# NAG C Library Function Document

## 1Purpose

nag_sparse_nherm_jacobi (f11dxc) computes the approximate solution of a complex, Hermitian or non-Hermitian, sparse system of linear equations applying a number of Jacobi iterations. It is expected that nag_sparse_nherm_jacobi (f11dxc) will be used as a preconditioner for the iterative solution of complex sparse systems of equations.

## 2Specification

 #include #include
 void nag_sparse_nherm_jacobi (Nag_SparseNsym_Store store, Nag_TransType trans, Nag_InitializeA init, Integer niter, Integer n, Integer nnz, const Complex a[], const Integer irow[], const Integer icol[], Nag_SparseNsym_CheckData check, const Complex b[], Complex x[], Complex diag[], NagError *fail)

## 3Description

nag_sparse_nherm_jacobi (f11dxc) computes the approximate solution of the complex sparse system of linear equations $Ax=b$ using niter iterations of the Jacobi algorithm (see also Golub and Van Loan (1996) and Young (1971)):
 $xk+1=xk+D-1b-Axk$ (1)
where $k=1,2,\dots ,{\mathbf{niter}}$ and ${x}_{0}=0$.
nag_sparse_nherm_jacobi (f11dxc) can be used both for non-Hermitian and Hermitian systems of equations. For Hermitian matrices, either all nonzero elements of the matrix $A$ can be supplied using coordinate storage (CS), or only the nonzero elements of the lower triangle of $A$, using symmetric coordinate storage (SCS) (see the f11 Chapter Introduction).
It is expected that nag_sparse_nherm_jacobi (f11dxc) will be used as a preconditioner for the iterative solution of complex sparse systems of equations. This may be with either the Hermitian or non-Hermitian suites of functions.
For Hermitian systems the suite consists of:
For non-Hermitian systems the suite consists of:

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

## 5Arguments

1:    $\mathbf{store}$Nag_SparseNsym_StoreInput
On entry: specifies whether the matrix $A$ is stored using symmetric coordinate storage (SCS) (applicable only to a Hermitian matrix $A$) or coordinate storage (CS) (applicable to both Hermitian and non-Hermitian 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 Hermitian matrix $A$ is stored in SCS format.
Constraint: ${\mathbf{store}}=\mathrm{Nag_SparseNsym_StoreCS}$ or $\mathrm{Nag_SparseNsym_StoreSCS}$.
2:    $\mathbf{trans}$Nag_TransTypeInput
On entry: if ${\mathbf{store}}=\mathrm{Nag_SparseNsym_StoreCS}$, specifies whether the approximate solution of $Ax=b$ or of ${A}^{\mathrm{H}}x=b$ is required.
${\mathbf{trans}}=\mathrm{Nag_NoTrans}$
The approximate solution of $Ax=b$ is calculated.
${\mathbf{trans}}=\mathrm{Nag_Trans}$
The approximate solution of ${A}^{\mathrm{H}}x=b$ is calculated.
Suggested value: if the matrix $A$ is Hermitian 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:    $\mathbf{init}$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:    $\mathbf{niter}$IntegerInput
On entry: the number of Jacobi iterations requested.
Constraint: ${\mathbf{niter}}\ge 1$.
5:    $\mathbf{n}$IntegerInput
On entry: $n$, the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 1$.
6:    $\mathbf{nnz}$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 ${\mathbf{store}}=\mathrm{Nag_SparseNsym_StoreSCS}$, $1\le {\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$.
7:    $\mathbf{a}\left[{\mathbf{nnz}}\right]$const ComplexInput
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_nherm_sort (f11znc) or nag_sparse_herm_sort (f11zpc) may be used to reorder the elements in this way for CS and SCS storage, respectively.
8:    $\mathbf{irow}\left[{\mathbf{nnz}}\right]$const IntegerInput
9:    $\mathbf{icol}\left[{\mathbf{nnz}}\right]$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:  $\mathbf{check}$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.
Constraint: ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ or $\mathrm{Nag_SparseNsym_NoCheck}$.
11:  $\mathbf{b}\left[{\mathbf{n}}\right]$const ComplexInput
On entry: the right-hand side vector $b$.
12:  $\mathbf{x}\left[{\mathbf{n}}\right]$ComplexOutput
On exit: the approximate solution vector ${x}_{{\mathbf{niter}}}$.
13:  $\mathbf{diag}\left[{\mathbf{n}}\right]$ComplexInput/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:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

A nonzero element has been supplied which does not lie within the matrix $A$, is out of order, or has duplicate row and column indices. Consider calling either nag_sparse_nsym_sort (f11zac) or nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates when ${\mathbf{store}}=\mathrm{Nag_SparseNsym_StoreCS}$ or ${\mathbf{store}}=\mathrm{Nag_SparseNsym_StoreSCS}$, respectively.
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
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: ${\mathbf{nnz}}\le {\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
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, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icol}}\left[i-1\right]=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{icol}}\left[i-1\right]\ge 1$ and ${\mathbf{icol}}\left[i-1\right]\le {\mathbf{n}}$.
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_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
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: $\mathit{I}=〈\mathit{\text{value}}〉$.

## 7Accuracy

In general, the Jacobi method cannot be used on its own to solve systems of linear equations. The rate of convergence is bound by its spectral properties (see, for example, Golub and Van Loan (1996)) and as a solver, the Jacobi method can only be applied to a limited set of matrices. One condition that guarantees convergence is strict diagonal dominance.
However, the Jacobi method can be used successfully as a preconditioner to a wider class of systems of equations. The Jacobi method has good vector/parallel properties, hence it can be applied very efficiently. Unfortunately, it is not possible to provide criteria which define the applicability of the Jacobi method as a preconditioner, and its usefulness must be judged for each case.

## 8Parallelism and Performance

nag_sparse_nherm_jacobi (f11dxc) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
nag_sparse_nherm_jacobi (f11dxc) 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.

### 9.1Timing

The time taken for a call to nag_sparse_nherm_jacobi (f11dxc) is proportional to ${\mathbf{niter}}×{\mathbf{nnz}}$.

### 9.2Use of check

It is expected that a common use of nag_sparse_nherm_jacobi (f11dxc) will be as preconditioner for the iterative solution of complex, Hermitian or non-Hermitian, linear systems. In this situation, nag_sparse_nherm_jacobi (f11dxc) is likely to be called many times. In the interests of both reliability and efficiency, you are recommended to set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_Check}$ for the first of such calls, and to set ${\mathbf{check}}=\mathrm{Nag_SparseNsym_NoCheck}$ for all subsequent calls.

## 10Example

This example solves the complex sparse non-Hermitian system of equations $Ax=b$ iteratively using nag_sparse_nherm_jacobi (f11dxc) as a preconditioner.

### 10.1Program Text

Program Text (f11dxce.c)

### 10.2Program Data

Program Data (f11dxce.d)

### 10.3Program Results

Program Results (f11dxce.r)