NAG CL Interface
d02tlc (bvp_​coll_​nlin_​solve)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{Y}(i,j)$ appears in this document, it refers to the array element $\mathbf{y}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{Y}(\mathit{i},\mathit{j})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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
$z\left(y\left(x\right)\right)=(y_1\left(x\right),y_1^1\left(x\right),\dots ,y_1^{(m_1-1)}\left(x\right),y_2\left(x\right),\dots ,y_n^{(m_n-1)}\left(x\right))$.

The specification of fjac is:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{Y}(i,j)$ appears in this document, it refers to the array element $\mathbf{y}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{Y}(\mathit{i},\mathit{j})$ 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 $\mathbf{neq}\times \mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{DFDY}(i,j,k)$ appears in this document, it refers to the array element $\mathbf{dfdy}\left[k\times \mathbf{neq}\times \mathbf{neq}+(j-1)\times \mathbf{neq}+i-1\right]$.

On entry: set to zero.

On exit: $\mathbf{DFDY}(\mathit{i},\mathit{j},\mathit{k})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{YA}(i,j)$ appears in this document, it refers to the array element $\mathbf{ya}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{YA}(\mathit{i},\mathit{j})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{YB}(i,j)$ appears in this document, it refers to the array element $\mathbf{yb}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{YB}(\mathit{i},\mathit{j})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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 $z\left(y\left(a\right)\right)=(y_1\left(a\right),y_1^1\left(a\right),\dots ,y_1^{(m_1-1)}\left(a\right),y_2\left(a\right),\dots ,y_n^{(m_n-1)}\left(a\right))$.

The specification of gajac is:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{YA}(i,j)$ appears in this document, it refers to the array element $\mathbf{ya}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{YA}(\mathit{i},\mathit{j})$ 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 $\mathbf{nlbc}\times \mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{DGADY}(i,j,k)$ appears in this document, it refers to the array element $\mathbf{dgady}\left[k\times \mathbf{nlbc}\times \mathbf{neq}+(j-1)\times \mathbf{nlbc}+i-1\right]$.

On entry: set to zero.

On exit: $\mathbf{DGADY}(\mathit{i},\mathit{j},\mathit{k})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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 $z\left(y\left(b\right)\right)=(y_1\left(b\right),y_1^1\left(b\right),\dots ,y_1^{(m_1-1)}\left(b\right),y_2\left(b\right),\dots ,y_n^{(m_n-1)}\left(b\right))$.

The specification of gbjac is:

copy

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{YB}(i,j)$ appears in this document, it refers to the array element $\mathbf{yb}\left[j\times \mathbf{neq}+i-1\right]$.

On entry: $\mathbf{YB}(\mathit{i},\mathit{j})$ 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 $\mathbf{nrbc}\times \mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{DGBDY}(i,j,k)$ appears in this document, it refers to the array element $\mathbf{dgbdy}\left[(k-1)\times \mathbf{nrbc}\times \mathbf{neq}+(j-1)\times \mathbf{nrbc}+i-1\right]$.

On entry: set to zero.

On exit: $\mathbf{DGBDY}(\mathit{i},\mathit{j},\mathit{k})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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:

copy

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; $x\in [a,b]$.

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 $\mathbf{neq}\times \max \left(\mathbf{m}\left[i\right]\right)$.

where $\mathbf{Y}(i,j)$ appears in this document, it refers to the array element $\mathbf{y}\left[j\times \mathbf{neq}+i-1\right]$.

On exit: $\mathbf{Y}(\mathit{i},\mathit{j})$ 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.

user – double *

iuser – Integer *

p – Pointer 

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).

6 Error 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.

NE_BAD_PARAM

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.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results