# NAG FL Interfacef07haf (dpbsv)

## ▸▿ Contents

Settings help

FL Name Style:

FL Specification Language:

## 1Purpose

f07haf computes the solution to a real system of linear equations
 $AX=B ,$
where $A$ is an $n×n$ symmetric positive definite band matrix of bandwidth $\left(2{k}_{d}+1\right)$ and $X$ and $B$ are $n×r$ matrices.

## 2Specification

Fortran Interface
 Subroutine f07haf ( uplo, n, kd, nrhs, ab, ldab, b, ldb, info)
 Integer, Intent (In) :: n, kd, nrhs, ldab, ldb Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), b(ldb,*) Character (1), Intent (In) :: uplo
#include <nag.h>
 void f07haf_ (const char *uplo, const Integer *n, const Integer *kd, const Integer *nrhs, double ab[], const Integer *ldab, double b[], const Integer *ldb, Integer *info, const Charlen length_uplo)
The routine may be called by the names f07haf, nagf_lapacklin_dpbsv or its LAPACK name dpbsv.

## 3Description

f07haf uses the Cholesky decomposition to factor $A$ as $A={U}^{\mathrm{T}}U$ if ${\mathbf{uplo}}=\text{'U'}$ or $A=L{L}^{\mathrm{T}}$ 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 $AX=B$.

## 4References

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 https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{uplo}$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: $\mathbf{n}$Integer Input
On entry: $n$, the number of linear equations, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3: $\mathbf{kd}$Integer Input
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: $\mathbf{nrhs}$Integer Input
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: $\mathbf{ab}\left({\mathbf{ldab}},*\right)$Real (Kind=nag_wp) array Input/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 symmetric 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 $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$ of the band matrix $A$, in the same storage format as $A$.
6: $\mathbf{ldab}$Integer Input
On entry: the first dimension of the array ab as declared in the (sub)program from which f07haf is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/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 $n×r$ right-hand side matrix $B$.
On exit: if ${\mathbf{info}}={\mathbf{0}}$, the $n×r$ solution matrix $X$.
8: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f07haf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
The leading minor of order $⟨\mathit{\text{value}}⟩$ of $A$ is not positive definite, so the factorization could not be completed, and the solution has not been computed.

## 7Accuracy

The computed solution for a single right-hand side, $\stackrel{^}{x}$, satisfies an equation of the form
 $(A+E) x^=b ,$
where
 $‖E‖1 = O(ε) ‖A‖1$
and $\epsilon$ is the machine precision. An approximate error bound for the computed solution is given by
 $‖x^-x‖1 ‖x‖1 ≤ κ(A) ‖E‖1 ‖A‖1 ,$
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.
f07hbf is a comprehensive LAPACK driver that returns forward and backward error bounds and an estimate of the condition number. Alternatively, f04bff solves $Ax=b$ and returns a forward error bound and condition estimate. f04bff calls f07haf to solve the equations.

## 8Parallelism and Performance

f07haf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f07haf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

When $n\gg k$, the total number of floating-point operations is approximately $n{\left(k+1\right)}^{2}+4nkr$, where $k$ is the number of superdiagonals and $r$ is the number of right-hand sides.
The complex analogue of this routine is f07hnf.

## 10Example

This example solves the equations
 $Ax=b ,$
where $A$ is the symmetric positive definite band matrix
 $A = ( 5.49 2.68 0.00 0.00 2.68 5.63 -2.39 0.00 0.00 -2.39 2.60 -2.22 0.00 0.00 -2.22 5.17 ) and b = ( 22.09 9.31 -5.24 11.83 ) .$
Details of the Cholesky factorization of $A$ are also output.

### 10.1Program Text

Program Text (f07hafe.f90)

### 10.2Program Data

Program Data (f07hafe.d)

### 10.3Program Results

Program Results (f07hafe.r)