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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)

```