f07 Chapter Contents
f07 Chapter Introduction
NAG Library Manual

# NAG Library Function Documentnag_zspsv (f07qnc)

## 1  Purpose

nag_zspsv (f07qnc) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ symmetric matrix stored in packed format and $X$ and $B$ are $n$ by $r$ matrices.

## 2  Specification

 #include #include
 void nag_zspsv (Nag_OrderType order, Nag_UploType uplo, Integer n, Integer nrhs, Complex ap[], Integer ipiv[], Complex b[], Integer pdb, NagError *fail)

## 3  Description

nag_zspsv (f07qnc) uses the diagonal pivoting method to factor $A$ as $A=UD{U}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, where $U$ (or $L$) is a product of permutation and unit upper (lower) triangular matrices, $D$ is symmetric and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## 4  References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia http://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5  Arguments

1:     orderNag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:     uploNag_UploTypeInput
On entry: if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, the upper triangle of $A$ is stored.
If ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
3:     nIntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     nrhsIntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5:     ap[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array ap must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: the $n$ by $n$ symmetric matrix $A$, packed by rows or columns.
The storage of elements ${A}_{ij}$ depends on the order and uplo arguments as follows:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(j-1\right)×j/2+i-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-j\right)×\left(j-1\right)/2+i-1\right]$, for $i\ge j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Upper}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(2n-i\right)×\left(i-1\right)/2+j-1\right]$, for $i\le j$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ and ${\mathbf{uplo}}=\mathrm{Nag_Lower}$,
${A}_{ij}$ is stored in ${\mathbf{ap}}\left[\left(i-1\right)×i/2+j-1\right]$, for $i\ge j$.
On exit: the block diagonal matrix $D$ and the multipliers used to obtain the factor $U$ or $L$ from the factorization $A=UD{U}^{\mathrm{T}}$ or $A=LD{L}^{\mathrm{T}}$ as computed by nag_zsptrf (f07qrc), stored as a packed triangular matrix in the same storage format as $A$.
6:     ipiv[n]IntegerOutput
On exit: details of the interchanges and the block structure of $D$. More precisely,
• if ${\mathbf{ipiv}}\left[i-1\right]=k>0$, ${d}_{ii}$ is a $1$ by $1$ pivot block and the $i$th row and column of $A$ were interchanged with the $k$th row and column;
• if ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ and ${\mathbf{ipiv}}\left[i-2\right]={\mathbf{ipiv}}\left[i-1\right]=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i-1\right)$th row and column of $A$ were interchanged with the $l$th row and column;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$ and ${\mathbf{ipiv}}\left[i-1\right]={\mathbf{ipiv}}\left[i\right]=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i+1\right)$th row and column of $A$ were interchanged with the $m$th row and column.
7:     b[$\mathit{dim}$]ComplexInput/Output
Note: the dimension, dim, of the array b must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdb}}×{\mathbf{nrhs}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdb}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
The $\left(i,j\right)$th element of the matrix $B$ is stored in
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{fail}}\mathbf{.}\mathbf{code}=$ NE_NOERROR, the $n$ by $r$ solution matrix $X$.
8:     pdbIntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
9:     failNagError *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.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
On entry, ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}>0$.
NE_INT_2
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{nrhs}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
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_SINGULAR
$D\left(⟨\mathit{\text{value}}⟩,⟨\mathit{\text{value}}⟩\right)$ is exactly zero. The factorization has been completed, but the block diagonal matrix $D$ is exactly singular, so the solution could not be computed.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) and Chapter 11 of Higham (2002) for further details.
nag_zspsvx (f07qpc) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_complex_sym_packed_lin_solve (f04djc) solves $AX=B$ and returns a forward error bound and condition estimate. nag_complex_sym_packed_lin_solve (f04djc) calls nag_zspsv (f07qnc) to solve the equations.

## 8  Parallelism and Performance

nag_zspsv (f07qnc) is not threaded by NAG in any implementation.
nag_zspsv (f07qnc) 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.

The total number of floating-point operations is approximately $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where $r$ is the number of right-hand sides.
The real analogue of this function is nag_dspsv (f07pac). The complex Hermitian analogue of this function is nag_zhpsv (f07pnc).

## 10  Example

This example solves the equations
 $Ax=b ,$
where $A$ is the complex symmetric matrix
 $A = -0.56+0.12i -1.54-2.86i 5.32-1.59i 3.80+0.92i -1.54-2.86i -2.83-0.03i -3.52+0.58i -7.86-2.96i 5.32-1.59i -3.52+0.58i 8.86+1.81i 5.14-0.64i 3.80+0.92i -7.86-2.96i 5.14-0.64i -0.39-0.71i$
and
 $b = -6.43+19.24i -0.49-01.47i -48.18+66.00i -55.64+41.22i .$
Details of the factorization of $A$ are also output.

### 10.1  Program Text

Program Text (f07qnce.c)

### 10.2  Program Data

Program Data (f07qnce.d)

### 10.3  Program Results

Program Results (f07qnce.r)