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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpftrs (f07ws)

## Purpose

nag_lapack_zpftrs (f07ws) solves a complex Hermitian positive definite system of linear equations with multiple right-hand sides,
 AX = B , $AX=B ,$
using the Cholesky factorization computed by nag_lapack_zpftrf (f07wr) stored in Rectangular Full Packed (RFP) format. The RFP storage format is described in Section [Rectangular Full Packed (RFP) Storage] in the F07 Chapter Introduction.

## Syntax

[b, info] = f07ws(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[b, info] = nag_lapack_zpftrs(transr, uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpftrs (f07ws) 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_zpftrf (f07wr) which computes the Cholesky factorization of A$A$, stored in RFP format. 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

Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

## Parameters

### Compulsory Input Parameters

1:     transr – string (length ≥ 1)
Specifies whether the normal RFP representation of A$A$ or its conjugate transpose is stored.
transr = 'N'${\mathbf{transr}}=\text{'N'}$
The matrix A$A$ is stored in normal RFP format.
transr = 'C'${\mathbf{transr}}=\text{'C'}$
The conjugate transpose of the RFP representation of the matrix A$A$ is stored.
Constraint: transr = 'N'${\mathbf{transr}}=\text{'N'}$ or 'C'$\text{'C'}$.
2:     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'$\text{'L'}$.
3:     a(n × (n + 1) / 2${\mathbf{n}}×\left({\mathbf{n}}+1\right)/2$) – complex array
The Cholesky factorization of A$A$ stored in RFP format, as returned by nag_lapack_zpftrf (f07wr).
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 a.
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: transr, 2: uplo, 3: n, 4: nrhs_p, 5: a, 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)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)$\mathrm{cond}\left(A,x\right)={‖|{A}^{-1}||A||x|‖}_{\infty }/{‖x‖}_{\infty }\le \mathrm{cond}\left(A\right)={‖|{A}^{-1}||A|‖}_{\infty }\le {\kappa }_{\infty }\left(A\right)$ and κ(A)${\kappa }_{\infty }\left(A\right)$ is the condition number when using the $\infty$-norm.
Note that cond(A,x)$\mathrm{cond}\left(A,x\right)$ can be much smaller than cond(A)$\mathrm{cond}\left(A\right)$.

The total number of real floating point operations is approximately 8n2r$8{n}^{2}r$.
The real analogue of this function is nag_lapack_dpftrs (f07we).

## Example

```function nag_lapack_zpftrs_example
a = [ 4.09 + 0.00i;
3.23 + 0.00i;
1.51 + 1.92i;
1.90 - 0.84i;
0.42 - 2.50i;
2.33 - 0.14i;
4.29 + 0.00i;
3.58 + 0.00i;
-0.23 - 1.11i;
-1.18 - 1.37i];
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];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = nag_lapack_zpftrf(transr, uplo, n, a);

if info == 0
% Compute solution
[bOut, info] = nag_lapack_zpftrs(transr, uplo, aOut, b);
fprintf('\n');
[ifail] = ...
nag_file_print_matrix_complex_gen_comp('g', ' ', bOut, 'b', 'f7.4', 'Solutions', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is not positive definite.\n');
end
```
```

Solutions
1                 2
1  ( 1.0000,-1.0000) (-1.0000, 2.0000)
2  ( 0.0000, 3.0000) ( 3.0000,-4.0000)
3  (-4.0000,-5.0000) (-2.0000, 3.0000)
4  ( 2.0000, 1.0000) ( 4.0000,-5.0000)

```
```function f07ws_example
a = [ 4.09 + 0.00i;
3.23 + 0.00i;
1.51 + 1.92i;
1.90 - 0.84i;
0.42 - 2.50i;
2.33 - 0.14i;
4.29 + 0.00i;
3.58 + 0.00i;
-0.23 - 1.11i;
-1.18 - 1.37i];
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];
transr = 'n';
uplo   = 'l';
n      = int64(4);

% Factorize a
[aOut, info] = f07wr(transr, uplo, n, a);

if info == 0
% Compute solution
[bOut, info] = f07ws(transr, uplo, aOut, b);
fprintf('\n');
[ifail] = x04db('g', ' ', bOut, 'b', 'f7.4', 'Solutions', 'i', 'i', int64(80), int64(0));
else
fprintf('\na is not positive definite.\n');
end
```
```

Solutions
1                 2
1  ( 1.0000,-1.0000) (-1.0000, 2.0000)
2  ( 0.0000, 3.0000) ( 3.0000,-4.0000)
3  (-4.0000,-5.0000) (-2.0000, 3.0000)
4  ( 2.0000, 1.0000) ( 4.0000,-5.0000)

```