F07HNF (ZPBSV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentF07HNF (ZPBSV)

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

F07HNF (ZPBSV) computes the solution to a complex system of linear equations
 $AX=B ,$
where $A$ is an $n$ by $n$ Hermitian positive definite band matrix of bandwidth $\left(2{k}_{d}+1\right)$ and $X$ and $B$ are $n$ by $r$ matrices.

## 2  Specification

 SUBROUTINE F07HNF ( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, INFO)
 INTEGER N, KD, NRHS, LDAB, LDB, INFO COMPLEX (KIND=nag_wp) AB(LDAB,*), B(LDB,*) CHARACTER(1) UPLO
The routine may be called by its LAPACK name zpbsv.

## 3  Description

F07HNF (ZPBSV) uses the Cholesky decomposition to factor $A$ as $A={U}^{\mathrm{H}}U$ if ${\mathbf{UPLO}}=\text{'U'}$ or $A=L{L}^{\mathrm{H}}$ if ${\mathbf{UPLO}}=\text{'L'}$, where $U$ is an upper triangular band matrix, and $L$ is a lower triangular band matrix, with the same number of superdiagonals or subdiagonals as $A$. The factored form of $A$ is then used to solve the system of equations $AX=B$.

## 4  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

## 5  Parameters

1:     UPLO – CHARACTER(1)Input
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the upper triangle of $A$ is stored.
If ${\mathbf{UPLO}}=\text{'L'}$, the lower triangle of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
2:     N – INTEGERInput
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{N}}\ge 0$.
3:     KD – INTEGERInput
On entry: ${k}_{d}$, the number of superdiagonals of the matrix $A$ if ${\mathbf{UPLO}}=\text{'U'}$, or the number of subdiagonals if ${\mathbf{UPLO}}=\text{'L'}$.
Constraint: ${\mathbf{KD}}\ge 0$.
4:     NRHS – INTEGERInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrix $B$.
Constraint: ${\mathbf{NRHS}}\ge 0$.
5:     AB(LDAB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array AB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the upper or lower triangle of the Hermitian band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{d}+1$, more precisely,
• if ${\mathbf{UPLO}}=\text{'U'}$, the elements of the upper triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left({k}_{d}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{d}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{d}\right)\text{.}$
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the triangular factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{H}}U$ or $A=L{L}^{\mathrm{H}}$ of the band matrix $A$, in the same storage format as $A$.
6:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F07HNF (ZPBSV) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KD}}+1$.
7:     B(LDB,$*$) – COMPLEX (KIND=nag_wp) arrayInput/Output
Note: the second dimension of the array B must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NRHS}}\right)$.
On entry: the $n$ by $r$ right-hand side matrix $B$.
On exit: if ${\mathbf{INFO}}={\mathbf{0}}$, the $n$ by $r$ solution matrix $X$.
8:     LDB – INTEGERInput
On entry: the first dimension of the array B as declared in the (sub)program from which F07HNF (ZPBSV) is called.
Constraint: ${\mathbf{LDB}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
9:     INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

## 6  Error Indicators and Warnings

Errors or warnings detected by the routine:
${\mathbf{INFO}}<0$
If ${\mathbf{INFO}}=-i$, the $i$th argument had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{INFO}}>0$
If ${\mathbf{INFO}}=i$, the leading minor of order $i$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## 7  Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $A+E x^=b ,$
where
 $E1 = Oε A1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $x^-x1 x1 ≤ κA E1 A1 ,$
where $\kappa \left(A\right)={‖{A}^{-1}‖}_{1}{‖A‖}_{1}$, the condition number of $A$ with respect to the solution of the linear equations. See Section 4.4 of Anderson et al. (1999) for further details.
F07HPF (ZPBSVX) is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, F04CFF solves $Ax=b$ and returns a forward error bound and condition estimate. F04CFF calls F07HNF (ZPBSV) to solve the equations.

## 8  Further Comments

When $n\gg k$, the total number of floating point operations is approximately $4n{\left(k+1\right)}^{2}+16nkr$, where $k$ is the number of superdiagonals and $r$ is the number of right-hand sides.
The real analogue of this routine is F07HAF (DPBSV).

## 9  Example

This example solves the equations
 $Ax=b ,$
where $A$ is the Hermitian positive definite band matrix
 $A = 9.39i+0.00 1.08-1.73i 0.00i+0.00 0.00i+0.00 1.08+1.73i 1.69i+0.00 -0.04+0.29i 0.00i+0.00 0.00i+0.00 -0.04-0.29i 2.65i+0.00 -0.33+2.24i 0.00i+0.00 0.00i+0.00 -0.33-2.24i 2.17i+0.00$
and
 $b = -12.42+68.42i -9.93+00.88i -27.30-00.01i 5.31+23.63i .$
Details of the Cholesky factorization of $A$ are also output.

### 9.1  Program Text

Program Text (f07hnfe.f90)

### 9.2  Program Data

Program Data (f07hnfe.d)

### 9.3  Program Results

Program Results (f07hnfe.r)

F07HNF (ZPBSV) (PDF version)
F07 Chapter Contents
F07 Chapter Introduction
NAG Library Manual