# NAG Library Routine Document

## 1Purpose

f07pbf (dspsvx) uses the diagonal pivoting factorization
 $A=UDUT or A=LDLT$
to compute 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. Error bounds on the solution and a condition estimate are also provided.

## 2Specification

Fortran Interface
 Subroutine f07pbf ( fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, ferr, berr, work, info)
 Integer, Intent (In) :: n, nrhs, ldb, ldx Integer, Intent (Inout) :: ipiv(n) Integer, Intent (Out) :: iwork(n), info Real (Kind=nag_wp), Intent (In) :: ap(*), b(ldb,*) Real (Kind=nag_wp), Intent (Inout) :: afp(*), x(ldx,*) Real (Kind=nag_wp), Intent (Out) :: rcond, ferr(nrhs), berr(nrhs), work(3*n) Character (1), Intent (In) :: fact, uplo
#include <nagmk26.h>
 void f07pbf_ (const char *fact, const char *uplo, const Integer *n, const Integer *nrhs, const double ap[], double afp[], Integer ipiv[], const double b[], const Integer *ldb, double x[], const Integer *ldx, double *rcond, double ferr[], double berr[], double work[], Integer iwork[], Integer *info, const Charlen length_fact, const Charlen length_uplo)
The routine may be called by its LAPACK name dspsvx.

## 3Description

f07pbf (dspsvx) performs the following steps:
 1 If ${\mathbf{fact}}=\text{'N'}$, the diagonal pivoting method is used 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 and $D$ is symmetric and block diagonal with $1$ by $1$ and $2$ by $2$ diagonal blocks. 2 If some ${d}_{ii}=0$, so that $D$ is exactly singular, then the routine returns with ${\mathbf{info}}=i$. Otherwise, the factored form of $A$ is used to estimate the condition number of the matrix $A$. If the reciprocal of the condition number is less than machine precision, ${\mathbf{info}}=\mathbf{n}+{\mathbf{1}}$ is returned as a warning, but the routine still goes on to solve for $X$ and compute error bounds as described below. 3 The system of equations is solved for $X$ using the factored form of $A$. 4 Iterative refinement is applied to improve the computed solution matrix and to calculate error bounds and backward error estimates for it.

## 4References

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

## 5Arguments

1:     $\mathbf{fact}$ – Character(1)Input
On entry: specifies whether or not the factorized form of the matrix $A$ has been supplied.
${\mathbf{fact}}=\text{'F'}$
afp and ipiv contain the factorized form of the matrix $A$. afp and ipiv will not be modified.
${\mathbf{fact}}=\text{'N'}$
The matrix $A$ will be copied to afp and factorized.
Constraint: ${\mathbf{fact}}=\text{'F'}$ or $\text{'N'}$.
2:     $\mathbf{uplo}$ – 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 $\text{'L'}$.
3:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5:     $\mathbf{ap}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
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 $i\ge j$.
6:     $\mathbf{afp}\left(*\right)$ – Real (Kind=nag_wp) arrayInput/Output
Note: the dimension of the array afp must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
On entry: if ${\mathbf{fact}}=\text{'F'}$, afp contains 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 $A$.
On exit: if ${\mathbf{fact}}=\text{'N'}$, afp contains 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 $A$.
7:     $\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer arrayInput/Output
On entry: if ${\mathbf{fact}}=\text{'F'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07pdf (dsptrf).
• 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.
On exit: if ${\mathbf{fact}}=\text{'N'}$, ipiv contains details of the interchanges and the block structure of $D$, as determined by f07pdf (dsptrf), as described above.
8:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput
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 $B$.
9:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f07pbf (dspsvx) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
10:   $\mathbf{x}\left({\mathbf{ldx}},*\right)$ – Real (Kind=nag_wp) arrayOutput
Note: the second dimension of the array x must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
11:   $\mathbf{ldx}$ – IntegerInput
On entry: the first dimension of the array x as declared in the (sub)program from which f07pbf (dspsvx) is called.
Constraint: ${\mathbf{ldx}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
12:   $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: the estimate of the reciprocal condition number of the matrix $A$. If ${\mathbf{rcond}}=0.0$, the matrix may be exactly singular. This condition is indicated by ${\mathbf{info}}>{\mathbf{0}} \text{and} {\mathbf{info}}\le \mathbf{n}$. Otherwise, if rcond is less than the machine precision, the matrix is singular to working precision. This condition is indicated by ${\mathbf{info}}=\mathbf{n}+{\mathbf{1}}$.
13:   $\mathbf{ferr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for each computed solution vector, such that ${‖{\stackrel{^}{x}}_{j}-{x}_{j}‖}_{\infty }/{‖{x}_{j}‖}_{\infty }\le {\mathbf{ferr}}\left(j\right)$ where ${\stackrel{^}{x}}_{j}$ is the $j$th column of the computed solution returned in the array x and ${x}_{j}$ is the corresponding column of the exact solution $X$. The estimate is as reliable as the estimate for rcond, and is almost always a slight overestimate of the true error.
14:   $\mathbf{berr}\left({\mathbf{nrhs}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: if ${\mathbf{info}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the component-wise relative backward error of each computed solution vector ${\stackrel{^}{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\stackrel{^}{x}}_{j}$ an exact solution).
15:   $\mathbf{work}\left(3×{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
16:   $\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer arrayWorkspace
17:   $\mathbf{info}$ – IntegerOutput
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0 \text{and} {\mathbf{info}}\le {\mathbf{n}}$
Element $〈\mathit{\text{value}}〉$ of the diagonal is exactly zero. The factorization has been completed, but the factor $D$ is exactly singular, so the solution and error bounds could not be computed. ${\mathbf{rcond}}=0.0$ is returned.
${\mathbf{info}}={\mathbf{n}}+1$
$D$ is nonsingular, but rcond is less than machine precision, meaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are computed because there are a number of situations where the computed solution can be more accurate than the value of rcond would suggest.

## 7Accuracy

For each right-hand side vector $b$, the computed solution $\stackrel{^}{x}$ is the exact solution of a perturbed system of equations $\left(A+E\right)\stackrel{^}{x}=b$, where
 $E1 = Oε A1 ,$
where $\epsilon$ is the machine precision. See Chapter 11 of Higham (2002) for further details.
If $\stackrel{^}{x}$ is the true solution, then the computed solution $x$ satisfies a forward error bound of the form
 $x-x^∞ x^∞ ≤ wc condA,x^,b$
where $\mathrm{cond}\left(A,\stackrel{^}{x},b\right)={‖\left|{A}^{-1}\right|\left(\left|A\right|\left|\stackrel{^}{x}\right|+\left|b\right|\right)‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖\left|{A}^{-1}\right|\left|A\right|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$. If $\stackrel{^}{x}$ is the $j$th column of $X$, then ${w}_{c}$ is returned in ${\mathbf{berr}}\left(j\right)$ and a bound on ${‖x-\stackrel{^}{x}‖}_{\infty }/{‖\stackrel{^}{x}‖}_{\infty }$ is returned in ${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f07pbf (dspsvx) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07pbf (dspsvx) 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The factorization of $A$ requires approximately $\frac{1}{3}{n}^{3}$ floating-point operations.
For each right-hand side, computation of the backward error involves a minimum of $4{n}^{2}$ floating-point operations. Each step of iterative refinement involves an additional $6{n}^{2}$ operations. At most five steps of iterative refinement are performed, but usually only one or two steps are required. Estimating the forward error involves solving a number of systems of equations of the form $Ax=b$; the number is usually $4$ or $5$ and never more than $11$. Each solution involves approximately $2{n}^{2}$ operations.
The complex analogues of this routine are f07ppf (zhpsvx) for Hermitian matrices, and f07qpf (zspsvx) for symmetric matrices.

## 10Example

This example solves the equations
 $AX=B ,$
where $A$ is the symmetric matrix
 $A = -1.81 2.06 0.63 -1.15 2.06 1.15 1.87 4.20 0.63 1.87 -0.21 3.87 -1.15 4.20 3.87 2.07 and B = 0.96 3.93 6.07 19.25 8.38 9.90 9.50 27.85 .$
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix $A$ are also output.

### 10.1Program Text

Program Text (f07pbfe.f90)

### 10.2Program Data

Program Data (f07pbfe.d)

### 10.3Program Results

Program Results (f07pbfe.r)