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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zsptrs (f07qs)

## Purpose

nag_lapack_zsptrs (f07qs) solves a complex symmetric system of linear equations with multiple right-hand sides,
 AX = B , $AX=B ,$
where A$A$ has been factorized by nag_lapack_zsptrf (f07qr), using packed storage.

## Syntax

[b, info] = f07qs(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zsptrs(uplo, ap, ipiv, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zsptrs (f07qs) is used to solve a complex symmetric system of linear equations AX = B$AX=B$, the function must be preceded by a call to nag_lapack_zsptrf (f07qr) which computes the Bunch–Kaufman factorization of A$A$, using packed storage.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where P$P$ is a permutation matrix, U$U$ is an upper triangular matrix and D$D$ is a symmetric block diagonal matrix with 1$1$ by 1$1$ and 2$2$ by 2$2$ blocks; the solution X$X$ is computed by solving PUDY = B$PUDY=B$ and then UTPTX = Y${U}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is a lower triangular matrix; the solution X$X$ is computed by solving PLDY = B$PLDY=B$ and then LTPTX = Y${L}^{\mathrm{T}}{P}^{\mathrm{T}}X=Y$.

## References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
Specifies how A$A$ has been factorized.
uplo = 'U'${\mathbf{uplo}}=\text{'U'}$
A = PUDUTPT$A=PUD{U}^{\mathrm{T}}{P}^{\mathrm{T}}$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = PLDLTPT$A=PLD{L}^{\mathrm{T}}{P}^{\mathrm{T}}$, where L$L$ is lower triangular.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     ap( : $:$) – complex array
Note: the dimension of the array ap must be at least max (1,n × (n + 1) / 2)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right)$.
The factorization of A$A$ stored in packed form, as returned by nag_lapack_zsptrf (f07qr).
3:     ipiv( : $:$) – int64int32nag_int array
Note: the dimension of the array ipiv must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
Details of the interchanges and the block structure of D$D$, as returned by nag_lapack_zsptrf (f07qr).
4:     b(ldb, : $:$) – complex array
The first dimension of the array b must be at least max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array must be at least max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
The n$n$ by r$r$ right-hand side matrix B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array ap and the second dimension of the array ap. (An error is raised if these dimensions are not equal.)
n$n$, the order of the matrix A$A$.
Constraint: n0${\mathbf{n}}\ge 0$.
2:     nrhs_p – int64int32nag_int scalar
Default: The second dimension of the array b.
r$r$, the number of right-hand sides.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

ldb

### Output Parameters

1:     b(ldb, : $:$) – complex array
The first dimension of the array b will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The second dimension of the array will be max (1,nrhs)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
The n$n$ by r$r$ solution matrix X$X$.
2:     info – int64int32nag_int scalar
info = 0${\mathbf{info}}=0$ unless the function detects an error (see Section [Error Indicators and Warnings]).

## Error Indicators and Warnings

info = i${\mathbf{info}}=-i$
If info = i${\mathbf{info}}=-i$, parameter i$i$ had an illegal value on entry. The parameters are numbered as follows:
1: uplo, 2: n, 3: nrhs_p, 4: ap, 5: ipiv, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

## Accuracy

For each right-hand side vector b$b$, the computed solution x$x$ is the exact solution of a perturbed system of equations (A + E)x = b$\left(A+E\right)x=b$, where
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, |E|c(n)εP|U||D||UT|PT$|E|\le c\left(n\right)\epsilon P|U||D||{U}^{\mathrm{T}}|{P}^{\mathrm{T}}$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |E|c(n)εP|L||D||LT|PT$|E|\le c\left(n\right)\epsilon P|L||D||{L}^{\mathrm{T}}|{P}^{\mathrm{T}}$,
c(n)$c\left(n\right)$ is a modest linear function of n$n$, and ε$\epsilon$ is the machine precision
If $\stackrel{^}{x}$ is the true solution, then the computed solution x$x$ satisfies a forward error bound of the form
 (‖x − x̂‖∞)/(‖x‖∞) ≤ c(n)cond(A,x)ε $‖x-x^‖∞ ‖x‖∞ ≤c(n)cond(A,x)ε$
where cond(A,x) = |A1||A||x| / xcond(A) = |A1||A|κ(A)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$.
Forward and backward error bounds can be computed by calling nag_lapack_zsprfs (f07qv), and an estimate for κ(A)${\kappa }_{\infty }\left(A\right)$ ( = κ1(A)$\text{}={\kappa }_{1}\left(A\right)$) can be obtained by calling nag_lapack_zspcon (f07qu).

The total number of real floating point operations is approximately 8n2r$8{n}^{2}r$.
This function may be followed by a call to nag_lapack_zsprfs (f07qv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dsptrs (f07pe).

## Example

function nag_lapack_zsptrs_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
-7.86 - 2.96i;
0.5278724801640799 - 0.3714660014825906i;
0.442558238872675 + 0.1936483698297402i;
-2.83 - 0.03i;
-0.6078391056683192 + 0.281079647893122i;
-0.4822822975185383 + 0.01498936219105284i;
4.407906236731014 + 5.399120676796941i;
-0.1070821880092683 - 0.3156780862488454i;
-2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
b = [ -55.64 + 41.22i,  -19.09 - 35.97i;
-48.18 + 66i,  -12.08 - 27.02i;
-0.49 - 1.47i,  6.95 + 20.49i;
-6.43 + 19.24i,  -4.59 - 35.53i];
[bOut, info] = nag_lapack_zsptrs(uplo, ap, ipiv, b)

bOut =

1.0000 - 1.0000i  -2.0000 - 1.0000i
-2.0000 + 5.0000i   1.0000 - 3.0000i
3.0000 - 2.0000i   3.0000 + 2.0000i
-4.0000 + 3.0000i  -1.0000 + 1.0000i

info =

0

function f07qs_example
uplo = 'L';
ap = [ -0.39 - 0.71i;
-7.86 - 2.96i;
0.5278724801640799 - 0.3714660014825906i;
0.442558238872675 + 0.1936483698297402i;
-2.83 - 0.03i;
-0.6078391056683192 + 0.281079647893122i;
-0.4822822975185383 + 0.01498936219105284i;
4.407906236731014 + 5.399120676796941i;
-0.1070821880092683 - 0.3156780862488454i;
-2.095414887840057 - 2.201139281440786i];
ipiv = [int64(-3);-3;3;4];
b = [ -55.64 + 41.22i,  -19.09 - 35.97i;
-48.18 + 66i,  -12.08 - 27.02i;
-0.49 - 1.47i,  6.95 + 20.49i;
-6.43 + 19.24i,  -4.59 - 35.53i];
[bOut, info] = f07qs(uplo, ap, ipiv, b)

bOut =

1.0000 - 1.0000i  -2.0000 - 1.0000i
-2.0000 + 5.0000i   1.0000 - 3.0000i
3.0000 - 2.0000i   3.0000 + 2.0000i
-4.0000 + 3.0000i  -1.0000 + 1.0000i

info =

0