# NAG CL Interfaced02tlc (bvp_​coll_​nlin_​solve)

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## 1Purpose

d02tlc solves a general two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations.

## 2Specification

 #include
void  d02tlc (
 void (*ffun)(double x, const double y[], Integer neq, const Integer m[], double f[], Nag_Comm *comm),
 void (*fjac)(double x, const double y[], Integer neq, const Integer m[], double dfdy[], Nag_Comm *comm),
 void (*gafun)(const double ya[], Integer neq, const Integer m[], Integer nlbc, double ga[], Nag_Comm *comm),
 void (*gbfun)(const double yb[], Integer neq, const Integer m[], Integer nrbc, double gb[], Nag_Comm *comm),
 void (*gajac)(const double ya[], Integer neq, const Integer m[], Integer nlbc, double dgady[], Nag_Comm *comm),
 void (*gbjac)(const double yb[], Integer neq, const Integer m[], Integer nrbc, double dgbdy[], Nag_Comm *comm),
 void (*guess)(double x, Integer neq, const Integer m[], double y[], double dym[], Nag_Comm *comm),
double rcomm[], Integer icomm[], Nag_Comm *comm, NagError *fail)
The function may be called by the names: d02tlc or nag_ode_bvp_coll_nlin_solve.

## 3Description

d02tlc and its associated functions (d02tvc, d02txc, d02tyc and d02tzc) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations
 $y1(m1) (x) = f1 (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1)) y2(m2) (x) = f2 (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1)) ⋮ yn(mn) (x) = fn (x,y1,y1(1),…,y1(m1-1),y2,…,yn(mn-1))$
over an interval $\left[a,b\right]$ subject to $p$ ($\text{}>0$) nonlinear boundary conditions at $a$ and $q$ ($\text{}>0$) nonlinear boundary conditions at $b$, where $p+q=\sum _{i=1}^{n}{m}_{i}$. Note that ${y}_{i}^{\left(m\right)}\left(x\right)$ is the $m$th derivative of the $i$th solution component. Hence ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$. The left boundary conditions at $a$ are defined as
 $gi(z(y(a)))=0, i=1,2,…,p,$
and the right boundary conditions at $b$ as
 $g¯j(z(y(b)))=0, j=1,2,…,q,$
where $y=\left({y}_{1},{y}_{2},\dots ,{y}_{n}\right)$ and
 $z(y(x)) = (y1(x), y1(1) (x) ,…, y1(m1-1) (x) ,y2(x),…, yn(mn-1) (x) ) .$
First, d02tvc must be called to specify the initial mesh, error requirements and other details. Note that the error requirements apply only to the solution components ${y}_{1},{y}_{2},\dots ,{y}_{n}$ and that no error control is applied to derivatives of solution components. (If error control is required on derivatives then the system must be reduced in order by introducing the derivatives whose error is to be controlled as new variables. See Section 9 in d02tvc.) Then, d02tlc can be used to solve the boundary value problem. After successful computation, d02tzc can be used to ascertain details about the final mesh and other details of the solution procedure, and d02tyc can be used to compute the approximate solution anywhere on the interval $\left[a,b\right]$.
A description of the numerical technique used in d02tlc is given in Section 3 in d02tvc.
d02tlc can also be used in the solution of a series of problems, for example in performing continuation, when the mesh used to compute the solution of one problem is to be used as the initial mesh for the solution of the next related problem. d02txc should be used in between calls to d02tlc in this context.
See Section 9 in d02tvc for details of how to solve boundary value problems of a more general nature.
The functions are based on modified versions of the codes COLSYS and COLNEW (see Ascher et al. (1979) and Ascher and Bader (1987)). A comprehensive treatment of the numerical solution of boundary value problems can be found in Ascher et al. (1988) and Keller (1992).

## 4References

Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput. 8 483–500
Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput. 33 659–679
Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York

## 5Arguments

1: $\mathbf{ffun}$function, supplied by the user External Function
ffun must evaluate the functions ${f}_{i}$ for given values $x,z\left(y\left(x\right)\right)$.
The specification of ffun is:
 void ffun (double x, const double y[], Integer neq, const Integer m[], double f[], Nag_Comm *comm)
1: $\mathbf{x}$double Input
On entry: $x$, the independent variable.
2: $\mathbf{y}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array y is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{Y}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{y}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
4: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
5: $\mathbf{f}\left[\mathit{dim}\right]$double Output
On exit: ${\mathbf{f}}\left[\mathit{i}-1\right]$ must contain ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to ffun.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by ffun when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: ffun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
2: $\mathbf{fjac}$function, supplied by the user External Function
fjac must evaluate the partial derivatives of ${f}_{i}$ with respect to the elements of
$z\left(y\left(x\right)\right)=\left({y}_{1}\left(x\right),{y}_{1}^{1}\left(x\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(x\right),{y}_{2}\left(x\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(x\right)\right)$.
The specification of fjac is:
 void fjac (double x, const double y[], Integer neq, const Integer m[], double dfdy[], Nag_Comm *comm)
1: $\mathbf{x}$double Input
On entry: $x$, the independent variable.
2: $\mathbf{y}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array y is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{Y}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{y}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
3: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
4: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
5: $\mathbf{dfdy}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array dfdy is ${\mathbf{neq}}×{\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{DFDY}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{dfdy}}\left[k×{\mathbf{neq}}×{\mathbf{neq}}+\left(j-1\right)×{\mathbf{neq}}+i-1\right]$.
On entry: set to zero.
On exit: ${\mathbf{DFDY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${f}_{\mathit{i}}$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$, $\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{m}}\left[\mathit{j}-1\right]-1$. Only nonzero partial derivatives need be set.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to fjac.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by fjac when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: fjac should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
3: $\mathbf{gafun}$function, supplied by the user External Function
gafun must evaluate the boundary conditions at the left-hand end of the range, that is functions ${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ for given values of $z\left(y\left(a\right)\right)$.
The specification of gafun is:
 void gafun (const double ya[], Integer neq, const Integer m[], Integer nlbc, double ga[], Nag_Comm *comm)
1: $\mathbf{ya}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array ya is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{YA}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{ya}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{YA}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
3: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4: $\mathbf{nlbc}$Integer Input
On entry: the number of boundary conditions at $a$.
5: $\mathbf{ga}\left[\mathit{dim}\right]$double Output
On exit: ${\mathbf{ga}}\left[\mathit{i}-1\right]$ must contain ${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nlbc}}$.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to gafun.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by gafun when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: gafun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
4: $\mathbf{gbfun}$function, supplied by the user External Function
gbfun must evaluate the boundary conditions at the right-hand end of the range, that is functions ${\overline{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ for given values of $z\left(y\left(b\right)\right)$.
The specification of gbfun is:
 void gbfun (const double yb[], Integer neq, const Integer m[], Integer nrbc, double gb[], Nag_Comm *comm)
1: $\mathbf{yb}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array yb is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{YB}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{yb}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{YB}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
3: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4: $\mathbf{nrbc}$Integer Input
On entry: the number of boundary conditions at $b$.
5: $\mathbf{gb}\left[\mathit{dim}\right]$double Output
On exit: ${\mathbf{gb}}\left[\mathit{i}-1\right]$ must contain ${\overline{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nrbc}}$.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to gbfun.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by gbfun when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: gbfun should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
5: $\mathbf{gajac}$function, supplied by the user External Function
gajac must evaluate the partial derivatives of ${g}_{i}\left(z\left(y\left(a\right)\right)\right)$ with respect to the elements of $z\left(y\left(a\right)\right)=\left({y}_{1}\left(a\right),{y}_{1}^{1}\left(a\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(a\right),{y}_{2}\left(a\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(a\right)\right)$.
The specification of gajac is:
 void gajac (const double ya[], Integer neq, const Integer m[], Integer nlbc, double dgady[], Nag_Comm *comm)
1: $\mathbf{ya}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array ya is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{YA}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{ya}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{YA}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(a\right)={y}_{i}\left(a\right)$.
2: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
3: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4: $\mathbf{nlbc}$Integer Input
On entry: the number of boundary conditions at $a$.
5: $\mathbf{dgady}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array dgady is ${\mathbf{nlbc}}×{\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{DGADY}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{dgady}}\left[k×{\mathbf{nlbc}}×{\mathbf{neq}}+\left(j-1\right)×{\mathbf{nlbc}}+i-1\right]$.
On entry: set to zero.
On exit: ${\mathbf{DGADY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${g}_{\mathit{i}}\left(z\left(y\left(a\right)\right)\right)$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(a\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nlbc}}$, $\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{m}}\left[\mathit{j}-1\right]-1$. Only nonzero partial derivatives need be set.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to gajac.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by gajac when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: gajac should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
6: $\mathbf{gbjac}$function, supplied by the user External Function
gbjac must evaluate the partial derivatives of ${\overline{g}}_{i}\left(z\left(y\left(b\right)\right)\right)$ with respect to the elements of $z\left(y\left(b\right)\right)=\left({y}_{1}\left(b\right),{y}_{1}^{1}\left(b\right),\dots ,{y}_{1}^{\left({m}_{1}-1\right)}\left(b\right),{y}_{2}\left(b\right),\dots ,{y}_{n}^{\left({m}_{n}-1\right)}\left(b\right)\right)$.
The specification of gbjac is:
 void gbjac (const double yb[], Integer neq, const Integer m[], Integer nrbc, double dgbdy[], Nag_Comm *comm)
1: $\mathbf{yb}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array yb is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{YB}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{yb}}\left[j×{\mathbf{neq}}+i-1\right]$.
On entry: ${\mathbf{YB}}\left(\mathit{i},\mathit{j}\right)$ contains ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(b\right)={y}_{i}\left(b\right)$.
2: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
3: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4: $\mathbf{nrbc}$Integer Input
On entry: the number of boundary conditions at $b$.
5: $\mathbf{dgbdy}\left[\mathit{dim}\right]$double Input/Output
Note: the dimension, dim, of the array dgbdy is ${\mathbf{nrbc}}×{\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{DGBDY}}\left(i,j,k\right)$ appears in this document, it refers to the array element ${\mathbf{dgbdy}}\left[\left(k-1\right)×{\mathbf{nrbc}}×{\mathbf{neq}}+\left(j-1\right)×{\mathbf{nrbc}}+i-1\right]$.
On entry: set to zero.
On exit: ${\mathbf{DGBDY}}\left(\mathit{i},\mathit{j},\mathit{k}\right)$ must contain the partial derivative of ${\overline{g}}_{\mathit{i}}\left(z\left(y\left(b\right)\right)\right)$ with respect to ${y}_{\mathit{j}}^{\left(\mathit{k}\right)}\left(b\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nrbc}}$, $\mathit{j}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{k}=0,1,\dots ,{\mathbf{m}}\left[\mathit{j}-1\right]-1$. Only nonzero partial derivatives need be set.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to gbjac.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by gbjac when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: gbjac should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
7: $\mathbf{guess}$function, supplied by the user External Function
guess must return initial approximations for the solution components ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}$ and the derivatives ${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$. Try to compute each derivative ${y}_{i}^{\left({m}_{i}\right)}$ such that it corresponds to your approximations to ${y}_{i}^{\left(\mathit{j}\right)}$, for $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[i-1\right]-1$. You should not call ffun to compute ${y}_{i}^{\left({m}_{i}\right)}$.
If d02tlc is being used in conjunction with d02txc as part of a continuation process, guess is not called by d02tlc after the call to d02txc.
The specification of guess is:
 void guess (double x, Integer neq, const Integer m[], double y[], double dym[], Nag_Comm *comm)
1: $\mathbf{x}$double Input
On entry: $x$, the independent variable; $x\in \left[a,b\right]$.
2: $\mathbf{neq}$Integer Input
On entry: the number of differential equations.
3: $\mathbf{m}\left[\mathit{dim}\right]$const Integer Input
On entry: ${\mathbf{m}}\left[\mathit{i}-1\right]$ contains ${m}_{\mathit{i}}$, the order of the $\mathit{i}$th differential equation, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
4: $\mathbf{y}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array y is ${\mathbf{neq}}×\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}}\left[i\right]\right)$.
where ${\mathbf{Y}}\left(i,j\right)$ appears in this document, it refers to the array element ${\mathbf{y}}\left[j×{\mathbf{neq}}+i-1\right]$.
On exit: ${\mathbf{Y}}\left(\mathit{i},\mathit{j}\right)$ must contain ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{\mathbf{m}}\left[\mathit{i}-1\right]-1$.
Note:  ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$.
5: $\mathbf{dym}\left[\mathit{dim}\right]$double Output
On exit: ${\mathbf{dym}}\left[\mathit{i}-1\right]$ must contain ${y}_{\mathit{i}}^{\left({m}_{\mathit{i}}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
6: $\mathbf{comm}$Nag_Comm *
Pointer to structure of type Nag_Comm; the following members are relevant to guess.
userdouble *
iuserInteger *
pPointer
The type Pointer will be void *. Before calling d02tlc you may allocate memory and initialize these pointers with various quantities for use by guess when called from d02tlc (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
Note: guess should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tlc. If your code inadvertently does return any NaNs or infinities, d02tlc is likely to produce unexpected results.
8: $\mathbf{rcomm}\left[\mathit{dim}\right]$double Communication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument rcomm in the previous call to d02tvc.
On entry: this must be the same array as supplied to d02tvc and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated functions.
9: $\mathbf{icomm}\left[\mathit{dim}\right]$Integer Communication Array
Note: the dimension, $\mathit{dim}$, of this array is dictated by the requirements of associated functions that must have been previously called. This array MUST be the same array passed as argument icomm in the previous call to d02tvc.
On entry: this must be the same array as supplied to d02tvc and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated functions.
10: $\mathbf{comm}$Nag_Comm *
The NAG communication argument (see Section 3.1.1 in the Introduction to the NAG Library CL Interface).
11: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONVERGENCE_SOL
All Newton iterations that have been attempted have failed to converge.
No results have been generated. Check the coding of the functions for calculating the Jacobians of system and boundary conditions.
Try to provide a better initial solution approximation.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MISSING_CALL
Either the setup function has not been called or the communication arrays have become corrupted. No solution will be computed.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_SING_JAC
Numerical singularity has been detected in the Jacobian used in the Newton iteration.
No results have been generated. Check the coding of the functions for calculating the Jacobians of system and boundary conditions.
NW_MAX_SUBINT
The expected number of sub-intervals required to continue the computation exceeds the maximum specified: $⟨\mathit{\text{value}}⟩$.
Results have been generated which may be useful.
Try increasing this number or relaxing the error requirements.
NW_NOT_CONVERGED
A Newton iteration has failed to converge. The computation has not succeeded but results have been returned for an intermediate mesh on which convergence was achieved.
These results should be treated with extreme caution.

## 7Accuracy

The accuracy of the solution is determined by the argument tols in the prior call to d02tvc (see Sections 3 and 9 in d02tvc for details and advice). Note that error control is applied only to solution components (variables) and not to any derivatives of the solution. An estimate of the maximum error in the computed solution is available by calling d02tzc.

## 8Parallelism and Performance

d02tlc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02tlc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

If d02tlc returns with ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_CONVERGENCE_SOL, NE_SING_JAC, NW_MAX_SUBINT or NW_NOT_CONVERGED and the call to d02tlc was a part of some continuation procedure for which successful calls to d02tlc have already been made, then it is possible that the adjustment(s) to the continuation parameter(s) between calls to d02tlc is (are) too large for the problem under consideration. More conservative adjustment(s) to the continuation parameter(s) might be appropriate.

## 10Example

The following example is used to illustrate the treatment of a high-order system, control of the error in a derivative of a component of the original system, and the use of continuation. See also d02tvc, d02txc, d02tyc and d02tzc, for the illustration of other facilities.
Consider the steady flow of an incompressible viscous fluid between two infinite coaxial rotating discs. See Ascher et al. (1979) and the references therein. The governing equations are
 $1R f′′′′ + ff′′′ + gg′ = 0 1R g′′ + fg′ - f′g = 0$
subject to the boundary conditions
 $f(0)=f′(0)= 0, g(0)=Ω0, f(1)=f′(1)= 0, g(1)=Ω1,$
where $R$ is the Reynolds number and ${\Omega }_{0},{\Omega }_{1}$ are the angular velocities of the disks.
We consider the case of counter-rotation and a symmetric solution, that is ${\Omega }_{0}=1,{\Omega }_{1}=-1$. This problem is more difficult to solve, the larger the value of $R$. For illustration, we use simple continuation to compute the solution for three different values of $R$ ($={10}^{6},{10}^{8},{10}^{10}$). However, this problem can be addressed directly for the largest value of $R$ considered here. Instead of the values suggested in Section 5 in d02txc for nmesh, ipmesh and mesh in the call to d02txc prior to a continuation call, we use every point of the final mesh for the solution of the first value of $R$, that is we must modify the contents of ipmesh. For illustrative purposes we wish to control the computed error in ${f}^{\prime }$ and so recast the equations as
 $y1′ = y2 y2′′′ = -R(y1y2′′+y3y3′) y3′′ = R(y2y3-y1y3′)$
subject to the boundary conditions
 $y1(0)=y2(0)= 0, y3(0)=Ω, y1(1)=y2(1)= 0, y3(1)=-Ω, Ω=1.$
For the symmetric boundary conditions considered, there exists an odd solution about $x=0.5$. Hence, to satisfy the boundary conditions, we use the following initial approximations to the solution in guess:
 $y1(x) = -x2(x-12) (x-1) 2 y2(x) = -x(x-1)(5⁢x2-5x+1) y3(x) = −8Ω (x-12) 3.$

### 10.1Program Text

Program Text (d02tlce.c)

### 10.2Program Data

Program Data (d02tlce.d)

### 10.3Program Results

Program Results (d02tlce.r)