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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpptrs (f07gs)

## Purpose

nag_lapack_zpptrs (f07gs) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
 AX = B , $AX=B ,$
where A$A$ has been factorized by nag_lapack_zpptrf (f07gr), using packed storage.

## Syntax

[b, info] = f07gs(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpptrs(uplo, ap, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpptrs (f07gs) is used to solve a complex Hermitian positive definite system of linear equations AX = B$AX=B$, the function must be preceded by a call to nag_lapack_zpptrf (f07gr) which computes the Cholesky factorization of A$A$, using packed storage. The solution X$X$ is computed by forward and backward substitution.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, A = UHU$A={U}^{\mathrm{H}}U$, where U$U$ is upper triangular; the solution X$X$ is computed by solving UHY = B${U}^{\mathrm{H}}Y=B$ and then UX = Y$UX=Y$.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, A = LLH$A=L{L}^{\mathrm{H}}$, where L$L$ is lower triangular; the solution X$X$ is computed by solving LY = B$LY=B$ and then LHX = Y${L}^{\mathrm{H}}X=Y$.

## 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 = UHU$A={U}^{\mathrm{H}}U$, where U$U$ is upper triangular.
uplo = 'L'${\mathbf{uplo}}=\text{'L'}$
A = LLH$A=L{L}^{\mathrm{H}}$, 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 Cholesky factor of A$A$ stored in packed form, as returned by nag_lapack_zpptrf (f07gr).
3:     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: b, 6: ldb, 7: 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)ε|UH||U|$|E|\le c\left(n\right)\epsilon |{U}^{\mathrm{H}}||U|$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, |E|c(n)ε|L||LH|$|E|\le c\left(n\right)\epsilon |L||{L}^{\mathrm{H}}|$,
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_zpprfs (f07gv), 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_zppcon (f07gu).

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_zpprfs (f07gv) to refine the solution and return an error estimate.
The real analogue of this function is nag_lapack_dpptrs (f07ge).

## Example

```function nag_lapack_zpptrs_example
uplo = 'L';
ap = [1.797220075561143;
0.8401864749527325 + 1.068316577423342i;
1.057188279741849 - 0.467388502622712i;
0.233694251311356 - 1.391037210186643i;
1.316353439509685 + 0i;
-0.4701749470106329 + 0.3130658155999466i;
0.08335250923944192 + 0.03676071443037458i;
1.560392977137124 + 0i;
0.9359617337923402 + 0.9899692192815736i;
0.6603332973655893 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
6.17 + 9.42i,  4.65 - 4.75i;
-7.17 - 21.83i,  -4.91 + 2.29i;
1.99 - 14.38i,  7.64 - 10.79i];
[bOut, info] = nag_lapack_zpptrs(uplo, ap, b)
```
```

bOut =

1.0000 - 1.0000i  -1.0000 + 2.0000i
-0.0000 + 3.0000i   3.0000 - 4.0000i
-4.0000 - 5.0000i  -2.0000 + 3.0000i
2.0000 + 1.0000i   4.0000 - 5.0000i

info =

0

```
```function f07gs_example
uplo = 'L';
ap = [1.797220075561143;
0.8401864749527325 + 1.068316577423342i;
1.057188279741849 - 0.467388502622712i;
0.233694251311356 - 1.391037210186643i;
1.316353439509685 + 0i;
-0.4701749470106329 + 0.3130658155999466i;
0.08335250923944192 + 0.03676071443037458i;
1.560392977137124 + 0i;
0.9359617337923402 + 0.9899692192815736i;
0.6603332973655893 + 0i];
b = [ 3.93 - 6.14i,  1.48 + 6.58i;
6.17 + 9.42i,  4.65 - 4.75i;
-7.17 - 21.83i,  -4.91 + 2.29i;
1.99 - 14.38i,  7.64 - 10.79i];
[bOut, info] = f07gs(uplo, ap, b)
```
```

bOut =

1.0000 - 1.0000i  -1.0000 + 2.0000i
-0.0000 + 3.0000i   3.0000 - 4.0000i
-4.0000 - 5.0000i  -2.0000 + 3.0000i
2.0000 + 1.0000i   4.0000 - 5.0000i

info =

0

```