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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zposv (f07fn)

## Purpose

nag_lapack_zposv (f07fn) 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.

## Syntax

[a, b, info] = f07fn(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)
[a, b, info] = nag_lapack_zposv(uplo, a, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zposv (f07fn) uses the Cholesky decomposition 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
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)
If uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the upper triangle of A$A$ is stored.
If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangle of 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 upper triangular part of a$a$ must be stored and the elements of the array below the diagonal are not referenced.
• If uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the lower triangular part of a$a$ must be stored and the elements of the array above the diagonal are 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
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 arrays a, b The second dimension of the array a.
n$n$, the number of linear equations, 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.
r$r$, the number of right-hand sides, 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 ${\mathbf{INFO}}={\mathbf{0}}$, 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)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$
Note: to solve the equations Ax = b$Ax=b$, where b$b$ is a single right-hand side, b may be supplied as a one-dimensional array with length ldb = max (1,n)$\mathit{ldb}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
ldbmax (1,n)$\mathit{ldb}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
If ${\mathbf{INFO}}={\mathbf{0}}$, the n$n$ by r$r$ solution matrix X$X$.
3:     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: a, 5: lda, 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.
INFO > 0${\mathbf{INFO}}>0$
If info = i${\mathbf{info}}=i$, the leading minor of order i$i$ of A$A$ is not positive definite, so the factorization could not be completed, and the solution has not 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. See Section 4.4 of Anderson et al. (1999) for further details.
nag_lapack_zposvx (f07fp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_solve (f04cd) solves Ax = b $Ax=b$ and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_solve (f04cd) calls nag_lapack_zposv (f07fn) to solve the equations.

The total number of floating point operations is approximately (4/3) n3 + 8n2r $\frac{4}{3}{n}^{3}+8{n}^{2}r$, where r $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dposv (f07fa).

## Example

```function nag_lapack_zposv_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;
6.17 + 9.42i;
-7.17 - 21.83i;
1.99 - 14.38i];
[aOut, bOut, info] = nag_lapack_zposv(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
-0.0000 + 3.0000i
-4.0000 - 5.0000i
2.0000 + 1.0000i

info =

0

```
```function f07fn_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;
6.17 + 9.42i;
-7.17 - 21.83i;
1.99 - 14.38i];
[aOut, bOut, info] = f07fn(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
-0.0000 + 3.0000i
-4.0000 - 5.0000i
2.0000 + 1.0000i

info =

0

```