F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07PAF (DSPSV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07PAF (DSPSV) computes the solution to a real 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

 SUBROUTINE F07PAF ( UPLO, N, NRHS, AP, IPIV, B, LDB, INFO)
 INTEGER N, NRHS, IPIV(N), LDB, INFO REAL (KIND=nag_wp) AP(*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name dspsv.

## 3  Description

F07PAF (DSPSV) uses the diagonal pivoting method to factor $A$ as $A=UD{U}^{\mathrm{T}}$ if ${\mathbf{UPLO}}=\text{'U'}$ or $A=LD{L}^{\mathrm{T}}$ if ${\mathbf{UPLO}}=\text{'L'}$, 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  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
4:     AP($*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the dimension 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 columns.
More precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+j\left(j-1\right)/2\right)$ for $i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{AP}}\left(i+\left(2n-j\right)\left(j-1\right)/2\right)$ 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 F07PDF (DSPTRF), stored as a packed triangular matrix in the same storage format as $A$.
5:     IPIV(N) – INTEGER arrayOutput
On exit: details of the interchanges and the block structure of $D$. More precisely,
• if ${\mathbf{IPIV}}\left(i\right)=k>0$, ${d}_{ii}$ 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 ${\mathbf{UPLO}}=\text{'U'}$ and ${\mathbf{IPIV}}\left(i-1\right)={\mathbf{IPIV}}\left(i\right)=-l<0$, $\left(\begin{array}{cc}{d}_{i-1,i-1}& {\stackrel{-}{d}}_{i,i-1}\\ {\stackrel{-}{d}}_{i,i-1}& {d}_{ii}\end{array}\right)$ 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 ${\mathbf{UPLO}}=\text{'L'}$ and ${\mathbf{IPIV}}\left(i\right)={\mathbf{IPIV}}\left(i+1\right)=-m<0$, $\left(\begin{array}{cc}{d}_{ii}& {d}_{i+1,i}\\ {d}_{i+1,i}& {d}_{i+1,i+1}\end{array}\right)$ 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.
6:     B(LDB,$*$) – REAL (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
7:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07PAF (DSPSV) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
8:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, ${d}_{ii}$ 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.
F07PBF (DSPSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04BJF solves $AX=B$ and returns a forward error bound and condition estimate. F04BJF calls F07PAF (DSPSV) to solve the equations.

The total number of floating point operations is approximately $\frac{1}{3}{n}^{3}+2{n}^{2}r$, where $r$ is the number of right-hand sides.
The complex analogues of F07PAF (DSPSV) are F07PNF (ZHPSV) for Hermitian matrices, and F07QNF (ZSPSV) for symmetric matrices.

## 9  Example

This example solves the equations
 $Ax=b ,$
where $A$ is the symmetric matrix
 $A = -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07 and b = 0.96 6.07 8.38 9.50 .$
Details of the factorization of $A$ are also output.

### 9.1  Program Text

Program Text (f07pafe.f90)

### 9.2  Program Data

Program Data (f07pafe.d)

### 9.3  Program Results

Program Results (f07pafe.r)