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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_lapack_zpbsv (f07hn)

## Purpose

nag_lapack_zpbsv (f07hn) 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 bandwidth (2kd + 1) $\left(2{k}_{d}+1\right)$ and X$X$ and B$B$ are n$n$ by r$r$ matrices.

## Syntax

[ab, b, info] = f07hn(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)
[ab, b, info] = nag_lapack_zpbsv(uplo, kd, ab, b, 'n', n, 'nrhs_p', nrhs_p)

## Description

nag_lapack_zpbsv (f07hn) 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 band matrix, and L$L$ is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as A$A$. 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:     kd – int64int32nag_int scalar
kd${k}_{d}$, the number of superdiagonals of the matrix A$A$ if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, or the number of subdiagonals if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$.
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 upper or lower triangle of the Hermitian band matrix A$A$.
The matrix is stored in rows 1$1$ to kd + 1${k}_{d}+1$, more precisely,
• if uplo = 'U'${\mathbf{uplo}}=\text{'U'}$, the elements of the upper triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(kd + 1 + ij,j)​ for ​max (1,jkd)ij${\mathbf{ab}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if uplo = 'L'${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of A$A$ within the band must be stored with element Aij${A}_{ij}$ in ab(1 + ij,j)​ for ​jimin (n,j + kd).${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
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 b The second dimension of the array ab.
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${\mathbf{nrhs}}\ge 0$.

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 ${\mathbf{INFO}}={\mathbf{0}}$, the triangular factor U$U$ or L$L$ from the Cholesky factorization A = UHU$A={U}^{\mathrm{H}}U$ or A = LLH$A=L{L}^{\mathrm{H}}$ of the band matrix A$A$, 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 ${\mathbf{INFO}}={\mathbf{0}}$, 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: kd, 4: nrhs_p, 5: ab, 6: ldab, 7: b, 8: ldb, 9: 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_zpbsvx (f07hp) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, nag_linsys_complex_posdef_band_solve (f04cf) solves Ax = b $Ax=b$ and returns a forward error bound and condition estimate. nag_linsys_complex_posdef_band_solve (f04cf) calls nag_lapack_zpbsv (f07hn) to solve the equations.

When nk $n\gg k$, the total number of floating point operations is approximately 4n(k + 1)2 + 16nkr $4n{\left(k+1\right)}^{2}+16nkr$, where k $k$ is the number of superdiagonals and r $r$ is the number of right-hand sides.
The real analogue of this function is nag_lapack_dpbsv (f07ha).

## Example

```function nag_lapack_zpbsv_example
uplo = 'U';
kd = int64(1);
ab = [0,  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;
-9.93 + 0.88i;
-27.3 - 0.01i;
5.31 + 23.63i];
[abOut, bOut, info] = nag_lapack_zpbsv(uplo, kd, ab, b)
```
```

0.0000 + 0.0000i   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
2.0000 - 3.0000i
-4.0000 - 5.0000i
7.0000 + 6.0000i

info =

0

```
```function f07hn_example
uplo = 'U';
kd = int64(1);
ab = [0,  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;
-9.93 + 0.88i;
-27.3 - 0.01i;
5.31 + 23.63i];
[abOut, bOut, info] = f07hn(uplo, kd, ab, b)
```
```

0.0000 + 0.0000i   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
2.0000 - 3.0000i
-4.0000 - 5.0000i
7.0000 + 6.0000i

info =

0

```