# NAG Library Routine Document

## 1Purpose

f04bff computes the solution to a real system of linear equations $AX=B$, where $A$ is an $n$ by $n$ symmetric positive definite band matrix of band width $2k+1$, and $X$ and $B$ are $n$ by $r$ matrices. An estimate of the condition number of $A$ and an error bound for the computed solution are also returned.

## 2Specification

Fortran Interface
 Subroutine f04bff ( uplo, n, kd, nrhs, ab, ldab, b, ldb,
 Integer, Intent (In) :: n, kd, nrhs, ldab, ldb Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (Inout) :: ab(ldab,*), b(ldb,*) Real (Kind=nag_wp), Intent (Out) :: rcond, errbnd Character (1), Intent (In) :: uplo
#include nagmk26.h
 void f04bff_ (const char *uplo, const Integer *n, const Integer *kd, const Integer *nrhs, double ab[], const Integer *ldab, double b[], const Integer *ldb, double *rcond, double *errbnd, Integer *ifail, const Charlen length_uplo)

## 3Description

The Cholesky factorization is used 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 with $k$ superdiagonals, and $L$ is a lower triangular band matrix with $k$ subdiagonals. 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 http://www.netlib.org/lapack/lug
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia

## 5Arguments

1:     $\mathbf{uplo}$ – Character(1)Input
On entry: if ${\mathbf{uplo}}=\text{'U'}$, the upper triangle of the matrix $A$ is stored.
If ${\mathbf{uplo}}=\text{'L'}$, the lower triangle of the matrix $A$ is stored.
Constraint: ${\mathbf{uplo}}=\text{'U'}$ or $\text{'L'}$.
2:     $\mathbf{n}$ – IntegerInput
On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{kd}$ – IntegerInput
On entry: the number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.
Constraint: ${\mathbf{kd}}\ge 0$.
4:     $\mathbf{nrhs}$ – IntegerInput
On entry: the number of right-hand sides $r$, 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) 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 $n$ by $n$ symmetric band matrix $A$. The upper or lower triangular part of the symmetric matrix is stored in the first ${\mathbf{kd}}+1$ rows of the array. The $j$th column of $A$ is stored in the $j$th column of the array ab as follows:
The matrix is stored in rows $1$ to $k+1$, 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 ${\mathbf{ab}}\left(k+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-k\right)\le i\le j$;
• if ${\mathbf{uplo}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{ab}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)\text{.}$
See Section 9 below for further details.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the factor $U$ or $L$ from the Cholesky factorization $A={U}^{\mathrm{T}}U$ or $A=L{L}^{\mathrm{T}}$, in the same storage format as $A$.
6:     $\mathbf{ldab}$ – IntegerInput
On entry: the first dimension of the array ab as declared in the (sub)program from which f04bff is called.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
7:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (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 $r$ matrix of right-hand sides $B$.
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, the $n$ by $r$ solution matrix $X$.
8:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f04bff is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
9:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the reciprocal of the condition number of the matrix $A$, computed as ${\mathbf{rcond}}=1/\left({‖A‖}_{1}{‖{A}^{-1}‖}_{1}\right)$.
10:   $\mathbf{errbnd}$ – Real (Kind=nag_wp)Output
On exit: if ${\mathbf{ifail}}={\mathbf{0}}$ or $\mathbf{n}+{\mathbf{1}}$, an estimate of the forward error bound for a computed solution $\stackrel{^}{x}$, such that ${‖\stackrel{^}{x}-x‖}_{1}/{‖x‖}_{1}\le {\mathbf{errbnd}}$, where $\stackrel{^}{x}$ is a column of the computed solution returned in the array b and $x$ is the corresponding column of the exact solution $X$. If rcond is less than machine precision, errbnd is returned as unity.
11:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}>0 \text{and} {\mathbf{ifail}}\le {\mathbf{n}}$
The principal minor of order $〈\mathit{\text{value}}〉$ of the matrix $A$ is not positive definite. The factorization has not been completed and the solution could not be computed.
${\mathbf{ifail}}={\mathbf{n}}+1$
A solution has been computed, but rcond is less than machine precision so that the matrix $A$ is numerically singular.
${\mathbf{ifail}}=-1$
On entry, uplo not one of 'U' or 'u' or 'L' or 'l': ${\mathbf{uplo}}=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 0$.
${\mathbf{ifail}}=-3$
On entry, ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{kd}}\ge 0$.
${\mathbf{ifail}}=-4$
On entry, ${\mathbf{nrhs}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
${\mathbf{ifail}}=-6$
On entry, ${\mathbf{ldab}}=〈\mathit{\text{value}}〉$ and ${\mathbf{kd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldab}}\ge {\mathbf{kd}}+1$.
${\mathbf{ifail}}=-8$
On entry, ${\mathbf{ldb}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
The integer allocatable memory required is n, and the real allocatable memory required is $3×{\mathbf{n}}$. Allocation failed before the solution could be computed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

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. f04bff uses the approximation ${‖E‖}_{1}=\epsilon {‖A‖}_{1}$ to estimate errbnd. See Section 4.4 of Anderson et al. (1999) for further details.

## 8Parallelism and Performance

f04bff is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f04bff 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.

The band storage scheme for the array ab is illustrated by the following example, when $n=6$, $k=2$, and ${\mathbf{uplo}}=\text{'U'}$:
On entry:
 $* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66$
On exit:
 $* * u13 u24 u35 u46 * u12 u23 u34 u45 u56 u11 u22 u33 u44 u55 u66$
Similarly, if ${\mathbf{uplo}}=\text{'L'}$ the format of ab is as follows:
On entry:
 $a11 a22 a33 a44 a55 a66 a21 a32 a43 a54 a65 * a31 a42 a53 a64 * *$
On exit:
 $l11 l22 l33 l44 l55 l66 l21 l32 l43 l54 l65 * l31 l42 l53 l64 * *$
Array elements marked $*$ need not be set and are not referenced by the routine.
Assuming that $n\gg k$, the total number of floating-point operations required to solve the equations $AX=B$ is approximately ${n\left(k+1\right)}^{2}$ for the factorization and $4nkr$ 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 complex analogue of f04bff is f04cff.

## 10Example

This example solves the equations
 $AX=B,$
where $A$ is the symmetric positive definite band matrix
 $A= 5.49 2.68 0 0 2.68 5.63 -2.39 0 0 -2.39 2.60 -2.22 0 0 -2.22 5.17 and B= 22.09 5.10 9.31 30.81 -5.24 -25.82 11.83 22.90 .$
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.

### 10.1Program Text

Program Text (f04bffe.f90)

### 10.2Program Data

Program Data (f04bffe.d)

### 10.3Program Results

Program Results (f04bffe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017