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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_posdef_solve (f04cd)

## Purpose

nag_linsys_complex_posdef_solve (f04cd) computes the solution to a complex system of linear equations AX = B$AX=B$, where A$A$ is an n$n$ by n$n$ Hermitian positive definite matrix and X$X$ and B$B$ are n$n$ by r$r$ matrices. An estimate of the condition number of A$A$ and an error bound for the computed solution are also returned.

## Syntax

[a, b, rcond, errbnd, ifail] = f04cd(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_solve(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

The Cholesky factorization is used to factor A$A$ as A = UHU$A={U}^{\mathrm{H}}U$, if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or A = LLH$A=L{L}^{\mathrm{H}}$, if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, where U$U$ is an upper triangular matrix and L$L$ is a lower triangular matrix. The factored form of A$A$ is then used to solve the system of equations AX = B$AX=B$.

## 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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## Parameters

### Compulsory Input Parameters

1:     uplo – string (length ≥ 1)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix A$A$ is stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix A$A$ is stored.
Constraint: uplo = 'U'${\mathbf{uplo}}=\text{'U'}$ or 'L'$\text{'L'}$.
2:     a(lda, : $:$) – complex array
The first dimension of the array a 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,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
The n$n$ by n$n$ Hermitian matrix A$A$.
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the leading n by n upper triangular part of a contains the upper triangular part of the matrix A$A$, and the strictly lower triangular part of a is not referenced.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the leading n by n lower triangular part of a contains the lower triangular part of the matrix A$A$, and the strictly upper triangular part of a is not referenced.
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$r$ matrix of right-hand sides B$B$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The first dimension of the array b.
The number of linear equations n$n$, i.e., 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.
The number of right-hand sides r$r$, i.e., the number of columns of the matrix B$B$.
Constraint: nrhs0${\mathbf{nrhs}}\ge 0$.

lda ldb

### Output Parameters

1:     a(lda, : $:$) – complex array
The first dimension of the array a 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,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldamax (1,n)$\mathit{lda}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the factor U$U$ or L$L$ from the Cholesky factorization A = UHU$A={U}^{\mathrm{H}}U$ or A = LLH$A=L{L}^{\mathrm{H}}$.
2:     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)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, the n$n$ by r$r$ solution matrix X$X$.
3:     rcond – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the reciprocal of the condition number of the matrix A$A$, computed as rcond = 1 / (A1A11)${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
4:     errbnd – double scalar
If ${\mathbf{ifail}}={\mathbf{0}}$ or n + 1$\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that x1 / x1errbnd${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and x$x$ is the corresponding column of the exact solution X$X$. If rcond is less than machine precision, then errbnd is returned as unity.
5:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

ifail < 0andifail999${\mathbf{ifail}}<0 \text{and} {\mathbf{ifail}}\ne -999$
If ifail = i${\mathbf{ifail}}=-i$, the i$i$th argument had an illegal value.
ifail = 999${\mathbf{ifail}}=-999$
Allocation of memory failed. The double allocatable memory required is n, and the complex allocatable memory required is 2 × n$2×{\mathbf{n}}$. Allocation failed before the solution could be computed.
ifail > 0andifailN${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{N}}$
If ifail = i${\mathbf{ifail}}=i$, the leading minor of order i$i$ of A$A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.
W ifail = N + 1${\mathbf{ifail}}={\mathbf{N}}+1$
rcond is less than machine precision, so that the matrix A$A$ is numerically singular. A solution to the equations AX = B$AX=B$ has nevertheless been computed.

## Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 (A + E) x̂ = b, $(A+E) x^=b,$
where
 ‖E‖1 = O(ε) ‖A‖1 $‖E‖1 = O(ε) ‖A‖1$
and ε$\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 (‖x̂ − x‖1)/(‖x‖1) ≤ κ(A) (‖E‖1)/(‖A‖1) , $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
where κ(A) = A11 A1 $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_complex_posdef_solve (f04cd) uses the approximation E1 = εA1${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

The total number of floating point operations required to solve the equations AX = B$AX=B$ is proportional to ((1/3)n3 + n2r)$\left(\frac{1}{3}{n}^{3}+{n}^{2}r\right)$. The condition number estimation typically requires between four and five solves and never more than eleven solves, following the factorization.
In practice the condition number estimator is very reliable, but it can underestimate the true condition number; see Section 15.3 of Higham (2002) for further details.
The real analogue of nag_linsys_complex_posdef_solve (f04cd) is nag_linsys_real_posdef_solve (f04bd).

## Example

```function nag_linsys_complex_posdef_solve_example
uplo = 'Upper';
a = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 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];
[aOut, bOut, rcond, errbnd, ifail] = nag_linsys_complex_posdef_solve(uplo, a, b)
```
```

aOut =

1.7972 + 0.0000i   0.8402 - 1.0683i   1.0572 + 0.4674i   0.2337 + 1.3910i
0.0000 + 0.0000i   1.3164 + 0.0000i  -0.4702 - 0.3131i   0.0834 - 0.0368i
0.0000 + 0.0000i   0.0000 + 0.0000i   1.5604 + 0.0000i   0.9360 - 0.9900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.6603 + 0.0000i

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

rcond =

0.0066

errbnd =

1.6806e-14

ifail =

0

```
```function f04cd_example
uplo = 'Upper';
a = [3.23,  1.51 - 1.92i,  1.9 + 0.84i,  0.42 + 2.5i;
0 + 0i,  3.58 + 0i,  -0.23 + 1.11i,  -1.18 + 1.37i;
0 + 0i,  0 + 0i,  4.09 + 0i,  2.33 - 0.14i;
0 + 0i,  0 + 0i,  0 + 0i,  4.29 + 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];
[aOut, bOut, rcond, errbnd, ifail] = f04cd(uplo, a, b)
```
```

aOut =

1.7972 + 0.0000i   0.8402 - 1.0683i   1.0572 + 0.4674i   0.2337 + 1.3910i
0.0000 + 0.0000i   1.3164 + 0.0000i  -0.4702 - 0.3131i   0.0834 - 0.0368i
0.0000 + 0.0000i   0.0000 + 0.0000i   1.5604 + 0.0000i   0.9360 - 0.9900i
0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.6603 + 0.0000i

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

rcond =

0.0066

errbnd =

1.6806e-14

ifail =

0

```