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Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_linsys_complex_posdef_band_solve (f04cf)

## Purpose

nag_linsys_complex_posdef_band_solve (f04cf) 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 band matrix of band width 2k + 1$2k+1$, 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

[ab, b, rcond, errbnd, ifail] = f04cf(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, b, rcond, errbnd, ifail] = nag_linsys_complex_posdef_band_solve(uplo, kd, ab, 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 band matrix with k$k$ superdiagonals, and L$L$ is a lower triangular band matrix with k$k$ subdiagonals. 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:     kd – int64int32nag_int scalar
The number of superdiagonals k$k$ (and the number of subdiagonals) of the band matrix A$A$.
Constraint: kd0${\mathbf{kd}}\ge 0$.
3:     ab(ldab, : $:$) – complex array
The first dimension of the array ab must be at least kd + 1${\mathbf{kd}}+1$
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$n$ symmetric band matrix A$A$. The upper or lower triangular part of the Hermitian matrix is stored in the first kd + 1${\mathbf{kd}}+1$ rows of the array. The j$j$th column of A$A$ is stored in the j$j$th column of the array ab as follows:
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, ab (k + 1 + ijj) = aij${\mathbf{ab}}\left(k+1+i-jj\right)={a}_{ij}$ for max (1,jk)ij$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, ab (1 + ijj) = aij${\mathbf{ab}}\left(1+i-jj\right)={a}_{ij}$ for jimin (n,j + k)$j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)$.
See Section [Further Comments] below for further details.
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$r$ 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${\mathbf{nrhs}}\ge 0$.

ldab ldb

### Output Parameters

1:     ab(ldab, : $:$) – complex array
The first dimension of the array ab will be kd + 1${\mathbf{kd}}+1$
The second dimension of the array will be max (1,n)$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$
ldabkd + 1$\mathit{ldab}\ge {\mathbf{kd}}+1$.
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$A=L{L}^{\mathrm{H}}$, in the same storage format as A$A$.
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) = A11A1$\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of A$A$ with respect to the solution of the linear equations. nag_linsys_complex_posdef_band_solve (f04cf) 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 band storage scheme for the array ab is illustrated by the following example, when n = 6$n=6$, k = 2$k=2$, and uplo = 'U'${\mathbf{uplo}}=\text{'U'}$:
On entry:
 * * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66
$* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66$
On exit:
 * * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66
$* * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66$
Similarly, if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$ the format of ab is as follows:
On entry:
 a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * *
$a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * *$
On exit:
 l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * *
$l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * *$
Array elements marked * $*$ need not be set and are not referenced by the function.
Assuming that nk$n\gg k$, the total number of floating point operations required to solve the equations AX = B$AX=B$ is approximately n(k + 1)2${n\left(k+1\right)}^{2}$ for the factorization and 4nkr$4nkr$ for the solution following the factorization. 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_band_solve (f04cf) is nag_linsys_real_posdef_band_solve (f04bf).

## Example

```function nag_linsys_complex_posdef_band_solve_example
uplo = 'U';
kd = int64(1);
ab = [ -1.375785829994644 - 1.375148775543909i,  1.08 - 1.73i, ...
-0.04 + 0.29i,  -0.33 + 2.24i;
9.39 + 0i,  1.69 + 0i,  2.65 + 0i,  2.17 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
[abOut, bOut, rcond, errbnd, ifail] = nag_linsys_complex_posdef_band_solve(uplo, kd, ab, b)
```
```

-1.3758 - 1.3751i   0.3524 - 0.5646i  -0.0358 + 0.2597i  -0.2054 + 1.3942i
3.0643 + 0.0000i   1.1167 + 0.0000i   1.6066 + 0.0000i   0.4289 + 0.0000i

bOut =

-1.0000 + 8.0000i   5.0000 - 6.0000i
2.0000 - 3.0000i   2.0000 + 3.0000i
-4.0000 - 5.0000i  -8.0000 + 4.0000i
7.0000 + 6.0000i  -1.0000 - 7.0000i

rcond =

0.0076

errbnd =

1.4676e-14

ifail =

0

```
```function f04cf_example
uplo = 'U';
kd = int64(1);
ab = [ -1.375785829994644 - 1.375148775543909i,  1.08 - 1.73i, ...
-0.04 + 0.29i,  -0.33 + 2.24i;
9.39 + 0i,  1.69 + 0i,  2.65 + 0i,  2.17 + 0i];
b = [ -12.42 + 68.42i,  54.3 - 56.56i;
-9.93 + 0.88i,  18.32 + 4.76i;
-27.3 - 0.01i,  -4.4 + 9.97i;
5.31 + 23.63i,  9.43 + 1.41i];
[abOut, bOut, rcond, errbnd, ifail] = f04cf(uplo, kd, ab, b)
```
```

-1.3758 - 1.3751i   0.3524 - 0.5646i  -0.0358 + 0.2597i  -0.2054 + 1.3942i
3.0643 + 0.0000i   1.1167 + 0.0000i   1.6066 + 0.0000i   0.4289 + 0.0000i

bOut =

-1.0000 + 8.0000i   5.0000 - 6.0000i
2.0000 - 3.0000i   2.0000 + 3.0000i
-4.0000 - 5.0000i  -8.0000 + 4.0000i
7.0000 + 6.0000i  -1.0000 - 7.0000i

rcond =

0.0076

errbnd =

1.4676e-14

ifail =

0

```