NAG Library Function Document

nag_sparse_nsym_jacobi (f11dkc)

1  Purpose

2  Specification

3  Description

4  References

5  Arguments

1:     store – Nag_SparseNsym_StoreInput

On entry: specifies whether the matrix $A$ is stored using symmetric coordinate storage (SCS) (applicable only to a symmetric matrix $A$) or coordinate storage (CS) (applicable to both symmetric and nonsymmetric matrices).

$\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$

The complete matrix $A$ is stored in CS format.

$\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$

The lower triangle of the symmetric matrix $A$ is stored in SCS format.

Constraint: $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$ or $\mathrm{Nag\_SparseNsym\_StoreSCS}$.

2:     trans – Nag_TransTypeInput

On entry: if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, specifies whether the approximate solution of $Ax=b$ or of $A^{\mathrm{T}}x=b$ is required.

$\mathbf{trans}=\mathrm{Nag\_NoTrans}$

The approximate solution of $Ax=b$ is calculated.

$\mathbf{trans}=\mathrm{Nag\_Trans}$

The approximate solution of $A^{\mathrm{T}}x=b$ is calculated.

Suggested value: if the matrix $A$ is symmetric and stored in CS format, it is recommended that $\mathbf{trans}=\mathrm{Nag\_NoTrans}$ for reasons of efficiency.

Constraint: $\mathbf{trans}=\mathrm{Nag\_NoTrans}$ or $\mathrm{Nag\_Trans}$.

3:     init – Nag_InitializeAInput

On entry: on first entry, init should be set to $\mathrm{Nag\_InitializeI}$, unless the diagonal elements of $A$ are already stored in the array diag. If diag already contains the diagonal of $A$, it must be set to $\mathrm{Nag\_InputA}$.

$\mathbf{init}=\mathrm{Nag\_InputA}$

diag must contain the diagonal of $A$.

$\mathbf{init}=\mathrm{Nag\_InitializeI}$

diag will store the diagonal of $A$ on exit.

Suggested value: $\mathbf{init}=\mathrm{Nag\_InitializeI}$ on first entry; $\mathbf{init}=\mathrm{Nag\_InputA}$, subsequently, unless diag has been overwritten.

Constraint: $\mathbf{init}=\mathrm{Nag\_InputA}$ or $\mathrm{Nag\_InitializeI}$.

4:     niter – IntegerInput

On entry: the number of Jacobi iterations requested.

Constraint: $\mathbf{niter}\ge 1$.

5:     n – IntegerInput

On entry: $n$, the order of the matrix $A$.

Constraint: $\mathbf{n}\ge 1$.

6:     nnz – IntegerInput

On entry: if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, the number of nonzero elements in the matrix $A$.

If $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$, the number of nonzero elements in the lower triangle of the matrix $A$.

Constraints:

• if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, $1\le \mathbf{nnz}\le {\mathbf{n}}^2$;
• if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$, $1\le \mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

7:     a[nnz] – const doubleInput

On entry: if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, the nonzero elements in the matrix $A$ (CS format).

If $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$, the nonzero elements in the lower triangle of the matrix $A$ (SCS format).

In both cases, the elements of either $A$ or of its lower triangle must be ordered by increasing row index and by increasing column index within each row. Multiple entries for the same row and columns indices are not permitted. The function nag_sparse_nsym_sort (f11zac) or nag_sparse_sym_sort (f11zbc) may be used to reorder the elements in this way for CS and SCS storage, respectively.

8:     irow[nnz] – const IntegerInput

9:     icol[nnz] – const IntegerInput

On entry: if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, the row and column indices of the nonzero elements supplied in a.

If $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$, the row and column indices of the nonzero elements of the lower triangle of the matrix $A$ supplied in a.

Constraints:

• $1\le \mathbf{irow}\left[\mathit{i}\right]\le \mathbf{n}$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$;
• if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreCS}$, $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{n}$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$;
• if $\mathbf{store}=\mathrm{Nag\_SparseNsym\_StoreSCS}$, $1\le \mathbf{icol}\left[\mathit{i}\right]\le \mathbf{irow}\left[\mathit{i}\right]$, for $\mathit{i}=0,1,\dots ,\mathbf{nnz}-1$;
• either $\mathbf{irow}\left[\mathit{i}-1\right]<\mathbf{irow}\left[\mathit{i}\right]$ or both $\mathbf{irow}\left[\mathit{i}-1\right]=\mathbf{irow}\left[\mathit{i}\right]$ and $\mathbf{icol}\left[\mathit{i}-1\right]<\mathbf{icol}\left[\mathit{i}\right]$, for $\mathit{i}=1,2,\dots ,\mathbf{nnz}-1$.

10:   check – Nag_SparseNsym_CheckDataInput

On entry: specifies whether or not the CS or SCS representation of the matrix $A$ should be checked.

$\mathbf{check}=\mathrm{Nag\_SparseNsym\_Check}$

Checks are carried out on the values of n, nnz, irow, icol; if $\mathbf{init}=\mathrm{Nag\_InputA}$, diag is also checked.

$\mathbf{check}=\mathrm{Nag\_SparseNsym\_NoCheck}$

None of these checks are carried out.

See also Section 9.2.

Constraint: $\mathbf{check}=\mathrm{Nag\_SparseNsym\_Check}$ or $\mathrm{Nag\_SparseNsym\_NoCheck}$.

11:   b[n] – const doubleInput

On entry: the right-hand side vector $b$.

12:   x[n] – doubleOutput

On exit: the approximate solution vector $x_{\mathbf{niter}}$.

13:   diag[n] – doubleInput/Output

On entry: if $\mathbf{init}=\mathrm{Nag\_InputA}$, the diagonal elements of $A$.

On exit: if $\mathbf{init}=\mathrm{Nag\_InputA}$, unchanged on exit.

If $\mathbf{init}=\mathrm{Nag\_InitializeI}$, the diagonal elements of $A$.

14:   fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6  Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.

NE_INT

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

On entry, $\mathbf{niter}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{niter}\ge 1$.

On entry, $\mathbf{nnz}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nnz}\ge 1$.

NE_INT_2

On entry, $\mathbf{nnz}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nnz}\le \mathbf{n}\times \left(\mathbf{n}+1\right)/2$.

On entry, $\mathbf{nnz}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{nnz}\le {\mathbf{n}}^2$.

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.

NE_INVALID_CS

On entry, $\mathit{I}=⟨\mathit{\text{value}}⟩$, $\mathbf{icol}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{irow}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{icol}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{icol}\left[\mathit{I}-1\right]\le \mathbf{irow}\left[\mathit{I}-1\right]$.

On entry, $\mathit{I}=⟨\mathit{\text{value}}⟩$, $\mathbf{icol}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{icol}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{icol}\left[\mathit{I}-1\right]\le \mathbf{n}$.

On entry, $\mathit{I}=⟨\mathit{\text{value}}⟩$, $\mathbf{irow}\left[\mathit{I}-1\right]=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{irow}\left[\mathit{I}-1\right]\ge 1$ and $\mathbf{irow}\left[\mathit{I}-1\right]\le \mathbf{n}$.

NE_NOT_STRICTLY_INCREASING

On entry, $\mathbf{a}\left[i-1\right]$ is out of order: $i=⟨\mathit{\text{value}}⟩$.

On entry, the location ($\mathbf{irow}\left[\mathit{I}-1\right],\mathbf{icol}\left[\mathit{I}-1\right]$) is a duplicate: $\mathit{I}=⟨\mathit{\text{value}}⟩$.

NE_ZERO_DIAG_ELEM

On entry, the diagonal element of the $\mathit{I}$-th row is zero or missing: $\mathit{I}=⟨\mathit{\text{value}}⟩$.

On entry, the element $\mathbf{diag}\left[\mathit{I}-1\right]$ is zero: $\mathit{I}=⟨\mathit{\text{value}}⟩$.

7  Accuracy

8  Parallelism and Performance

9  Further Comments

9.1  Timing

9.2  Use of check

10  Example

10.1  Program Text

Program Text (f11dkce.c)

10.2  Program Data

Program Data (f11dkce.d)

10.3  Program Results

Program Results (f11dkce.r)