7$F`imF__]m7$FcimFb_]m7$FfimFe_]m7$FiimFh_]mFid]m7'7$F]jmF`^]m7$F`jmFc^]m7$Fcjm$"3Ia7R-WCf*)F/7$FfjmF[_]mF^e]m7'7$FjjmF]]]m7$F][nFb]]m7$F`[nFg]]m7$Fc[nF\^]mFee]m7'7$$"3kRHY0v'Qd*F/F^\]m7$Fj[nFa\]m7$F]\nFd\]m7$F`\nFg\]mF\f]m7'7$Fd\nFa[]m7$Fg\nFd[]m7$Fj\nFg[]m7$F]]nFj[]mFaf]m7'7$Fa]nFdj\m7$Fd]nFgj\m7$Fg]nFjj\m7$Fj]nF][]mFff]m7'7$F^^nFgi\m7$Fa^nFji\m7$Fd^nF]j\m7$Fg^nF`j\mF[g]m7'7$F[_nFjh\m7$F^_nF]i\m7$Fa_nF`i\m7$Fd_nFci\mF`g]m7'7$Fh_nF]h\m7$F[`nF`h\m7$F^`nFch\m7$Fa`nFfh\mFeg]m7'7$Fe`nF`g\m7$Fh`nFcg\m7$F[anFfg\m7$F^anFig\mFjg]m7'7$F*$"3TO(H:"\MT5F,7$F1$"3QSEq%)3b'e*F/7$F6$"3mb%f$=$fEv*F/7$F;$"3gTb1)))*\y(*F/Fah]m7'7$FA$"3%\URg987/"F,7$FF$"37cdgR&oye*F/7$FK$"33TLt$\**4v*F/7$FP$"3ko$y)**Ql"y*F/F^i]m7'7$FV$"3@/*)\Qw+T5F,7$Fen$"3Mi4,:O#**e*F/7$Fjn$"33aF8h%Q!\(*F/7$F_o$"3=n<yN`%fy*F/F[j]m7'7$Feo$"3N)*4%3Yu1/"F,7$Fjo$"3/A+f"RbKf*F/7$F_p$"3!oFo20%yY(*F/7$Fdp$"3ujr9'*y(>z*F/Fhj]m7'7$Fjp$"3/Y(ReX5,/"F,7$F_q$"3-WDgTa*))f*F/7$Fdq$"3n">8f9wVu*F/7$Fiq$"35MyVu9y+)*F/Fe[^m7'7$F_r$"3)Q')z*R76R5F,7$Fdr$"3nl8?+w))3'*F/7$Fir$"3hHf[^dKU(*F/7$F^s$"3M;v(GsiT")*F/Fb\^m7'7$Fds$"3#o*\i*zns."F,7$Fis$"3LP+v.?KF'*F/7$F^t$"3vn2Jg:6U(*F/7$Fct$"3peKPf5GN)*F/F_]^m7'7$$!3lEU?QRD?))F/$"3d7Q*3$R"Q."F,7$$!35MCYGFTYyF/$"3%)z=1"pg=m*F/7$$!3-OOl*=DjA)F/$"3-#GK7B9uu*F/7$$!3i8br!)e=))yF/$"3!e+]CQW"p)*F/F^^^m7'7$F^v$"3_8zhE=oF5F,7$Fcv$"3Ep3#Qt"=B(*F/7$Fhv$"3+M$H=U#>l(*F/7$F]w$"3;'4blp-4#**F/Fa_^m7'7$Fcw$"3T)\7)**Qj=5F,7$Fhw$"3A>](=+hO")*F/7$F]x$"3m:&eely6!)*F/7$Fbx$"3m)\$)Rl<v)**F/F^`^m7'7$Fhx$"3hxd2p(Q!45F,7$F]y$"3OGAC4Bh4**F/7$Fby$"3abDLv^1Z)*F/7$$!3m!))p[V,p#GF/$"3iaP&zuV]+"F,F[a^m7'7$F]z$"3I0w"3D6B+"F,7$Fbz$"3I]R#=\()o(**F/7$$!31$HyC(4Y17F/$"3'z`WY@*)G))*F/7$$!3aTAIk%[L="F/$"3')**G#)y,445F,Fja^m7'7$Fc[l$"3A+++++++5F,7$Fi[lFib^m7$$"3j'=)*3:%*[s%Fdz$"3#fLLLLLe*)*F/7$$"3z!>)*3:%*[s%Fdz$"3'pmmmm;/,"F,F[c^m7'7$Fe\lFha^m7$Fh\lF[b^m7$$"3&*R]&3Oso7#F/F`b^m7$$"3g"4J!p[)*\@F/Feb^mFhc^m7'7$Fb]lFi`^m7$Fg]lF\a^m7$Fj]lF_a^m7$$"3k&y'zJ_wRQF/Fda^mFad^m7'7$Fa^lF\`^m7$Ff^lF_`^m7$Fi^lFb`^m7$F\_lFe`^mFhd^m7'7$F`_lF__^m7$Fc_lFb_^m7$Ff_lFe_^m7$Fi_lFh_^mF]e^m7'7$F]`lF\^^m7$F``lFa^^m7$Fc`lFf^^m7$Ff`lF[_^mFbe^m7'7$Fj`lF]]^m7$F]alF`]^m7$F`alFc]^m7$FcalFf]^mFge^m7'7$FgalF`\^m7$FjalFc\^m7$F]blFf\^m7$F`blFi\^mF\f^m7'7$FdblFc[^m7$FgblFf[^m7$FjblFi[^m7$F]clF\\^mFaf^m7'7$FaclFfj]m7$FdclFij]m7$FgclF\[^m7$FjclF_[^mFff^m7'7$F^dlFii]m7$FadlF\j]m7$FddlF_j]m7$FgdlFbj]mF[g^m7'7$F[elF\i]m7$F^elF_i]m7$FaelFbi]m7$FdelFei]mF`g^m7'7$FhelF_h]m7$F[flFbh]m7$F^flFeh]m7$FaflFhh]mFeg^m-%&COLORG6&%$RGBG$Fg[lFf[lF\h^m$"#5Ff[l-F&6%7en7$$!"#Fg[l$"+$3YW+"!"*7$$!+#p0k&>Fgh^m$"+?1&QS)!#57$$!+$Q6G">Fgh^m$"+(QK8&pF]i^m7$$!+3-)[(=Fgh^m$"+)e3c"eF]i^m7$$!+M!\p$=Fgh^m$"+)p[=y%F]i^m7$$!+h9H%z"Fgh^m$"+J#3\s$F]i^m7$$!+))Qj^<Fgh^m$"+88[lFF]i^m7$$!+`Np3<Fgh^m$"+ba/')=F]i^m7$$!+=Kvl;Fgh^m$"+[7P$3"F]i^m7$$!+C2G!e"Fgh^m$!+r%yW;$!#67$$!+"yO5]"Fgh^m$!+V$*H79F]i^m7$$!+oU)*=9Fgh^m$!+<dsrBF]i^m7$$!+Ta7M8Fgh^m$!+Ukd)>$F]i^m7$$!+e(Q&\7Fgh^m$!+)po:(QF]i^m7$$!+d4`i6Fgh^m$!+C;9=WF]i^m7$$!+QW*e3"Fgh^m$!+jAm$y%F]i^m7$$!*q)>'***Fgh^m$!+_r&*p]F]i^m7$$!*]5*H"*Fgh^m$!+n&eqA&F]i^m7$$!*I"3&H)Fgh^m$!+W%omD&F]i^m7$$!*Twp`(Fgh^m$!+w*36=&F]i^m7$$!*P;bj'Fgh^m$!+&*o.m\F]i^m7$$!*Zh=(eFgh^m$!+9MmzYF]i^m7$$!*G\N)\Fgh^m$!+I"34B%F]i^m7$$!*ZUs>%Fgh^m$!+-v3OPF]i^m7$$!*GRXL$Fgh^m$!+>^u(4$F]i^m7$$!*$=/8DFgh^m$!+MN#)4CF]i^m7$$!*U&*el"Fgh^m$!+fI&fi"F]i^m7$$!)Wn(o)Fgh^m$!+u>4W')Ff[_m7$$!(eV(>Fgh^m$!+pGNu>!#77$$")f`@')Fgh^m$"+U$R*y&)Ff[_m7$$"*nZ)H;Fgh^m$"+*f'G,;F]i^m7$$"*Ky*eCFgh^m$"+6%z@O#F]i^m7$$"*S^bJ$Fgh^m$"+j(pE3$F]i^m7$$"*0TN:%Fgh^m$"+xh01PF]i^m7$$"*7RV'\Fgh^m$"+w.))>UF]i^m7$$"*:#fkeFgh^m$"+5#)[wYF]i^m7$$"*`4Nn'Fgh^m$"+oO"y(\F]i^m7$$"*],s`(Fgh^m$"+vd9"=&F]i^m7$$"*zM)>$)Fgh^m$"+du\d_F]i^m7$$"*qfa<*Fgh^m$"+:s,A_F]i^m7$$"*1O0)**Fgh^m$"+7*\R2&F]i^m7$$"+#G2A3"Fgh^m$"+`[e)z%F]i^m7$$"+$)G[k6Fgh^m$"+sdV2WF]i^m7$$"+7yh]7Fgh^m$"+9F)Q'QF]i^m7$$"+()fdL8Fgh^m$"+Z0U.KF]i^m7$$"+-FT=9Fgh^m$"+Pj$yP#F]i^m7$$"+Epa-:Fgh^m$"+vT/$R"F]i^m7$$"+Sv&)z:Fgh^m$"+-i&yA$Ff[_m7$$"+GUYo;Fgh^m$!+/'R>8"F]i^m7$$"+o'*33<Fgh^m$!+NeCu=F]i^m7$$"+2^rZ<Fgh^m$!+9Rz"o#F]i^m7$$"+!4k**y"Fgh^m$!+9qNBOF]i^m7$$"+sI@K=Fgh^m$!+AqDfYF]i^m7$$"+S2ls=Fgh^m$!+#Qo@v&F]i^m7$$"+2%)38>Fgh^m$!+'yR+'pF]i^m7$$"+/Uac>Fgh^m$!+hiu3%)F]i^m7$$""#Fg[l$!+$3YW+"Fgh^m-%*THICKNESSG6#F`j_m-Fig^m6&F[h^m$"1_MmX%)eqk!#;$"2wmoV()eqk"F,F[[`m-%+AXESLABELSG6'Q"x6"Q%y(x)Fa[`m-%%FONTG6$%*HELVETICAGF^h^m%+HORIZONTALGFg[`m-%*GRIDSTYLEG6#%,RECTANGULARG-%*LINESTYLEG6#Fg[lFc[`m-%,ORIENTATIONG6$$"#XFg[lFb\`m-%+PROJECTIONG6#F]h^m-Fig^m6#%%NONEG-%%VIEWG6$;$!$;#Fdh^m$"$;#Fdh^m;$!$3"Fdh^m$"$3"Fdh^m

(c) We can find the first positive zero of the solution using fsolve. fsolve('fn'(x),x=1..2);

NiMkIis3dyY0ZyIhIio=

Note that quotes must be placed around fn to delay evaluation. We can check that 1.600957612 is the closest approximation for the zero with a 10 digit value (under the assumption that fn gives values which are correct to about 10 digits). fn(1.600957611); fn(1.600957612); fn(1.600957613);

NiMkIiwrKT0oKVxAISM+NiMkIitqZzFvZSEjPg==NiMkISs0bmVpKCohIz4=

As a further check on the accuracy of this result the whole calculation can be repeated using a higher precision. Digits := 15: fn2 := desolveRK({de,ic},x=0..2): fsolve('fn2'(x),x=1..2); Digits := 10:

NiMkIjBDJlE3dyY0ZyIhIzk=

The special root-finding procedures secant or bisect could be used.secant('fn'(x),x=1..2,info=true);

NiQlN2FwcHJveGltYXRpb25+MX5+LT5+fn5HJCIuVzVkJlJOOCEjNw==NiQlN2FwcHJveGltYXRpb25+Mn5+LT5+fn5HJCIuLSdSIypbJlwiISM3NiQlN2FwcHJveGltYXRpb25+M35+LT5+fn5HJCIueC4nZXFNOyEjNw==NiQlN2FwcHJveGltYXRpb25+NH5+LT5+fn5HJCIuQG4uQ3NmIiEjNw==NiQlN2FwcHJveGltYXRpb25+NX5+LT5+fn5HJCIucFZxSDNnIiEjNw==NiQlN2FwcHJveGltYXRpb25+Nn5+LT5+fn5HJCIuITQtImU0ZyIhIzc=NiQlN2FwcHJveGltYXRpb25+N35+LT5+fn5HJCIuNkJoZDRnIiEjNw==NiQlN2FwcHJveGltYXRpb25+OH5+LT5+fn5HJCIudkJoZDRnIiEjNw==NiQlN2FwcHJveGltYXRpb25+OX5+LT5+fn5HJCIudkJoZDRnIiEjNw==NiMkIis3dyY0ZyIhIio=

bisect('fn'(x),x=1..2);

NiMkIis3dyY0ZyIhIio=

;(d) Now consider the problem of finding the coordinates of first maximum point on the graph of the solution.The maximum point occurs where the derivative is zero.We can refer back to the original differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLChGJkYmKiQlInhHIiIjRigqJCUieUdGLEYo to obtain a numerical procedure for the derivative of the numerical solution.The following procedure "dgn" gives the derivative of the solution.dfn := x -> 1-x^2-fn(x)^2;

NiM+JSRkZm5HZio2IyUieEc2IjYkJSlvcGVyYXRvckclJmFycm93R0YoLCgiIiJGLSokKTkkIiIjRi0hIiIqJCktJSNmbkc2I0YwRjFGLUYyRihGKEYo

Again quotes must be used with the procedure dfn in a plot command.plot('dfn'(x),x=0..2);

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

xmax := fsolve('dfn'(x),x=0.8); fn(xmax);

NiM+JSV4bWF4RyQiKyI+KmYvJikhIzU=NiMkIitAJilSZ18hIzU=

The coordinates of the maximum point on the graph of the solution are approximately ( 0.8504599191, 0.5260398521 ). Alternatively, the numerical procedure findmax from the relevant worksheet can be used.findmax('fn'(x),x=0.85);

NiM3JCQiKyI+KmYvJikhIzUkIitAJilSZ19GJg==

;

;In this example we construct a procedure to evaluate the function NiMvLSUiZkc2IyUieEctJSRJbnRHNiQqJiIiIkYsLCYlInVHRiwtJSRleHBHNiNGLkYsISIiLyUidEc7IiIhRic=. LSUlUExPVEc2MC0lJ0NVUlZFU0c2JDdVNyQkIiIhRiskIiIiRis3JCQiM0FMTCQzRldZcyMhIz4kIjNvPjJdM1InKXolKiEjPTckJCIzV21tbVQmKUdcYUYxJCIzVkg7Pm8sKFwrKkY0NyQkIjNPKipcN2BwKSo+eUYxJCIzTVNjdUQoPlRpKUY0NyQkIjNQTCRla2BvIT41RjQkIjM5SSRlPT89K0YpRjQ3JCQiM29tbSJ6PilHXzpGNCQiMyUpZm83JkcqcGR2RjQ3JCQiMy1ubVQmUVUhKjMjRjQkIjNOW0d6NT5eUXBGNDckJCIzSEwkZVJaWEtpI0Y0JCIzTlYnSC5GSTRTJ0Y0NyQkIjN4bTt6PixfPUpGNCQiMylmWDgyIilwLCdmRjQ3JCQiM3YqKlw3RyRbOGokRjQkIjNjRiVIdDohZV9iRjQ3JCQiMzVuO3olKmZyaFRGNCQiM0dBPz1lRTh2XkY0NyQkIjNBK11pbEZRIXAlRjQkIjNmN3dRUEJ4T1tGNDckJCIzQE1MJDNfIj1NX0Y0JCIzMFxSKikpKjNVQVhGNDckJCIzSG5tVGcoZkpyJkY0JCIzUyF6biV5ZyxxVUY0NyQkIjNrKytdN2VQX2lGNCQiM11iJTRhdFIoNFNGNDckJCIzUSsrXVBmIVF6J0Y0JCIzIVwiKXloeChvcVBGNDckJCIzQCsrXSg9dWJKKEY0JCIzSWRzXXcwKSllTkY0NyQkIjM3bjt6VygqUSp5KEY0JCIzMWNtV2YyYiFRJEY0NyQkIjMjUUxMM0YtR04pRjQkIjNtJikpR1l4aVI9JEY0NyQkIjM9TUxMJGUnM0kpKUY0JCIzI3pLVWNTRSNISUY0NyQkIjM/K103LjxHJlEqRjQkIjMreUo4NSZSOSdHRjQ3JCQiMylITEwkZU1zdykqRjQkIjNlKipwOyRvSEdzI0Y0NyQkIjM7K0RKJkgiZlQ1ISM8JCIzIz0oM3lgZ1YhZSNGNDckJCIzNSt2JGYpWyRINCJGZnIkIjNrPF0nWz86TVgjRjQ3JCQiM2NMJGVrYDFsOSJGZnIkIjMnUUokPj05K0hCRjQ3JCQiM09MZSpbLi1kPiJGZnIkIjNdVCwjPUxVOUEjRjQ3JCQiM2ttOy9FZ3dbN0ZmciQiM1o3US9hJj4/NiNGNDckJCIzem0ieiUqZiUpUUkiRmZyJCIzdlRoJz1eRl0rI0Y0NyQkIjMvK3ZvemEnPU4iRmZyJCIzWVptTXhrKnAiPkY0NyQkIjMob20ieldoby45RmZyJCIzW1FfUUFRJ28jPUY0NyQkIjMtKytdaT5BZDlGZnIkIjMoRzNZYm9jKFE8RjQ3JCQiMzIrXWk6amY0OkZmciQiM1NPIVJGNFFybCJGNDckJCIzOStESiY+ci1jIkZmciQiMydSUThCYiM9I2UiRjQ3JCQiMysrXVA0cWA7O0ZmciQiM1hiTk8jUnJLXSJGNDckJCIzO0xMJGVNJTRuO0ZmciQiM0BRakJQdSlmViJGNDckJCIzNysrdiQ0djVzIkZmciQiM2EpcEQ2cShvbjhGNDckJCIzY20ielduKikqcDxGZnIkIjM3Mz0yI1xtKDM4RjQ3JCQiM0grK0RKaVlCPUZmciQiM10jKnosP3ZRWjdGNDckJCIzQ0xlay5OeXQ9RmZyJCIzQSk9cidwd08jPiJGNDckJCIzPStEY14memojPkZmciQiM2FpKSllITNHdjgiRjQ3JCQiM0NMTDMtPSF5KD5GZnIkIjN5UGpmVCUzazMiRjQ3JCQiMykpKlw3RzhPOy4jRmZyJCIzK2BVPF8xUk41RjQ3JCQiMyFwbW07KlxbJDMjRmZyJCIzaXphP3AoSGEpKSpGMTckJCIzKnBtVCZRel1PQEZmciQiM3Y3KSopenFWJEclKkYxNyQkIjNpTGVrRz00Kj0jRmZyJCIzXV4hZmAhWyllKiopRjE3JCQiM0YrK11pNFRQQUZmciQiM3F2Km9hJGYqZWgpRjE3JCQiM3FMJDNGOSF6I0gjRmZyJCIzSTdha193JykqPilGMTckJCIzJ3BtbTslPktVQkZmciQiM3pAaSJcIXliV3lGMTckJCIzLytESnFKOCZSI0ZmciQiMyEqUkoyc2hNI1soRjE3JCQiM0crdm9hLW9YQ0ZmciQiMzlcI2VgdyEqNDooRjE3JCQiMysrKysrKysrREZmciQiM2I3MyQqPj8kMyJvRjEtJSZDT0xPUkc2JiUkUkdCRyQiIzUhIiIkRitGW1xsRlxcbC0lKVBPTFlHT05TRzYlN1U3JEYqRipGKTckJCIrdlF1aD4hIzYkIitvMm4/JyohIzU3JCQiK0pua29PRmVcbCQiKy9zXjUkKkZoXGw3JCQiKzh2QillJkZlXGwkIitiZCY9KSopRmhcbDckJCIrKGVfMF8oRmVcbCQiKyRRJ29xJylGaFxsNyQkIisxUG9WJSpGZVxsJCIrekMpKXkkKUZoXGw3JCQiK1ZzbUE2RmhcbCQiKz9DJ0g3KUZoXGw3JCQiKylSJkcyOEZoXGwkIitcd2hyeUZoXGw3JCQiK2V4QClcIkZoXGwkIitFUDpEd0ZoXGw3JCQiKyd6UCYpbyJGaFxsJCIrJ1FyP1IoRmhcbDckJCIrW2BJJSk9RmhcbCQiK0NPVmtyRmhcbDckJCIrOXZ0Yz9GaFxsJCIrdE5QdHBGaFxsNyQkIisjSGIzRCNGaFxsJCIrWiI+Im9uRmhcbDckJCIrUCx4WENGaFxsJCIrTCl5PGQnRmhcbDckJCIrM25nTEVGaFxsJCIrZyZ5NlInRmhcbDckJCIrMy49L0dGaFxsJCIrZzYtTWlGaFxsNyQkIis8KTNxKyRGaFxsJCIrOCF6XTAnRmhcbDckJCIrcTYkKXlKRmhcbCQiKyRlPyk0ZkZoXGw3JCQiKzg5cXlMRmhcbCQiKyZwI3BaZEZoXGw3JCQiK1gvaWJORmhcbCQiK0AiZSo0Y0ZoXGw3JCQiK2onRyhcUEZoXGwkIitHYXFrYUZoXGw3JCQiKyplbFgkUkZoXGwkIisvX3dKYEZoXGw3JCQiK0pOVUZURmhcbCQiK2A1Syk+JkZoXGw3JCQiK0V0Xy9WRmhcbCQiK3I8RCEzJkZoXGw3JCQiKyVwZGJcJUZoXGwkIisiKm9WZFxGaFxsNyQkIitlWClScCVGaFxsJCIrJWUkZU1bRmhcbDckJCIrRmRybVtGaFxsJCIreF9NSlpGaFxsNyQkIitALEZgXUZoXGwkIitzJj5OaSVGaFxsNyQkIitsISoqZkMmRmhcbCQiK1Y6JGZeJUZoXGw3JCQiK09uYU1hRmhcbCQiK0A5QjlXRmhcbDckJCIrLmoocGgmRmhcbCQiK0c5LD5WRmhcbDckJCIrTUtgPmVGaFxsJCIrTCJwbkAlRmhcbDckJCIrWCdSOisnRmhcbCQiKzJoKHk3JUZoXGw3JCQiK1EuKGU+J0ZoXGwkIitnTyNmLiVGaFxsNyQkIitHRyc+UCdGaFxsJCIrdjg3YlJGaFxsNyQkIitMJXlXYydGaFxsJCIrJVsrJXBRRmhcbDckJCIrODFpWG5GaFxsJCIrajc4InokRmhcbDckJCIrJ1FtXCRwRmhcbCQiKyMqR282UEZoXGw3JCQiKyhbJzM/ckZoXGwkIit5XkNPT0ZoXGw3JCQiK3krKlFKKEZoXGwkIit1c2BmTkZoXGw3JCQiK3FmYSt2RmhcbCQiK20lZnhbJEZoXGw3JCQiK3omRzlwKEZoXGwkIiskKSkpUjtNRmhcbDckJCIrJGVJMil5RmhcbCQiKyNSJmVaTEZoXGw3JCQiK2wlelkwKUZoXGwkIitzQysnRyRGaFxsNyQkIis5WC9hIylGaFxsJCIrenZHPEtGaFxsNyQkIishKiplQlYpRmhcbCQiK1xAWGRKRmhcbDckJCIrOCV6Q2kpRmhcbCQiKyNweV80JEZoXGw3JCQiKzwqW1chKSlGaFxsJCIrVz5HUElGaFxsNyQkIisrKysrISpGaFxsJCIrcT9hd0hGaFxsNyQkIiIqRltcbEYqLSUmU1RZTEVHNiMlLFBBVENITk9HUklERy1GZltsNiZGaFtsJCIxIz10OiF6Zz4hKiEjO0ZdXG1GXVxtLUYmNiQ3JDckJCIzQSsrKysrKyshKkY0Rio3JEZkXG0kIjMvKysrcT9hd0hGNC1GZltsNiZGaFtsRlxcbEZcXGxGXFxsLSUlVEVYVEc2JTckJCIjISohIiMkISImRmFdbVEieDYiRmlcbS1GXF1tNiU3JCQiI1hGYV1tJCIyLysrKysrKyskRmZyUSVmKHgpRmVdbUZpXG0tJStBWEVTTEFCRUxTRzYnUSFGZV1tRmFebS0lJUZPTlRHNiQlKkhFTFZFVElDQUdGaltsJStIT1JJWk9OVEFMR0ZmXm0tJSpBWEVTVElDS1NHNiUiIiNGal5tLUZjXm02IyUoREVGQVVMVEctJSpHUklEU1RZTEVHNiMlLFJFQ1RBTkdVTEFSRy0lKkxJTkVTVFlMRUc2I0YrRmJebS0lLE9SSUVOVEFUSU9ORzYkJCIjV0YrJEZqXW1GKy0lK1BST0pFQ1RJT05HNiNGaVtsLUZmW2w2IyUlTk9ORUctJSVWSUVXRzYkO0ZcXGwkIiNERltcbDskITErKysrKysrckZmciQiJUA1ISIkThe function NiMtJSJmRzYjJSJ4Rw== gives the area under the graph NiMvJSJ5RyomIiIiRiYsJiUieEdGJi0lJGV4cEc2I0YoRiYhIiI= starting at the y axis, and going as far as the vertical line through x on the x axis. NiMtJSJmRzYjJSJ4Rw== is the solution of the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiKiZGJkYmLCYlInhHRiYtJSRleHBHNiNGK0YmRig=, with the initial condition NiMvLSUiZkc2IyIiIUYn.The function f can certainly be evaluated by means numerical integration using evalf/Int.Every evaluation of f would then require an integral to be evaluated numerically.A procedure constructed using desolveRK may be faster to evaluate.dsolve applied to the initial value problem above, automatically gives the decription of f as an integral.de := diff(y(x),x)=1/(x+exp(x)); ic := y(0)=0; dsolve({de,ic},y(x)): f := unapply(rhs(%),x);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiIiIkYuLCZGLEYuLSUkZXhwR0YrRi4hIiI=NiM+JSNpY0cvLSUieUc2IyIiIUYpNiM+JSJmR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lJEludEc2JComIiIiRjAsJiUidUdGMC0lJGV4cEc2I0YyRjAhIiIvRjI7IiIhOSRGKEYoRig=

We use desolveRK to construct a procedure for NiMtJSJmRzYjJSJ4Rw== which can be evaluated over the interval from 0 to 5. de := diff(y(x),x)=1/(x+exp(x)); ic := y(0)=0; fn := evalf(desolveRK({de,ic},x=0..5,method=rk45,output=rkinterp),15);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiIiIkYuLCZGLEYuLSUkZXhwR0YrRi4hIiI=NiM+JSNpY0cvLSUieUc2IyIiIUYp

evalf(f(Pi),15); evalf(fn(Pi),15);

NiMkIjAjKXA7b19HbSghIzo=NiMkIjAkKXA7b19HbSghIzo=

;The construction of the numerical function fn is rather time consuming, but, once it has been formed, it evaluates quite rapidly compared to the function f defined by the integral.Here are some time trials.First the function f, which is evaluated by perfoming numerical integration.st := time(): xx := Pi; for i from 1 to 10 do xx := evalf(f(xx),15) od: time() - st;

NiM+JSN4eEclI1BpRw==NiMkIiVfUyEiJA==

Now the function fn, which is evaluated by interpolation between precalculated values in the interval NiM3JCIiISIiJg==. st := time(): xx := Pi; for i from 1 to 10 do xx := evalf(fn(xx),15) od: time() - st;

NiM+JSN4eEclI1BpRw==NiMkIiNSISIk

A graph can be plotted using fn.plot('fn'(x),x=0..5);

LSUlUExPVEc2Ky0lJ0NVUlZFU0c2JDdVNyQkIiIhRitGKjckJCIrWCYpR1xhISM2JCIrLnAmKnBeRi83JCQiKzR4JikqMyIhIzUkIiszJ1tIJSkqRi83JCQiKyNSKCpSYyJGNSQiKygqUVhjOEY1NyQkIit1cThRP0Y1JCIrOl4hM3EiRjU3JCQiKyhSd1g1JEY1JCIrTihIKSpRI0Y1NyQkIitzWjN5VEY1JCIrbCc9ZilIRjU3JCQiK100XFlfRjUkIitzaiU9XSRGNTckJCIrVS0vUGlGNSQiKzVkdEJSRjU3JCQiK2ZtcGlzRjUkIitkTHA3VkY1NyQkIisjKj5WQiQpRjUkIismRy45biVGNTckJCIrTWJ3IVEqRjUkIiskXHU2KlxGNTckJCIrMGokby8iISIqJCIrZnMyJ0cmRjU3JCQiK18+alU2RmJvJCIrZWx6P2JGNTckJCIral5aXTdGYm8kIitxZ0hnZEY1NyQkIispPWgoZThGYm8kIispcHJ0KGZGNTckJCIrUVs2ajlGYm8kIisvIz5vOydGNTckJCIrXHooeWIiRmJvJCIrOCl6UEsnRjU3JCQiK2IvY3E7RmJvJCIrNCkzT1wnRjU3JCQiKzx0LG08RmJvJCIrKjROWGknRjU3JCQiK1RqMHg9RmJvJCIrXk5Nam5GNTckJCIrI3BXYCg+RmJvJCIrSVQ/dm9GNTckJCIrImYjPSQzI0ZibyQiK2QhUnIpcEY1NyQkIit0KHBlPSNGYm8kIitYOTslMyhGNTckJCIrdUksJEgjRmJvJCIrc0NNd3JGNTckJCIrclNTIlIjRmJvJCIrQyJlTkQoRjU3JCQiK2A/YChcI0ZibyQiKyllZyZIdEY1NyQkIisrI3B4ZyNGYm8kIis8MT8sdUY1NyQkIitnNHQuRkZibyQiKypwRyFldUY1NyQkIishSHN0IUdGYm8kIitZYTY5dkY1NyQkIitFUlc5SEZibyQiK3YxeW12RjU3JCQiK0tFPj5JRmJvJCIrb3JjOHdGNTckJCIrI1JVMDckRmJvJCIrd2F3YXdGNTckJCIrP1MyTEtGYm8kIit0ZkIncChGNTckJCIrJHApPU1MRmJvJCIrOT8rSXhGNTckJCIrKj1dQFckRmJvJCIrUjN0aXhGNTckJCIrXSR6KlJORmJvJCIrITQibyp5KEY1NyQkIitrQyRwayRGYm8kIitYOlk7eUY1NyQkIiszcWNaUEZibyQiKzhRS1J5RjU3JCQiKy8iZkYmUUZibyQiKyVlPDUneUY1NyQkIiswT2diUkZibyQiK2RLQyEpeUY1NyQkIituQUZqU0ZibyQiK142WSkqeUY1NyQkIismKSpwcDslRmJvJCIrRiVIViJ6RjU3JCQiK3llLHRVRmJvJCIrSkQsSHpGNTckJCIrZk89eVZGYm8kIitmVjxVekY1NyQkIitFPiNbWiVGYm8kIis/PDpgekY1NyQkIisoRyFlJmUlRmJvJCIrJUhSWCd6RjU3JCQiKyYpUWslbyVGYm8kIisoKkh1dHpGNTckJCIrVWpFIXolRmJvJCIrOzpqIyl6RjU3JCQiKzYwTyIqW0ZibyQiKyFSRi4qekY1NyQkIiImRiskIisvb3ooKnpGNS0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiRGK0ZpW2xGaltsLSUrQVhFU0xBQkVMU0c2JFEieDYiUSFGX1xsLSUqR1JJRFNUWUxFRzYjJSxSRUNUQU5HVUxBUkctJSpMSU5FU1RZTEVHNiNGKy0lJUZPTlRHNiQlKkhFTFZFVElDQUdGaFtsLSUsT1JJRU5UQVRJT05HNiQkIiNYRitGX11sLSUrUFJPSkVDVElPTkc2I0ZnW2wtRmRbbDYjJSVOT05FRy0lJVZJRVdHNiQ7RmpbbCQiI11GaVtsOyQhLDNPZiYqZiIhIzckIi0zU0Z2ZCIpRmBebA==

Now consider the problem of solving the equation NiMvLSUkSW50RzYkKiYiIiJGKCwmJSJ1R0YoLSUkZXhwRzYjRipGKCEiIi8lInRHOyIiISUieEcsJkYoRihGM0Yu. This is the x coordinate of the point of intersection of the graphs NiMvJSJ5Ry0lImZHNiMlInhH and NiMvJSJ5RywmIiIiRiYlInhHISIi. plot(['fn'(x),1-x],x=0..2,y=-0.5..1);

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

fsolve('fn'(x)=1-x,x=0.6);

NiMkIitkY2JBaCEjNQ==

The original function NiMtJSJmRzYjJSJ4Rw== defined by the integral can be used; fsolve just takes a little longer since each evaluation of NiMtJSJmRzYjJSJ4Rw== required by fsolve involves the numerical evaluation of an integral. fsolve(f(x)=1-x,x=0.6);

NiMkIitkY2JBaCEjNQ==

Note: The function NiMtJSJmRzYjJSJ4Rw== is defined for negative values of x. An observation in this connection is that NiMvKiYlI2R5RyIiIiUjZHhHISIiKiZGJkYmLCYlInhHRiYtJSRleHBHNiNGK0YmRig= tends to infinity as the denominator NiMsJiUieEciIiItJSRleHBHNiNGJEYl tends to zero, which occurs for a negative value of x.To find this critical value of x we can solve the equation NiMsJiUieEciIiItJSRleHBHNiNGJEYl using fsolve.fsolve(x+exp(x)=0,x=-0.5);

NiMkISsvSFZyYyEjNQ==

In the light of this last result, we could try to construct a numerical function to evaluate NiMvLSUiZkc2IyUieEctJSRJbnRHNiQqJiIiIkYsLCYlInVHRiwtJSRleHBHNiNGLkYsISIiLyUidEc7IiIhRic= on the interval from x = -0.5671432904 to NiMvJSJ4RyIiJg==.This time we use the default precision of 10 digits. de := diff(y(x),x)=1/(x+exp(x)); ic := y(0)=0; fn2 := desolveRK({de,ic},x=-0.5671432904..5,method=rk45,output=rkinterp);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwqJiIiIkYuLCZGLEYuLSUkZXhwR0YrRi4hIiI=NiM+JSNpY0cvLSUieUc2IyIiIUYp

plot('fn2'(x),x=-0.567..5);

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

The value of fn2 is a bit suspect when x is close to -0.5671432904.fn2(-0.56); evalf(f(-0.56));

NiMkISsqPTVScyMhIio=NiMkISsqPTVScyMhIio=

fn2(-0.5671); evalf(f(-0.5671));

NiMkIStwaEMiKWYhIio=NiMkIStyaEMiKWYhIio=

fn2(-0.56714); evalf(f(-0.56714));

NiMkIStCc3dEdyEiKg==NiMkIStCS2VEdyEiKg==

;

;In this example we consider the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCQqJiUieEdGJiUieUdGKEYo, subject to the initial condition y(0) = 1. We construct a discrete numerical solution over the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIiIg== using the combined 7th and 8th order Runge-Kutta method with adaptive step-size control. The numerical solution is constructed using the option ''errorcontrol=accumulative'' in an attempt to minimise the global error. We compare the numrical solution with the analytical solution NiMvLSUiZ0c2IyUieEctJSVzcXJ0RzYjLCYiIiJGLCokRiciIiMhIiI=, both in a table and graphically. de := diff(y(x),x)=-x/y(x); ic := y(0)=1; soln := desolveRK({de,ic},x=0..1,method=rk78, errorcontrol=accumulative,output=points): g := x -> sqrt(1-x^2); comparewithfcn(soln,g); plot([soln,g(x)],x=0..1,style=[point,line],color=[black,coral], symbol=circle,scaling=constrained);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYwNiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lJXNxcnRHNiMsJiIiIkYwKiQpOSQiIiNGMCEiIkYoRihGKA== 0 1 function val: 1 rel err: 0.0000e-01 .01 .999949999 function val: .999949999 rel err: 0.0000e-01 .06 .998198377 function val: .998198377 rel err: 0.0000e-01 .152417527 .988316193 function val: .988316193 rel err: 0.0000e-01 .279076821 .960268779 function val: .960268779 rel err: 1.0414e-10 .435141605 .900362029 function val: .900362029 rel err: 0.0000e-01 .549440671 .835532733 function val: .835532733 rel err: 0.0000e-01 .641529597 .767098283 function val: .767098283 rel err: 0.0000e-01 .716000056 .698100222 function val: .698100222 rel err: 0.0000e-01 .775756132 .631032823 function val: .631032823 rel err: 1.5847e-10 .823378676 .56749234 function val: .56749234 rel err: 1.7621e-10 .861144294 .508360606 function val: .508360606 rel err: 0.0000e-01 .890986499 .4540298 function val: .4540298 rel err: 2.2025e-10 .914505606 .404573228 function val: .404573228 rel err: 0.0000e-01 .933004953 .359863526 function val: .359863527 rel err: 2.7788e-10 .947534164 .319654515 function val: .319654515 rel err: 0.0000e-01 .958932231 .283635288 function val: .283635288 rel err: 0.0000e-01 .967866081 .251466198 function val: .251466198 rel err: 3.9767e-10 .974863769 .222801776 function val: .222801776 rel err: 8.9766e-10 .980342016 .197305681 function val: .197305681 rel err: 0.0000e-01 .984629022 .174658777 function val: .174658777 rel err: 0.0000e-01 .987982758 .15456413 function val: .15456413 rel err: 6.4698e-10 .990605753 .136748827 function val: .136748827 rel err: 2.1938e-09 .992656843 .120964428 function val: .120964428 rel err: 8.2669e-10 .994260476 .106986474 function val: .106986473 rel err: 1.8694e-09 .995514129 .094612995 function val: .094612996 rel err: 5.9188e-09 .996494092 .083663163 function val: .083663163 rel err: 1.7929e-09 .997260061 .073975475 function val: .073975475 rel err: 6.3535e-09 .997858733 .065406038 function val: .065406038 rel err: 3.9752e-09 .998326627 .057826857 function val: .057826857 rel err: 1.0376e-09 .9986923 .051124254 function val: .051124254 rel err: 3.7164e-09 .998978076 .045197376 function val: .045197377 rel err: 1.8806e-08 .999201408 .039956804 function val: .039956805 rel err: 3.1284e-08 .999375937 .035323316 function val: .035323317 rel err: 2.0949e-08 .999512326 .031226751 function val: .031226751 rel err: 4.4833e-09 .999618909 .027605018 function val: .027605018 rel err: 2.5720e-08 .999702199 .024403157 function val: .024403158 rel err: 3.2783e-08 .999767286 .021572547 function val: .021572547 rel err: 4.6355e-09 .999818148 .019070181 function val: .019070178 rel err: 1.2690e-07 .999857893 .016858026 function val: .016858025 rel err: 9.5503e-08 .999888953 .014902437 function val: .014902439 rel err: 1.4427e-07 .999913223 .013173677 function val: .013173682 rel err: 3.9624e-07 .99993219 .011645442 function val: .011645445 rel err: 2.7822e-07 .99994701 .010294479 function val: .010294484 rel err: 5.1095e-07 .999958592 .009100228 function val: .009100225 rel err: 2.5010e-07 .999967642 .008044511 function val: .008044514 rel err: 3.5204e-07 .999974715 .007111266 function val: .007111273 rel err: 8.9928e-07 .999980241 .006286286 function val: .006286287 rel err: 1.1263e-07 .99998456 .005557008 function val: .005557014 rel err: 9.9118e-07 .999987934 .004912332 function val: .004912342 rel err: 1.9217e-06 .999992934 .003759136 function val: .003759149 rel err: 3.4385e-06 .999997934 .002032503 function val: .002032535 rel err: 1.5681e-05 1 8.084348051e-05 function val: 0 rel err: infinityNiMlIUc= Maximum relative error: 1.5681e-05 obtained for the input value: .999997934 excluding any cases where the function value is zero.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

There is some difficulty in obtaining values with small relative error when x is close to 1, so the majority of the points for the discrete numerical solution are crowded towards the pointNiMtJSFHNiQiIiIiIiE=.A continuous solution can also be obtained.de := diff(y(x),x)=-x/y(x); ic := y(0)=1; gn := desolveRK({de,ic},x=0..1,errorcontrol=accumulative,method=rk78): g := x -> sqrt(1-x^2); plot([g,gn],0..1,color=[red,green],thickness=[1,2],scaling=constrained);

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJComRiwiIiJGKSEiIkYwNiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiM+JSJnR2YqNiMlInhHNiI2JCUpb3BlcmF0b3JHJSZhcnJvd0dGKC0lJXNxcnRHNiMsJiIiIkYwKiQpOSQiIiNGMCEiIkYoRihGKA==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

Values given by the continuous numerical solution gn are not very accurate when x is close to 1. xx := 0.999; evalf(gn(xx)); evalf(g(xx));

NiM+JSN4eEckIiQqKiohIiQ=NiMkIitveCxyVyEjNg==NiMkIisieTw1WiUhIzY=

xx := 0.99999; evalf(gn(xx)); evalf(g(xx));

NiM+JSN4eEckIiYqKioqKiEiJg==NiMkIit2TTdzVyEjNw==NiMkIit2WjdzVyEjNw==

evalf[15](plot(g(x)-'gn'(x),x=0.997..0.99999,color=blue));

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

;

;

The Maple procedure dsolve can be used to obtain a continuous numerical solution for a differential equation of the formNiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiZkc2JCUieEclInlH by an adaptive combined 4th and 5th order Runge-Kutta method.The options "type=numeric" together with "method=rkf45" achieve this.The following subsection contains the information from the relevant Maple help file.;

The following information comes from the Maple help files.Calling Sequence: dsolve(deqns, vars, type=numeric, method=rkf45, options) dsolve(deqns, vars, numeric, method=rkf45, options)Parameters: deqns - set or list; ordinary differential equation(s) and initial conditions type=numeric - instruct dsolve to find a numerical solution method=rkf45 - equation; numerical method to use vars - (optional) dependent variable or a set or list of dependent variables for deqns options - (optional) equations of the form keyword = valueDescription:Invoking the dsolve function with the options numeric and method=rkf45 causes dsolve to find a numerical solution using a Fehlberg fourth-fifth order Runge-Kutta method with degree four interpolant. This is the default method of the type=numeric solution for initial value problems when the stiff argument is not used. The rkf45 method has two distinct modes of operation (for procedure-type outputs):With range:When used with the range option, the method computes the solution for the IVP over the specified range, storing the solution information internally, and uses that information to rapidly interpolate the desired solution value for any call to the returned procedure(s).In this mode of operation, steppast defaults to false, indicating that the method should adjust the step sizes at the end-points of the integration (in either direction) so as to not step past the end-points of the range.Though possible, it is not recommended that the returned procedure be called for points outside the specified range.This method can be used in combination with the refine option of odeplot to produce an adaptive plot (that is, odeplot uses the precomputed points to produce the plot when refine is specified.)It is not recommended that this method be used for problems in which the solution can become singular, as each step is stored, and many steps may be taken when near a singularity, so memory usage can become a significant issue.The storage of the interpolant in use by this method can be disabled with the interpolation=false option described below. This may be desirable for high accuracy solutions where storage of the interpolant (in addition to the discrete solution) requires too much memory. Disabling the interpolant is not generally recommended as the solution values will be obtained from an interpolation of the 5 closest points, and does not necessarily provide an interpolant with order 4 error. Without range:When used without the range option, the IVP solution values are not stored, but rather computed when requested.In this mode of operation, steppast defaults to true, and output solution values are computed using the method interpolant. This approach provides more efficient continuation of the solution if multiple points are to be requested. The use of the interpolant with this method can be disabled with the interpolation=false option described below, but also requires that steppast=false is specified. Since not all solution values are stored, computation must restart at the initial values whenever a point is requested between the initial point and the most recently computed point (to avoid reversal of the integration direction), so it is advisable to collect solution values moving away from the initial value. By setting infolevel[dsolve] to 2, information on the last mesh point evaluation is provided in the event of an error.The following options are available for the rkf45 method. 'output' = keyword or array 'abserr' = numeric 'relerr' = numeric 'initstep' = numeric 'maxfun' = integer 'number' = integer 'procedure' = procedure 'start' = numeric 'initial' = array 'procvars' = list 'startinit' = boolean 'implicit' = boolean 'range' = numeric..numeric 'interpolate' = boolean 'steppast' = boolean 'stop_cond' = list The output option specifies the desired output from dsolve. The keywords procedurelist or listprocedure provide procedure-type output, the keyword piecewise provides output in the form of piecewise functions over a specified range of independent variable values, and a 1-d array or Array provide output at fixed values of the independent variable. This is discussed in more detail dsolve[numeric].The abserr, relerr, and initstep options specify the desired accuracy of the solution, and the starting step size for the method, and are discussed in dsolve[Error_Control]. The default values for rkf45 are abserr=Float(1,-7) and relerr=Float(1,-6). The value for initstep, if not specified, is determined by the method, taking into account the local behavior of the ODE system.The maxfun option specifies a maximum on the number of evaluations of the right-hand side of the first order ODE system. This option is disabled by specifying maxfun=0. The default value for rkf45 is 30000.The number, procedure, start, initial, procvars, startinit, and implicit parameters are used for specification of the IVP using procedures, and the behavior of the computation, and are discussed in dsolve[numeric,IVP].The range argument determines the range of values of the independent variable for which solution values are required. Use of this option significantly changes the behavior of the method for the procedure-style output types discussed in dsolve[numeric] (see description of with/without range above).The interpolate argument accepts a boolean value that indicates if interpolation should be performed to compute additional solution values between steps during the Runge-Kutta integration. This is true by default. For loose error tolerances (such as the default) this provides more computed points, and results in a smoother plot, and more consistent accuracy for the interpolated solution. It may be desirable to set this to false for problems with tighter error tolerances with a specified range, reducing the amount of memory needed to store the computed solution.The steppast argument accepts a boolean value that tells the method whether the computation can utilize values outside the range (when range is specified), or compute values past the requested point (when range is not specified). The default value depends on the presence of the range argument. It is false if range is used, and true otherwise. When storage is not in use, this option must agree with interpolate, and provides more consistent solution values, and more efficient solution continuation. In this case, steppast=false, interpolate=false means that the method must adjust the step size whenever near an output point, and must step back to before the step size adjustment to continue the computation. The stop_cond argument accepts a list of stopping criteria. This is a list of criteria that are monitored during numerical integration of the ODE solution. When one of the criteria is met, the integration stops, and the current point is given as the solution. This option is discussed in detail in dsolve[Stop_Conditions]. The rkf45 method is capable of computing high-accuracy solutions for IVPs because the precision of the computation can be increased by changing the Digits environment variable. However, it is recommended that the higher order dverk78 or gear methods be used instead.Results can be plotted by using the function odeplot in the plots package.ReferencesE. Fehlberg. "Klassische Runge-Kutta-Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Waermeleitungsprobleme" Computing, vol. 6 (1970) 61-71. W.H. Enright, K.R. Jackson, S.P. Norsett, and P.G. Thomsen. "Interpolants for Runge-Kutta Formulas" ACM TOMS, vol. 12 (1986) 193-218. L.F. Shampine and R.M. Corless. "Initial Value Problems for ODEs in Problem Solving Environments" J. Comp. Appl. Math, vol. 125(1-2) (2000) 31-40. G.E. Forsythe, M.A. Malcolm, and C.B. Moler. Computer Methods for Mathematical Computations

The following commands give a numerical solution for the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLChGJkYmKiQlInhHIiIjRigqJCUieUdGLEYo as a numerical procedure fn. de := diff(y(x),x)=1-x^2-y(x)^2; ic := y(0)=0; dsol := dsolve({de,ic},y(x),type=numeric,method=rkf45, output=listprocedure): fn := subs(dsol,y(x));

NiM+SSNkZUc2Ii8tSSVkaWZmR0kqcHJvdGVjdGVkR0YpNiQtSSJ5R0YlNiNJInhHRiVGLiwoIiIiRjAqJEYuIiIjISIiKiRGK0YyRjM=NiM+SSNpY0c2Ii8tSSJ5R0YlNiMiIiFGKg==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

;Note that there is no requirement to specify a solution interval, so the function fn which gives the solution cannot be using interpolation between precomputed values in the way that the solution function constructed with desolveRK does. In order to obtain the first value asked for, say NiMtJSNmbkc2IyUieEc=, the procedure must be constructing points along the solution curve all the way from the initial pointNiMtJSFHNiQiIiFGJg== to the pointNiMtJSFHNiQlInhHLSUjZm5HNiNGJg==. For subsequent evaluations of the solution function, it appears likely that any previously computed values are accessible, and do not have to be recomputed. The graph of the solution can be plotted as in a previous example.plot('fn'(x),x=0..2,color=red);

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

With Maple 6 fsolve gave no value for the following, but Maple 7 gives a value which differs somewhat from the value given when the numerical solution for the differential equation is constructed using desolveRK. fsolve('fn(x)'=0,x=1..2);

NiMkIisvdSY0ZyIhIio=

If the precision used in the construction of the solution procedure hn by dsolve is increased to16, and a parameter which controls the relative error is adjusted, fsolve produces a different answer, but it still disagrees with the one given by desolveRK. de := diff(y(x),x)=1-x^2-y(x)^2; ic := y(0)=0; saveDigits := Digits: Digits := 16: dsol := dsolve({de,ic},y(x),type=numeric, method=rkf45,output=listprocedure,relerr=Float(1,-14)): Digits := saveDigits: fn := subs(dsol,y(x));

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsKCIiIkYuKiQpRiwiIiNGLiEiIiokKUYpRjFGLkYyNiM+JSNpY0cvLSUieUc2IyIiIUYp

fsolve('fn'(x)=0,x=1..2);

NiMkIitTdiY0ZyIhIio=

Note: A quick way of obtaining pictures of various solution curves along with the direction field is to insert a list of initial conditions after the plotting range in a DEplot command.with(DEtools): de := diff(y(x),x)=1-x^2-y(x)^2; DEplot(de,y(x),x=0..2,y=-1..1,[[0,0],[0,0.5]],arrows=medium,color=blue, dirgrid=[25,25],linecolor=[coral,green]);

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cm7'7$F`en$"3]tFAY[**\&*F<7$Feen$"3_j06([QLy)F<7$Fjen$"3%Ht2Wg'\3*)F<7$F_fn$"3-CaH6B:!**)F<Fcgcm7'7$Fefn$"3apn"yb0fb*F<7$Fjfn$"3]nl^vxUx()F<7$F_gn$"3*o#Q(*eW!)3*)F<7$Fdgn$"3?"=O*f=9$)*)F<F`hcm7'7$Fjgn$"3%**y=.771c*F<7$F_hn$"35ZX,87ss()F<7$Fdhn$"3GHhr"Gw$4*)F<7$Fihn$"35)3")["HBx*)F<F]icm7'7$F_in$"3'[DDa+#Qk&*F<7$Fdin$"3=#33zK^*o()F<7$Fiin$"3h*[N044,"*)F<7$F^jn$"3WW.7_^As*)F<Fjicm7'7$Fdjn$"3!Rz8U(4Un&*F<7$Fijn$"39V&>"fB"fw)F<7$F^[o$"3)e:_/!>$4"*)F<7$Fc[o$"3S#y)zGi&z'*)F<Fgjcm7'7$Fi[o$"35BV)*\p))p&*F<7$F^\o$"3%R,\LQYMw)F<7$$"3G:9RGF(e*>F8$"3VRt)fY'z6*)F<7$Fh\o$"3YO\Q4`Hk*)F<Fd[dm7'7$F*$"3A+++++++5F87$F1Fa\dm7$$"3o&3\a2ZCO#F,$"3#fLLLLLe*)*F<7$$"3Q'3\a2ZCO#F,$"3'pmmmm;/,"F8Fc\dm7'7$F?$"3;j]([M*G+5F87$FD$"3ur$\7b1r***F<7$FI$"3A%)zG'H&>%*)*F<7$$"3;f\y1dHq5F<$"3MNLjgBD55F8Fb]dm7'7$FT$"32:]8hp:,5F87$FY$"3)[&)\')QI%))**F<7$Fhn$"3?+"p^n8$*))*F<7$F]o$"3+vr%zmc(45F8Fa^dm7'7$Fco$"32mkD&4*f-5F87$Fho$"3yV`VZ!4S(**F<7$F]p$"3Y%[r(\'*H"))*F<7$Fbp$"3Uv)GqsA*35F8F^_dm7'7$Fhp$"3A_ke88g/5F87$F]q$"3G#[NT'o)R&**F<7$Fbq$"3)GaB2e"Qq)*F<7$Fgq$"3$zQ6#pSu25F8F[`dm7'7$F]r$"3Qg")4%=Fr+"F87$Fbr$"3n+%=!f"G(G**F<7$Fgr$"3CqrV?$ep&)*F<7$F\s$"3F=kCA@A15F8Fh`dm7'7$Fbs$"3t!=V5l0,,"F87$Fgs$"39(>o&*[V*)*)*F<7$F\t$"3VnAwJdkT)*F<7$Fat$"3w(ei_(eP/5F8Feadm7'7$Fgt$"3B20<NCU85F87$F\u$"3EL\H[cxl)*F<7$Fau$"3)z]q!)o#GD)*F<7$Ffu$"3dW-pI5D-5F8Fbbdm7'7$F\v$"3lgb3OC#p,"F87$Fav$"3+*RW"RcxI)*F<7$Ffv$"3[!>$)QBj)3)*F<7$$"3B?b$*)*e&[#pF<$"3_Wd%)R1C****F<F_cdm7'7$Faw$"3zc8l^vU?5F87$Ffw$"3UNk[$[Cdz*F<7$F[x$"3_H>Pe#*Q$z*F<7$F`x$"3IxGF&\n\(**F<F^ddm7'7$Ffx$"3>1B1#\mP-"F87$F[y$"3cUpPz]Li(*F<7$F`y$"3sJmPPwoz(*F<7$Fey$"3cN_A!Q13&**F<F[edm7'7$F[z$"3<ou@l[!o-"F87$F`z$"3!RKDyM^>t*F<7$Fez$"3s*yCn!)p#o(*F<7$Fjz$"3y3$f6$)px#**F<Fhedm7'7$F`[l$"3zR%\Dyi%H5F87$Fe[l$"3j2c]u@P0(*F<7$Fj[l$"3%f'>!*3GHf(*F<7$F_\l$"3yj"\;s1m!**F<Fefdm7'7$Fe\l$"3(ozTm.:<."F87$Fj\l$"3yN?eL'\Go*F<7$F_]l$"3G*f:F6@Ev*F<7$Fd]l$"3'[(feG%Qx))*F<Fbgdm7'7$Fj]l$"3/c/()3%yN."F87$F_^l$"3)GW&H6f@k'*F<7$Fd^l$"3A[-Ee+%zu*F<7$Fi^l$"37$RN`J*Gr)*F<F_hdm7'7$F__l$"3$*HUpTY4N5F87$Fd_l$"3A1x0$e`!\'*F<7$Fi_l$"3J;lc7k'[u*F<7$F^`l$"3a]!))fEpr&)*F<F\idm7'7$Fd`l$"3x*=!QZcJO5F87$Fi`l$"3p1")>EN%oj*F<7$F^al$"3LC.+8g-V(*F<7$Fcal$"3I3rDrP;X)*F<Fiidm7'7$Fial$"3<Q'zDd$HP5F87$F^bl$"3(Qi.UFkqi*F<7$Fcbl$"3(>(z6>X4U(*F<7$Fhbl$"3'os2=%e+N)*F<Ffjdm7'7$$"3MRW"RcxI["F8$"3.6D>"[v!Q5F87$$"3xfb3OC#p^"F8$"3E&*[2)=X#>'*F<7$$"3ADa,Of2+:F8$"3S\6tn2"=u*F<7$$"3i!o6mn8">:F8$"3X]*e"[HUE)*F<Fe[em7'7$Fcdl$"3t&)H8)H,(Q5F87$Fhdl$"3:Z,n=q)Hh*F<7$F]el$"3Tn@1jW(>u*F<7$Fbel$"3$Q8/xki">)*F<Fh\em7'7$Fhel$"3?(>h@k.#R5F87$F]fl$"3RK!)QyN'zg*F<7$Fbfl$"3^6`g]qVU(*F<7$Fgfl$"3@ma\1O+8)*F<Fe]em7'7$F]gl$"3PCrog&3'R5F87$Fbgl$"3sg(GJR9Rg*F<7$Fggl$"3aJC,t44V(*F<7$F\hl$"3/h*o3,ex!)*F<Fb^em7'7$Fbhl$"3C)3htdO*R5F87$Fghl$"3!47*QEUj+'*F<7$F\il$"39'f6oyfQu*F<7$Fail$"3l$p:8dpK!)*F<F__em7'7$Fgil$"3KdyU\P?S5F87$F\jl$"3EJ9s0D'zf*F<7$Fajl$"3VdE!z,!pW(*F<7$Ffjl$"3a(Gfon4%*z*F<F\`em7'7$F\[m$"3[AF</EUS5F87$Fa[m$"3y!ys#eRx&f*F<7$Ff[m$"3)4Lo6%[aX(*F<7$F[\m$"3_LUQ'4tgz*F<Fi`em-%*THICKNESSG6#"""-%&COLORG6&%$RGBG$F/F.Fjaem$"#5F.-F&6M7$7$$F/F/Fabem7$$"+++++5!#5$"3#)>#H7)=eL**F,7$Fbbem7$$"+++++?Febem$"3O'R&RZ))\Z>F<7$Fibem7$$"+++++IFebem$"3PCHv>;=EGF<7$F_cem7$$"+++++SFebem$"3]yr7JH^)f$F<7$Fecem7$$"+++++]Febem$"3EFL`Z^HSUF<7$F[dem7$$"+++++gFebem$"3v,*RxFSUt%F<7$Fadem7$$"+++++qFebem$"3;&*4@Jb?p]F<7$Fgdem7$$"+++++!)Febem$"3/pu">Gl(Q_F<7$F]eem7$$"+++++!*Febem$"33b$[j&eXR_F<7$Fceem7$$Fdbem!"*$"3?7`ga8&*o]F<7$Fieem7$$"+++++6F[fem$"3.6gSFTICZF<7$F_fem7$$"+++++7F[fem$"33_f?FOD+UF<7$Fefem7$$"+++++8F[fem$"3+"4#3=r[([$F<7$F[gem7$$"+++++9F[fem$"3m!3ZcB81d#F<7$Fagem7$$"+++++:F[fem$"3xI<1dcWD9F<7$Fggem7$$"+++++;F[fem$"3e7Uo37."\"Fgc_l7$F]hem7$$"+++++<F[fem$!3+")>m%pzwr"F<7$Fchem7$$"+++++=F[fem$!3q')3!o>z/'QF<7$Fihem7$$"+++++>F[fem$!3yo!4)z0_blF<7$F_iem7$$F[cemF[fem$!37BJ\HvY/5F87$7$Fabem$""&F.7$Fcbem$"3+?Rp&e!y3dF<7$F]jem7$Fjbem$"3a-koJIAAjF<7$Fajem7$F`cem$"3NB'fNTd_#oF<7$Fejem7$Ffcem$"3TrqwyV,3sF<7$Fijem7$F\dem$"3yqu1Q]#[Y(F<7$F][fm7$Fbdem$"3C\wZz5+$f(F<7$Fa[fm7$Fhdem$"3E/S(*)[(e"f(F<7$Fe[fm7$F^eem$"3S(QV932-Y(F<7$Fi[fm7$Fdeem$"3'HbyF`T!)>(F<7$F]\fm7$Fjeem$"3'*ed!HHKH!oF<7$Fa\fm7$F`fem$"3Qf&)[9(Q0F'F<7$Fe\fm7$Fffem$"3Ik=9ILX$f&F<7$Fi\fm7$F\gem$"3uE\)*zI8gZF<7$F]]fm7$Fbgem$"3'=!Qu+?W`PF<7$Fa]fm7$Fhgem$"3yg5u9`[[DF<7$Fe]fm7$F^hem$"3lu'y*H'*=46F<7$Fi]fm7$Fdhem$!3A"\\hrUP<'F,7$F]^fm7$Fjhem$!3Me")R&\64r#F<7$Fa^fm7$F`iem$!3E7/56SI'H&F<7$Fe^fm7$Ffiem$!3q]Y<7JP!e)F<-Fcaem6#""$-%&STYLEG6#%%LINEG-Fgaem6erFiaemF[bem$"1k9.e@R!)\!#;FjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemF[bemFe_fmFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaemFjaemF[bemFjaem-%+AXESLABELSG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;

An adaptive combined 7th and 8th order Runge-Kutta method can be applied to construct a continuous numerical solution for a differential equation of the formNiMvKiYlI2R5RyIiIiUjZHhHISIiLSUiZkc2JCUieEclInlH with dsolve by including the options "type=numeric" and "method=dverk78".The following subsection contains the information from the relevant Maple help file.;

The following information comes from the Maple help files.Calling Sequence: dsolve(deqns, vars, type=numeric, method=dverk78, options) dsolve(deqns, vars, numeric, method=dverk78, options)Parameters: deqns - set or list; ordinary differential equation(s) and initial conditions type=numeric - instruct dsolve to find a numerical solution method=dverk78 - literal equation; which numerical method to use vars - (optional) dependent variable or a set or list of dependent variables for deqns options - (optional) equations of the form keyword = valueDescription:Invoking the dsolve function with options numeric and method=dverk78 causes dsolve to find a numerical solution using a seventh-eighth order continuous Runge-Kutta method.The following options are available for the basic use form of the dverk78 method: 'output' = keyword or array 'number' = integer 'procedure' = procedure 'start' = numeric 'initial' = array 'procvars' = list 'startinit' = boolean 'implicit' = boolean 'abserr' = numeric 'relerr' = numeric 'errorest' = symbol 'initstep' = numeric 'minstep' = numeric 'maxstep' = numeric 'maxfun' = integer Note: See also information about the advanced use form of the dverk78 method in dsolve[dverk78,advanced]. The output option specifies the desired output from dsolve. This option is discussed in dsolve[numeric].The number, procedure, start, initial, procvars, startinit, and implicit parameters are used for specification of the IVP using procedures, and the behavior of the computation. These options are discussed in dsolve[numeric,IVP].The abserr, relerr, initstep, minstep, and maxstep options specify the desired accuracy of the solution, and allow for more detailed control of the step size. These options are discussed in detail in dsolve[Error_Control]. The default values are abserr=Float(1,-8), relerr=Float(1,-8), minstep=0, maxstep=2.0, and initstep=maxstep*(relerr)^(1/6).The dverk78 method is capable of working in arbitrary precision based on the setting of Digits, and can be used to obtain high accuracy solutions for ODE systems. As a note of caution, however, it is often necessary to work with a greater number of Digits than would be expected for the requested error tolerance, to prevent round-off error from giving an inaccurate error estimate. When the error tolerance is too strict for the current setting of Digits, it is detected by the algorithm, and an error is issued. Asymptotically dverk78 requires Digits set so that tol = O(Float(1,-Digits)^(9/8)). Note: just as for all other numeric dsolve methods, any setting of Digits that is less than or equal to hardware precision works in hardware precision, so for Digits < evalhf(Digits), the algorithm works in trunc(evalhf(Digits)) Digits (see evalhf).The errorest option allows control over the method used to obtain error estimates, and can have the values interpolant or pair. The interpolant option is the default, and tells dverk78 to estimate the error using the interpolant of the method. The pair option tells dverk to estimate the error using the difference between the computed solution and a lower order estimate of the solution (hence 'pair'). Typically, the interpolant option gives a better error estimate than the pair option, but requires more work to compute. This may also be a better option for expensive systems, as dverk78 with the interpolant option performs 20 function evaluations per step, while the pair option only requires 13.The maxfun option is an integer that specifies the maximum on the number of evaluations of the ODE or ODE system in one call to the returned procedure. This limits the amount of work done on any individual call (See xrefdsolve[maxfun]). This option is relevant for the procedure-style output only. By default, this value is set to zero (disabled).Much finer control over the behavior of the dverk78 method is possible through use of the control option, including the capability of running partial calculations, specification of the scale of the problem, and obtaining information on the last step size used, the next step size to be attempted, the local error, etc. This option is somewhat complex, and is discussed in detail in dsolve[dverk78,advanced].Results can be plotted using the function odeplot in the plots package.Credits:Much appreciation and thanks go to Dr. Jim Verner who provided us with the coefficients for the dverk78 method, and the related interpolant. Without these coefficients the method would be restricted to hardware precision.Reference:W.H. Enright. "The Relative Efficiency of Alternative Defect Control Schemes for High Order Continuous Runge-Kutta Formulas." Technical Report 252/91, Dept. of Computer Science, University of Toronto, June, 1991. J.H. Verner. "Explicit Runge-Kutta Methods with Estimates of the Local Truncation Error." SIAM Journal of Numerical Analysis, Aug. 1978.

The following commands construct a numerical solution to the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLChGJkYmKiQlInhHIiIjRigqJCUieUdGLEYo , with the initial condition NiMvLSUieUc2IyIiIUYn. The numerical solution is given as a function gn.de := diff(y(x),x)=1-x^2-y(x)^2; ic := y(0)=0; dsol := dsolve({de,ic},y(x),type=numeric, method=dverk78,output=listprocedure,tolerance=Float(1,-10)): gn := subs(dsol,y(x));

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsKCIiIkYuKiQpRiwiIiNGLiEiIiokKUYpRjFGLkYyNiM+JSNpY0cvLSUieUc2IyIiIUYp

The value obtained for the positive zero of the solution now agrees with the value obtained using desolveRK. fsolve('gn'(x),x=1..2);

NiMkIis3dyY0ZyIhIio=

;

;

;

The differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJiUieUdGJi0lJGNvc0c2IyUieEdGJkYmKiZGJkYmIiImRihGKA== with the initial condition NiMvLSUieUc2IyIiISIiIg== has the analytical solutionNiMvLSUieUc2IyUieEcsJiooLCQqJiIiIkYsIiImISIiRi5GLC0lJGV4cEc2Iy0lJHNpbkdGJkYsLSUkSW50RzYkLUYwNiMsJC1GMzYjJSJ1R0YuL0Y8OyIiIUYnRixGLEYvRiw=.de := diff(y(x),x)=y(x)*cos(x)-1/5; ic := y(0)=1; dsolve({de,ic},y(x));

NiM+JSNkZUcvLSUlZGlmZkc2JC0lInlHNiMlInhHRiwsJiomRikiIiItJSRjb3NHRitGL0YvI0YvIiImISIiNiM+JSNpY0cvLSUieUc2IyIiISIiIg==NiMvLSUieUc2IyUieEcsJiomLSUkZXhwRzYjLSUkc2luR0YmIiIiLSUkSW50RzYkLCQtRis2IywkLUYuNiMlInVHISIiI0Y6IiImL0Y5OyIiIUYnRi9GL0YqRi8=

;(a) Construct a discrete numerical solution for differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCYqJiUieUdGJi0lJGNvc0c2IyUieEdGJkYmKiZGJkYmIiImRihGKA== with the initial condition NiMvLSUieUc2IyIiISIiIg== over the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIjOA== using a combined 7th and 8th order Runge-Kutta method via desolveRK. Compare the numerical solution with the analytical solution, both in a table and graphically.(b) Construct and plot a continuous solution over the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIjOA== along with the direction field. Take x from 0 to 13 and y from -3 to 3 for the plot.(c) Compare the values of the continuous numerical solution and the analytical solution when NiMvJSJ4RyIjOA==.____________________________________________________________________;

Consider the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLSUkc2luRzYjLCYqJiIiI0YmJSJ4R0YmRiYlInlHRig= with the initial condition NiMvLSUieUc2IyIiISwkIiIiISIi.(a) Plot the direction field for this differential equation for x between 0 and 5 and y between -1 and 2.(b) Use a Runge-Kutta method with adaptive step-size to obtain a numerical solution of this equation in the form of a Maple procedure which can be evaluated anywhere in the interval NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIiJg== to give a value which is accurate to about 10 digits. (c) Find the first positive zero of the solution to an accuracy of about 10 digits. (d) Find the coordinates of the first maximum point on the graph of the solution to an accuracy of about 10 digits.____________________________________________________________________;

(a) Find a continuous numerical solution for the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiKiYsJiUieEdGJi0lJWNvc2hHNiMlInlHRihGJiwmRitGJkYvRiZGKA== with the initial condition NiMvLSUieUc2IyIiISIiIw==, in the form of a Maple procedure which can be evaluated anywhere throughout the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIiKQ== to give a value which is accurate to about 10 digits.(b) Plot the graph of the solution found in (a).(c) Find the value of x (other than NiMvJSJ4RyIiIQ==) for which the solution has the value 2.(d) Find the minimum value of the solution NiMtJSJ5RzYjJSJ4Rw== in the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIiKQ==, and the value of NiMlInhH where this minimum occurs.____________________________________________________________________;

By solving an appropriate initial value problem numerically, construct a procedure which evaluates the function NiMvLSUiZkc2IyUieEctJSRJbnRHNiQtJSNsbkc2IywmLSUnYXJjdGFuRzYjJSJ0RyIiIkYzRjMvRjI7IiIhRic= for NiMxIiIhJSJ4Rw==NiMxJSFHIiIl, to an accuracy of about 10 digits.The function NiMtJSJmRzYjJSJ4Rw== gives the area under the graph NiMvJSJ5Ry0lI2xuRzYjLCYtJSdhcmN0YW5HNiMlInhHIiIiRi1GLQ== starting at the y axis, and going as far as the vertical line through x on the x axis. 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 the graph of f over the interval NiMxIiIhJSJ4Rw==NiMxJSFHIiIl. ____________________________________________________________________;

(a) Find a differential equation which is satisfied by the function NiMvLSUiZkc2IyUieEctJSRJbnRHNiQtJSVzcXJ0RzYjLCYiIiJGLy0lJGNvc0c2IyokJSJ0RyIiI0YvL0Y0OyIiIUYn. When x is positive, the function NiMtJSJmRzYjJSJ4Rw== gives the area under the graph NiMvJSJ5Ry0lJXNxcnRHNiMsJiIiIkYpLSUkY29zRzYjKiQlInhHIiIjRik= starting at the y axis, and going as far as the vertical line through x on the x axis. 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(b) Using an appropriate initial condition, construct a numerical solution for the differential equation found in (a) as a procedure which can be evaluated throughout the interval from NiMvJSJ4RywkIiImISIi to NiMvJSJ4RyIiJg== to give values which are correct to about 10 digits. (c) Use the procdure found in (b) to help in calculating all the solutions of the equation NiMvLSUkSW50RzYkLSUlc3FydEc2IywmIiIiRistJSRjb3NHNiMqJCUidEciIiNGKy9GMDsiIiElInhHLCYiIilGKyokRjVGMSEiIg== correct to about 10 digits. ____________________________________________________________________;

By solving an appropriate initial value problem numerically, construct a procedure which evaluates the function NiMvLSUiZ0c2IyUieEcsJiomIiIiRioiIiMhIiJGKiomRipGKi0lJXNxcnRHNiMqJkYrRiolI3BpR0YqRixGKg== NiMtJSRJbnRHNiQtJSRleHBHNiMsJComJSJ0RyIiI0YsISIiRi0vRis7IiIhJSJ4Rw==, for NiMxLCQiIiYhIiIlInhHNiMxJSFHIiIm, to an accuracy of about 15 digits.Plot the graph of g over the interval NiMxLCQiIiYhIiIlInhHNiMxJSFHIiIm.What is the connection between the function g and probabilty theory?____________________________________________________________________;

This question is concerned with the differential equation NiMvKiYlI2R5RyIiIiUjZHhHISIiLCgqJCUieUciIiNGJi0lJHNpbkc2IyUieEdGKCokLSUkY29zR0YvRixGKA==.(a) Use DEplot top plot the direction field for NiMxIiIhJSJ4Rw==NiMxJSFHIiM1 and NiMxLCQiIiMhIiIlInlHNiMxJSFHIiIj, together with the two solutions which satisfy the initial conditions NiMvLSUieUc2IyIiISIiIg== and NiMvLSUieUc2IyIiISwkIiIiISIi.(b) DEplot does not perform accurate enough calculations to construct a correct picture if we change the interval for x to NiMxIiIhJSJ4Rw==NiMxJSFHIiM/. Construct numerical solutions using desolveRK with the initial conditions NiMvLSUieUc2IyIiISIiIg== and NiMvLSUieUc2IyIiISwkIiIiISIi over the interval from NiMvJSJ4RyIiIQ== to NiMvJSJ4RyIjPw==, and plot their graphs in the same picture along with the direction field.____________________________________________________________________;

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fn := x -> 1/(x+exp(x)): p1 := plot(fn(x),x=0..2.5): p2 := plots[polygonplot]([[0,0],op(op(1,op(1,plot(fn(x),x=0..0.9)))), [.9,0]],style=patchnogrid,color=COLOR(RGB,.9,.9,.9)): p3 := plot([[.9,0],[.9,fn(.9)]],color=black): t1 := plots[textplot]([[.9,-.05,`x`],[.45,.3,`f(x)`]],color=black): plots[display]([p1,p2,p3,t1],labels=[``,``],tickmarks=[2,2]);fn := x -> ln(arctan(x)+1): p1 := plot(fn(x),x=0..4): p2 := plots[polygonplot]([[0,0],op(op(1,op(1,plot(fn(x),x=0..1.6)))), [1.6,0]],style=patchnogrid,color=COLOR(RGB,.9,.9,.9)): p3 := plot([[1.6,0],[1.6,fn(1.6)]],color=black): t1 := plots[textplot]([[1.6,-.05,`x`],[.94,.3,`f(x)`]],color=black): t2 := plots[textplot]([[3.5,.75,`y = ln(arctan(x)+1)`]],color=red): plots[display]([p1,p2,p3,t1,t2],labels=[``,``],tickmarks=[2,2]);fn := x -> sqrt(1+cos(x^2)): p1 := plot(fn(x),x=0..1.772): p2 := plots[polygonplot]([[0,0],op(op(1,op(1,plot(fn(x),x=0..0.7)))), [.7,0]],style=patchnogrid,color=COLOR(RGB,.9,.9,.9)): p3 := plot([[.7,0],[.7,fn(.7)]],color=black): t1 := plots[textplot]([[.7,-.05,`x`],[.35,.75,`f(x)`]],color=black): t2 := plots[textplot]([[1.4,1.2,` y = 1+cos(x )`], [1.4,1.23,`/`]],color=red,font=[HELVETICA,10]): t3 := plots[textplot]([[1.4,1.35,` ________`]],color=red, font=[HELVETICA,10]): t4 := plots[textplot]([[1.4,1.23,` 2`],[1.38,1.18,`v`]],color=red, font=[HELVETICA,8]): plots[display]([p1,p2,p3,t1,t2,t3,t4],labels=[``,``],tickmarks=[2,2]);;