NAG FL Interface
f07pbf (dspsvx)
1
Purpose
f07pbf uses the diagonal pivoting factorization
to compute the solution to a real system of linear equations
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.
2
Specification
Fortran Interface
Subroutine f07pbf ( 
fact, uplo, n, nrhs, ap, afp, ipiv, b, ldb, x, ldx, rcond, ferr, berr, work, iwork, 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 

C Header Interface
#include <nag.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) 

C++ Header Interface
#include <nag.h> extern "C" {
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 the names f07pbf, nagf_lapacklin_dspsvx or its LAPACK name dspsvx.
3
Description
f07pbf 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.
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
https://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:
$\mathbf{fact}$ – 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:
$\mathbf{n}$ – Integer
Input

On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{n}}\ge 0$.

4:
$\mathbf{nrhs}$ – Integer
Input

On entry: $r$, the number of righthand 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) array
Input

Note: the dimension of the array
ap
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right)$.
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 ${\mathbf{ap}}\left(i+j\left(j1\right)/2\right)$ for $i\le j$;
 if ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of $A$ must be stored with element ${A}_{ij}$ in ${\mathbf{ap}}\left(i+\left(2nj\right)\left(j1\right)/2\right)$ for $i\ge j$.

6:
$\mathbf{afp}\left(*\right)$ – Real (Kind=nag_wp) array
Input/Output

Note: the dimension of the array
afp
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\times \left({\mathbf{n}}+1\right)/2\right)$.
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, 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, stored as a packed triangular matrix in the same storage format as
$A$.

7:
$\mathbf{ipiv}\left({\mathbf{n}}\right)$ – Integer array
Input/Output

On entry: if
${\mathbf{fact}}=\text{'F'}$,
ipiv contains details of the interchanges and the block structure of
$D$, as determined by
f07pdf.
 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 ${\mathbf{ipiv}}\left(i1\right)={\mathbf{ipiv}}\left(i\right)=l<0$, $\left(\begin{array}{cc}{d}_{i1,i1}& {\overline{d}}_{i,i1}\\ {\overline{d}}_{i,i1}& {d}_{ii}\end{array}\right)$ is a $2$ by $2$ pivot block and the $\left(i1\right)$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, as described above.

8:
$\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) array
Input

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 $r$ righthand side matrix $B$.

9:
$\mathbf{ldb}$ – Integer
Input

On entry: the first dimension of the array
b as declared in the (sub)program from which
f07pbf 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) array
Output

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:
$\mathbf{ldx}$ – Integer
Input

On entry: the first dimension of the array
x as declared in the (sub)program from which
f07pbf 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
${\mathbf{info}}>{\mathbf{0}}\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le \mathbf{n}$. 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) array
Output

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
${\Vert {\hat{x}}_{j}{x}_{j}\Vert}_{\infty}/{\Vert {x}_{j}\Vert}_{\infty}\le {\mathbf{ferr}}\left(j\right)$ where
${\hat{x}}_{j}$ 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) array
Output

On exit: if ${\mathbf{info}}={\mathbf{0}}$ or ${\mathbf{n}+{\mathbf{1}}}$, an estimate of the componentwise relative backward error of each computed solution vector ${\hat{x}}_{j}$ (i.e., the smallest relative change in any element of $A$ or $B$ that makes ${\hat{x}}_{j}$ an exact solution).

15:
$\mathbf{work}\left(3\times {\mathbf{n}}\right)$ – Real (Kind=nag_wp) array
Workspace


16:
$\mathbf{iwork}\left({\mathbf{n}}\right)$ – Integer array
Workspace


17:
$\mathbf{info}$ – Integer
Output

On exit:
${\mathbf{info}}=0$ unless the routine detects an error (see
Section 6).
6
Error 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.
 ${\mathbf{info}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{info}}\le {\mathbf{n}}$

Element $\u2329\mathit{\text{value}}\u232a$ 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.
7
Accuracy
For each righthand side vector
$b$, the computed solution
$\hat{x}$ is the exact solution of a perturbed system of equations
$\left(A+E\right)\hat{x}=b$, where
where
$\epsilon $ is the
machine precision. See Chapter 11 of
Higham (2002) for further details.
If
$\hat{x}$ is the true solution, then the computed solution
$x$ satisfies a forward error bound of the form
where
$\mathrm{cond}\left(A,\hat{x},b\right)={\Vert \left{A}^{1}\right\left(\leftA\right\left\hat{x}\right+\leftb\right\right)\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}\le \mathrm{cond}\left(A\right)={\Vert \left{A}^{1}\right\leftA\right\Vert}_{\infty}\le {\kappa}_{\infty}\left(A\right)$.
If
$\hat{x}$ is the
$j$th column of
$X$, then
${w}_{c}$ is returned in
${\mathbf{berr}}\left(j\right)$ and a bound on
${\Vert x\hat{x}\Vert}_{\infty}/{\Vert \hat{x}\Vert}_{\infty}$ is returned in
${\mathbf{ferr}}\left(j\right)$. See Section 4.4 of
Anderson et al. (1999) for further details.
8
Parallelism and Performance
f07pbf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07pbf 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 implementationspecific information.
The factorization of $A$ requires approximately $\frac{1}{3}{n}^{3}$ floatingpoint operations.
For each righthand side, computation of the backward error involves a minimum of $4{n}^{2}$ floatingpoint 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 for Hermitian matrices, and
f07qpf for symmetric matrices.
10
Example
This example solves the equations
where
$A$ is the symmetric matrix
Error estimates for the solutions, and an estimate of the reciprocal of the condition number of the matrix $A$ are also output.
10.1
Program Text
10.2
Program Data
10.3
Program Results