# bvode

boundary value problems for ODE using collocation method

# bvodeS

Simplified call to bvode

### Syntax

zu = bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,fsub,dfsub,gsub,dgsub,guess)

zu = bvodeS(xpoints,m,N,x_low,x_up,fsub,gsub,zeta, <optional_args>)

### Arguments

- zu
a column vector of size

`M`

. The solution of the ode evaluated on the mesh given by points. It contains`z(u(x))`

for each requested points.- xpoints
an array which gives the points for which we want to observe the solution.

- N
a scalar with integer value, number of differential equations (

`N`

<= 20).- m
a vector of size

`N`

with integer elements. It is the vector of order of each differential equation:`m(i)`

gives the order of the`i`

-th differential equation. In the following,`M`

will represent the sum of the elements of`m`

.- x_low
a scalar: left end of interval

- x_up
a scalar: right end of interval

- zeta
a vector of size

`M`

,`zeta(j)`

gives`j`

-th side condition point (boundary point). One must have`x_low<=zeta(j) <=zeta(j+1)<=x_up`

All side condition points must be mesh points in all meshes used, see description of

`ipar(11)`

and`fixpnt`

below.- ipar
an array with 11 integer elements:

`[nonlin, collpnt, subint, ntol, ndimf, ndimi, iprint, iread, iguess, rstart,nfxpnt]`

- nonlin: ipar(1)
0 if the problem is linear, 1 if the problem is nonlinear

- collpnt: ipar(2)
Gives the number of collocation points per subinterval where

`max(m(j)) <= collpnt <= 7`

.If

`ipar(2)=0`

then`collpnt`

is set to`max( max(m(j))+1, 5-max(m(j)) )`

.- subint: ipar(3)
Gives the number of subintervals in the initial mesh. If

`ipar(3) = 0`

then`bvode`

arbitrarily sets`subint = 5`

.- ntol: ipar(4)
Gives the number of solution and derivative tolerances. We require

`0 < ntol <= M`

.`ipar(4)`

must be set to the dimension of the`tol`

argument or to`0`

. In the latter case the actual value will automatically be set to`size(tol,'*')`

.- ndimf: ipar(5)
Gives the dimension of

`fspace`

(a real work array). Its value provides a constraint on`nmax`

the maximum number of subintervals.The

`ipar(5)`

value must respect the constraint`ipar(5)>=nmax*nsizef`

where`nsizef=4 + 3*M + (5+collpnt*N)*(collpnt*N+M) + (2*M-nrec)*2*M`

(`nrec`

is the number of right end boundary conditions).- ndimi: ipar(6)
Gives the dimension of

`ispace`

(an integer work array). Its value provides a constraint on`nmax`

, the maximum number of subintervals.The

`ipar(6)`

value must respect the constraint`ipar(6)>=nmax*nsizei`

where`nsizei= 3 + collpnt*N + M`

.- iprint: ipar(7)
output control, may take the following values:

- -1
for full diagnostic printout

- 0
for selected printout

- 1
for no printout

- iread: ipar(8)
- = 0
causes

`bvode`

to generate a uniform initial mesh.- = xx
Other values are not implemented yet in Scilab

- = 1
if the initial mesh is provided by the user it is defined in

`fspace`

as follows: the mesh will occupy`fspace(1), ..., fspace(n+1)`

. The user needs to supply only the interior mesh points`fspace(j) = x(j),j = 2, ..., n`

.- = 2
if the initial mesh is supplied by the user as with

`ipar(8)=1`

, and in addition no adaptive mesh selection is to be done.

- iguess: ipar(9)
- = 0
if no initial guess for the solution is provided.

- = 1
if an initial guess is provided by the user through the argument

`guess`

.- = 2
if an initial mesh and approximate solution coefficients are provided by the user in

`fspace`

(the former and new mesh are the same).- = 3
if a former mesh and approximate solution coefficients are provided by the user in

`fspace`

, and the new mesh is to be taken twice as coarse; i.e. every second point from the former mesh.- = 4
if in addition to a former initial mesh and approximate solution coefficients, a new mesh is provided in

`fspace`

as well (see description of output for further details on iguess = 2, 3 and 4).

- ireg: ipar(10)
- = 0
if the problem is regular

- = 1
if the first relaxation factor is equal to

`ireg`

, and the nonlinear iteration does not rely on past convergence (use for an extra-sensitive nonlinear problem only)- = 2
if we are to return immediately upon (a) two successive nonconvergences, or (b) after obtaining an error estimate for the first time.

- nfxpnt: ipar(11)
Gives the number of fixed points in the mesh other than

`x_low`

and`x_up`

(the dimension of`fixpnt`

).`ipar(11)`

must be set to the dimension of the`fixpnt`

argument or to`0`

. In the latter case the actual value will automatically be set to`size(fixpnt,'*')`

.

- ltol
an array of dimension

`ntol=ipar(4)`

.`ltol(j) = l`

specifies that the`j`

-th tolerance in the`tol`

array controls the error in the`l`

-th component of . It is also required that:`1 <= ltol(1) < ltol(2) < ... < ltol(ntol) <= M`

- tol
an array of dimension

`ntol=ipar(4)`

.`tol(j)`

is the error tolerance on the`ltol(j)`

-th component of . Thus, the code attempts to satisfy on each subintervalwhere is the approximate solution vector and is the exact solution (unknown).

- fixpnt
an array of dimension

`nfxpnt=ipar(11)`

. It contains the points, other than`x_low`

and`x_up`

, which are to be included in every mesh. The code requires that all side condition points other than`x_low`

and`x_up`

(see description of`zeta`

) be included as fixed points in`fixpnt`

.- fsub
an external used to evaluate the column vector

`f=`

for any`x`

such as`x_low`

<=`x`

<=`x_up`

and for any`z=z(u(x))`

(see description below).The external must have the headings:

In Fortran the calling sequence must be:

subroutine fsub(x,zu,f) double precision zu(*), f(*),x

In C the function prototype must be:

void fsub(double *x, double *zu, double *f)

And in Scilab:

function f = fsub(x,zu,parameters)

- dfsub
an external used to evaluate the Jacobian of

`f(x,z(u))`

at a point`x`

. Where`z(u(x))`

is defined as for`fsub`

and the`N`

by`M`

array`df`

should be filled by the partial derivatives of`f`

:The external must have the headings:

In Fortran the calling sequence must be:

subroutine dfsub(x,zu,df) double precision zu(*), df(*),x

In C the function prototype must be:

void dfsub(double *x, double *zu, double *df)

And in Scilab:

function df = dfsub(x,zu,parameters)

- gsub
an external used to evaluate given z=

`z = zeta(i)`

for`1<=i<=M.`

The external must have the headings:

In Fortran the calling sequence must be:

subroutine gsub(i,zu,g) double precision zu(*), g(*) integer i

In C the function prototype must be:

void gsub(int *i, double *zu, double *g)

And in Scilab:

function g = gsub(i,zu,parameters)

Note that in contrast to

`f`

in`fsub`

, here only one value per call is returned in`g`

.

- dgsub
an external used to evaluate the

`i`

-th row of the Jacobian of`g(x,u(x))`

. Where`z(u)`

is as for`fsub`

,`i`

as for`gsub`

and the`M`

-vector`dg`

should be filled with the partial derivatives of`g`

, viz, for a particular call one calculatesThe external must have the headings:

In Fortran the calling sequence must be:

subroutine dgsub(i,zu,dg) double precision zu(*), dg(*)

In C the function prototype must be

void dgsub(int *i, double *zu, double *dg)

And in Scilab

function dg = dgsub(i,zu,parameters)

- guess
An external used to evaluate the initial approximation for

`z(u(x))`

and`dmval(u(x))`

the vector of the`mj`

-th derivatives of`u(x)`

. Note that this subroutine is used only if`ipar(9) = 1`

, and then all`M`

components of`zu`

and`N`

components of`dmval`

should be computed for any`x`

such as`x_low`

<=`x`

<=`x_up`

.The external must have the headings:

In Fortran the calling sequence must be:

subroutine guess(x,zu,dmval) double precision x,z(*), dmval(*)

In C the function prototype must be

void fsub(double *x, double *zu, double *dmval)

And in Scilab

function [dmval,zu] = fsub(x,parameters)

- <optional_args>
It should be either:

any left part of the ordered sequence of values:

`guess, dfsub, dgsub, fixpnt, ndimf, ndimi, ltol, tol, ntol,nonlin, collpnt, subint, iprint, ireg, ifail`

or any sequence of

`arg_name=argvalue`

with`arg_name`

in:`guess`

,`dfsub`

,`dgsub`

,`fixpnt`

,`ndimf`

,`ndimi`

,`ltol`

,`tol`

,`ntol`

,`nonlin`

,`collpnt`

,`subint`

,`iprint`

,`ireg`

,`ifail`

where all these arguments excepted

`ifail`

are described above.`ifail`

can be used to display the bvode call corresponding to the selected optional arguments. If`guess`

is given`iguess`

is set to 1

### Description

These functions solve a multi-point boundary value problem for a mixed order system of ode-s given by

where

The argument `zu`

used by the external functions
and returned by `bvode`

is the column vector formed by
the components of `z(u(x))`

for a given `x`

.

The method used to approximate the solution `u`

is collocation at
gaussian points, requiring `m(i)-1`

continuous derivatives in the `i`

-th
component, `i = 1:N`

. here, `k`

is the number of collocation points (stages)
per subinterval and is chosen such that *k ≥ max(m(i))*.
A runge-kutta-monomial solution representation is utilized.

### Examples

The first two problems below are taken from the paper [1] of the Bibliography.

**The problem 1**describes a uniformly loaded beam of variable stiffness, simply supported at both end.It may be defined as follow :

Solve the fourth order differential equation:

Subjected to the boundary conditions:

The exact solution of this problem is known to be:

N = 1; // just one differential equation m = 4; // a fourth order differential equation M = sum(m); x_low = 1; x_up = 2; // the x limits zeta = [x_low,x_low,x_up,x_up]; // two constraints (on the value of u and its second derivative) on each bound. // The external functions // These functions are called by the solver with zu=[u(x);u'(x);u''(x);u'''(x)] // - The function which computes the right hand side of the differential equation function f=fsub(x, zu) f = (1-6*x^2*zu(4)-6*x*zu(3))/x^3 endfunction // - The function which computes the derivative of fsub with respect to zu function df=dfsub(x, zu) df = [0,0,-6/x^2,-6/x] endfunction // - The function which computes the ith constraint for a given i function g=gsub(i, zu), select i case 1 then // x = zeta(1) = 1 g = zu(1) // u(1) = 0 case 2 then // x = zeta(2) = 1 g = zu(3) // u''(1) = 0 case 3 then // x = zeta(3) = 2 g = zu(1) // u(2) = 0 case 4 then // x = zeta(4) = 2 g = zu(3) // u''(2) = 0 end endfunction // - The function which computes the derivative of gsub with respect to z function dg=dgsub(i, z) select i case 1 then // x = zeta(1) = 1 dg = [1,0,0,0] case 2 then // x = zeta(2) = 1 dg = [0,0,1,0] case 3 then // x = zeta(3) = 2 dg = [1,0,0,0] case 4 then // x = zeta(4) = 2 dg = [0,0,1,0] end endfunction // - The function which computes the initial guess, unused here function [zu, mpar]=guess(x) zu = 0; mpar = 0; endfunction // define the function which computes the exact value of u for a given x ( for testing purposes) function zu=trusol(x) zu = 0*ones(4,1) zu(1) = 0.25*(10*log(2)-3)*(1-x) + 0.5 *( 1/x + (3+x)*log(x) - x) zu(2) = -0.25*(10*log(2)-3) + 0.5 *(-1/x^2 + (3+x)/x + log(x) - 1) zu(3) = 0.5*( 2/x^3 + 1/x - 3/x^2) zu(4) = 0.5*(-6/x^4 - 1/x/x + 6/x^3) endfunction fixpnt = []; // All boundary conditions are located at x_low and x_up // nonlin collpnt n ntol ndimf ndimi iprint iread iguess rstart nfxpnt ipar = [0 0 1 2 2000 200 1 0 0 0 0 ] ltol = [1,3]; // set tolerance control on zu(1) and zu(3) tol = [1.e-11,1.e-11]; // set tolerance values for these two controls xpoints = x_low:0.01:x_up; zu = bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,... fsub,dfsub,gsub,dgsub,guess) // check the constraints zu([1,3],[1 $]) // should be zero plot(xpoints,zu(1,:)) // the evolution of the solution u zu1 = []; for x = xpoints zu1 = [zu1,trusol(x)]; end norm(zu-zu1)

Same problem using

`bvodeS`

and an initial guess.**The problem 2**describes the small finite deformation of a thin shallow spherical cap of constant thickness subject to a quadratically varying axisymmetric external pressure distribution. Here*φ*is the meridian angle change of the deformed shell and*ψ*is a stress function. For*ε = μ = 10*, two different solutions may found depending on the starting point^{-3}Subject to the boundary conditions

for

`x=0`

and`x=1`

N = 2; // two differential equations m = [2 2]; // each differential equation is of second order M = sum(m); x_low = 0;x_up=1; // the x limits zeta = [x_low,x_low, x_up x_up]; // two constraints on each bound. // The external functions // These functions are called by the solver with zu=[u1(x);u1'(x);u2(x);u2'(x)] // - The function which computes the right hand side of the differential equation function f=fsub2(x, zu, eps, dmu, eps4mu, gam, xt), f = [zu(1)/x^2-zu(2)/x+(zu(1)-zu(3)*(1-zu(1)/x)-gam*x*(1-x^2/2))/eps4mu //phi'' zu(3)/x^2-zu(4)/x+zu(1)*(1-zu(1)/(2*x))/dmu];//psi'' endfunction // - The function which computes the derivative of fsub with respect to zu function df=dfsub2(x, zu, eps, dmu, eps4mu, gam, xt), df = [1/x^2+(1+zu(3)/x)/eps4mu, -1/x, -(1-zu(1)/x)/eps4mu, 0 (1-zu(1)/x)/dmu 0 1/x^2 -1/x]; endfunction // - The function which computes the ith constraint for a given i function g=gsub2(i, zu), select i case 1 then // x = zeta(1) = 0 g = zu(1) // u(0) = 0 case 2 then // x = zeta(2) = 0 g = -0.3*zu(3) // x*psi'-0.3*psi+0.7x = 0 case 3 then // x = zeta(3) = 1 g = zu(1) // u(1) = 0 case 4 then // x = zeta(4) = 1 g = 1*zu(4)-0.3*zu(3)+0.7*1 // x*psi'-0.3*psi+0.7x = 0 end endfunction // - The function which computes the derivative of gsub with respect to z function dg=dgsub2(i, z) select i case 1 then // x = zeta(1) = 1 dg = [1,0,0,0] case 2 then // x = zeta(2) = 1 dg = [0,0,-0.3,0] case 3 then // x = zeta(3) = 2 dg = [1,0,0,0] case 4 then // x = zeta(4) = 2 dg = [0,0,-0.3,1] end endfunction [gam, eps, dmu] = (1.1, 1d-3, 1d-3); eps4mu = eps^4/dmu xt = sqrt(2*(gam-1)/gam) fixpnt = []; // All boundary conditions are located at x_low and x_up collpnt = 4; nsizef = 4 + 3*M + (5+collpnt*N)*(collpnt*N+M)+(2*M-2)*2*M ; nsizei = 3 + collpnt*N+M; nmax = 200; // nonlin collpnt n ntol ndimf ndimi iprint iread iguess rstart nfxpnt ipar = [1 collpnt 10 4 nmax*nsizef nmax*nsizei -1 0 0 0 0 ]; ltol = 1:4; // set tolerance control on zu(1), zu(2), zu(3) and zu(4) tol = [1.e-5,1.e-5,1.e-5,1.e-5]; // set tolreance values for these four controls xpoints = x_low:0.01:x_up; // - The function which computes the initial guess, unused here function [zu, dmval]=guess2(x, gam), cons = gam*x*(1-x^2/2) dcons = gam*(1-3*x^2/2) d2cons = -3*gam*x dmval = zeros(2,1) if x>xt then zu = [0 0 -cons -dcons] dmval(2) = -d2cons else zu = [2*x;2; -2*x+cons; -2*dcons] dmval(2) = d2cons end endfunction zu = bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,... fsub2,dfsub2,gsub2,dgsub2,guess2); scf(1); clf(); plot(xpoints,zu([1 3],:)) // the evolution of the solution phi and psi // using an initial guess ipar(9) = 1; // iguess zu2 = bvode(xpoints,N,m,x_low,x_up,zeta,ipar,ltol,tol,fixpnt,... fsub2,dfsub2,gsub2,dgsub2,guess2); scf(2); clf(); plot(xpoints,zu2([1 3],:)) // the evolution of the solution phi and psi

**An eigenvalue problem:**// y''(x)=-la*y(x) // BV: y(0)=y'(0); y(1)=0 // Eigenfunctions and eigenvalues are y(x,n)=sin(s(n)*(1-x)), la(n)=s(n)^2, // where s(n) are the zeros of f(s,n)=s+atan(s)-(n+1)*pi, n=0,1,2,... // To get a third boundary condition, we choose y(0)=1 // (With y(x) also c*y(x) is a solution for each constant c.) // We solve the following ode system: // y''=-la*y // la'=0 // BV: y(0)=y'(0), y(0)=1; y(1)=0 // z=[y(x) ; y'(x) ; la] function rhs=fsub(x, z) rhs = [-z(3)*z(1);0] endfunction function g=gsub(i, z) g = [z(1)-z(2) z(1)-1 z(1)] g = g(i) endfunction // The following start function is good for the first 8 eigenfunctions. function [z, lhs]=ystart(x, z, la0) z = [1;0;la0] lhs = [0;0] endfunction [a, b, m, n] =(0, 1, [2;1], 2); zeta = [a a b]; N = 101; x = linspace(a,b,N)'; // We have s(n)-(n+1/2)*pi -> 0 for n to infinity. la0 = evstr(x_dialog('n-th eigenvalue: n= ?','10')); la0 = (%pi/2+la0*%pi)^2; z = bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0)); // The same call without any display z = bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0),iprint=1); // The same with a lot of display z = bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,la0),iprint=-1); clf() plot(x,[z(1,:)' z(2,:)']) xtitle(['Startvalue = '+string(la0);'Eigenvalue = '+string(z(3,1))],'x',' ') legend(['y(x)';'y''(x)']);

**A boundary value problem with more than one solution.**// DE: y''(x) = -exp(y(x)) // BV: y(0)=0; y(1)=0 // This boundary value problem has more than one solution. // It is demonstrated how to find two of them with the help of // some preinformation of the solutions y(x) to build the function ystart. // z = [y(x);y'(x)] [a, b, m, n] = (0, 1, 2, 1); zeta = [a b]; N = 101; tol = 1e-8*[1 1]; x = linspace(a,b,N); function rhs=fsub(x, z) rhs = -exp(z(1)) endfunction function g=gsub(i, z) g = [z(1) z(1)] g = g(i) endfunction function [z, lhs]=ystart(x, z, M) // z = [4*x*(1-x)*M ; 4*(1-2*x)*M] z = [M;0] //lhs = [-exp(4*x*(1-x)*M)] lhs = 0 endfunction for M = [1 4] if M==1 z = bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol); else z1 = bvodeS(x,m,n,a,b,fsub,gsub,zeta,ystart=list(ystart,M),tol=tol); end end // Integrating the ode yield e.g. the two solutions yex and yex1. function y=f(c) y = c.*(1-tanh(sqrt(c)/4).^2)-2; endfunction c = fsolve(2,f); function y=yex(x, c) y = log(c/2*(1-tanh(sqrt(c)*(1/4-x/2)).^2)) endfunction function y=f1(c1), y=2*c1^2+tanh(1/4/c1)^2-1; endfunction c1 = fsolve(0.1,f1); function y=yex1(x, c1) y = log((1-tanh((2*x-1)/4/c1).^2)/2/c1/c1) endfunction disp('norm(yex(x)-z(1,:))= ', norm(z(1,:)-yex(x))) disp('norm(yex1(x)-z1(1,:))= ', norm(z1(1,:)-yex1(x))) clf(); subplot(2,1,1) plot2d(x,z(1,:),style=[5]) xtitle('Two different solutions','x',' ') subplot(2,1,2) plot2d(x,z1(1,:),style=[5]) xtitle(' ','x',' ')

**A multi-point boundary value problem.**// DE y'''(x)=1 // z = [y(x);y'(x);y''(x)] // BV: y(-1)=2 y(1)=2 // Side condition: y(0)=1 a = -1; b = 1; c = 0; // The side condition point c must be included in the array fixpnt. n = 1; m = 3; function rhs=fsub(x, z) rhs = 1 endfunction function g=gsub(i, z) g = [z(1)-2 z(1)-1 z(1)-2] g = g(i) endfunction N = 10; zeta = [a c b]; x = linspace(a,b,N); z = bvodeS(x,m,n,a,b,fsub,gsub,zeta,fixpnt=c); function y=yex(x) y = x.^3/6 + x.^2 - x/6 + 1 endfunction disp(norm(yex(x)-z(1,:)),'norm(yex(x)-z(1,:))= ')

**Quantum Neumann equation, with 2 "eigenvalues" (c_1 and c2). Continuation being used.**// Quantum Neumann equation, with 2 "eigenvalues" c_1 and c_2 // (c_1=v-c_2-c_3, v is a parameter, used in continuation) // // diff(f,x,2) + (1/2)*(1/x + 1/(x-1) + 1/(x-y))*diff(f,x) // - (c_1/x + c_2/(x-1) + c_3/(x-y))* f(x) = 0 // diff(c_2,x)=0, diff(c_3,x) = 0 // // and 4 "boundary" conditions: diff(f,x)(a_k)=2*c_k*f(a_k) for // k=1,2,3, a_k=(0, 1 , y) and normalization f(1) = 1 // // The z-vector is z_1=f, z_2=diff(f,x), z_3=c_2 and z_4=c_3 // The guess is chosen to have one node in [0,1], f(x)=2*x-1 // such that f(1)=1, c_2 and c_3 are chosen to cancel poles in // the differential equation at 1.0 and y, z_3=1, z_4=1/(2*y-1) // Ref: http://arxiv.org/pdf/hep-th/0407005 y = 1.9d0; eigens = zeros(3,40); // To store the results // General setup for bvode // Number of differential equations ncomp = 3; // Orders of equations m = [2, 1, 1]; // Non-linear problem ipar(1) = 1; // Number of collocation points ipar(2) = 3; // Initial uniform mesh of 4 subintervals ipar(3) = 4; ipar(8) = 0; // Size of fspace, ispace, see colnew.f to choose size ipar(5) = 30000; ipar(6) = 2000; // Medium output ipar(7) = 0; // Initial approx is provided ipar(9) = 1; // fixpnt is an array containing all fixed points in the mesh, in // particular "boundary" points, except aleft and aright, ipar[11] its // size, here only one interior "boundary point" ipar(11) = 1; fixpnt = [1.0d0]; // Tolerances on all components z_1, z_2, z_3, z_4 ipar(4) = 4; // Tolerance check on f and diff(f,x) and on c_2 and c_3 ltol = [1, 2, 3, 4]; tol = [1d-5, 1d-5, 1d-5, 1d-5]; // Define the differential equations function [f]=fsub(x, z) f = [ -.5*(1/x+1/(x-1)+1/(x-y))*z(2) +... z(1) * ((v-z(3)-z(4))/x + z(3)/(x-1) + z(4)/(x-y)),... 0,0]; endfunction function [df]=dfsub(x, z) df = [(v-z(3)-z(4))/x + z(3)/(x-1) + z(4)/(x-y),... -.5*(1/x+1/(x-1)+1/(x-y)),z(1)/(x*(x-1)),z(1)*y/(x*(x-y));... 0,0,0,0;0,0,0,0]; endfunction // Boundary conditions function [g]=gsub(i, z) select i case 1, g = z(2) - 2*z(1)*(v-z(3)-z(4)) case 2, g = z(2) - 2*z(1)*z(3) case 3, g = z(1)-1. case 4, g = z(2) - 2*z(1)*z(4) end endfunction function [dg]=dgsub(i, z) select i case 1, dg = [-2*(v-z(3)-z(4)),1.,2*z(1),2*z(1)] case 2, dg = [-2*z(3),1.,-2*z(1),0] case 3, dg = [1,0,0,0] case 4, dg = [-2*z(4),1.,0,-2*z(1)] end endfunction // Start computation // Locations of side conditions, sorted zeta = [0.0d0, 1.0d0, 1.0d0, y]; // Interval ends aleft = 0.0d0; aright = y; // Array of 40 values of v explored by continuation, and array of 202 // points where to evaluate function f. valv = [linspace(0,.9,10) logspace(0,2,30)]; res = [linspace(0,.99,100) linspace(1,y,101)]; // eigenstates are characterized by number of nodes in [0,1] and in // [1,y], here guess selects one node (zero) in [0,1] with linear // f(x)=2*x-1 and constant c_2, c_3, so dmval=0. Notice that the z-vector // has mstar = 4 components, while dmval has ncomp = 3 components. function [z, dmval]=guess(x) z = [2*x-1, 2., 1., 1/(2*y-1)] dmval = [0,0,0] endfunction // First execution has ipar(9)=1 and uses the guess // Subsequent executions have ipar(9)=3 and use continuation. This is // run in tight closed loop to not disturb the stack for i = 1:40 v = valv(i); sol = bvode(res,ncomp,m,aleft,aright,zeta,ipar,ltol,tol,fixpnt,... fsub,dfsub,gsub,dgsub,guess); eigens(:,i) = [v;sol(3,101);sol(4,101)]; // c_2 and c_3 are constant! ipar(9) = 3; end // To see the evolution of the eigenvalues with v, disp(eigens) // Note they evolve smoothly. // To see the solution f for v=40, disp(sol(1,:)). Note that it vanishes // exactly once in [0,1] at x close to 0.98, and becomes very small // when x -> 0 and very large when x -> y. // This is markedly different from the case at small v. // The continuation procedure allows to explore these exponential behaviours // without skipping to other eigenstates.

### Used Functions

This function is based on the Fortran routine
`colnew`

developed by

U. Ascher, Department of Computer Science, University of British Columbia, Vancouver, B.C. V6T 1W5, Canada

G. Bader, institut f. Angewandte mathematik university of Heidelberg; im Neuenheimer feld 294d-6900 Heidelberg 1

### Bibliography

U. Ascher, J. Christiansen and R.D. Russell, collocation software for boundary-value ODEs, acm trans. math software 7 (1981), 209-222. this paper contains EXAMPLES where use of the code is demonstrated.

G. Bader and U. Ascher, a new basis implementation for a mixed order boundary value ode solver, siam j. scient. stat. comput. (1987).

U. Ascher, J. Christiansen and R.D. Russell, a collocation solver for mixed order systems of boundary value problems, math. comp. 33 (1979), 659-679.

U. Ascher, J. Christiansen and R.D. russell, colsys - a collocation code for boundary value problems, lecture notes comp.sc. 76, springer verlag, b. children et. al. (eds.) (1979), 164-185.

C. Deboor and R. Weiss, solveblok: a package for solving almost block diagonal linear systems, acm trans. math. software 6 (1980), 80-87.

### See also

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