NAG Library Routine Document
F04BFF
1 Purpose
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.
2 Specification
SUBROUTINE F04BFF ( 
UPLO, N, KD, NRHS, AB, LDAB, B, LDB, RCOND, ERRBND, IFAIL) 
INTEGER 
N, KD, NRHS, LDAB, LDB, IFAIL 
REAL (KIND=nag_wp) 
AB(LDAB,*), B(LDB,*), RCOND, ERRBND 
CHARACTER(1) 
UPLO 

3 Description
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 $AX=B$.
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
Higham N J (2002) Accuracy and Stability of Numerical Algorithms (2nd Edition) SIAM, Philadelphia
5 Parameters
 1: $\mathrm{UPLO}$ – 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: $\mathrm{N}$ – INTEGERInput

On entry: the number of linear equations $n$, i.e., the order of the matrix $A$.
Constraint:
${\mathbf{N}}\ge 0$.
 3: $\mathrm{KD}$ – INTEGERInput

On entry: the number of superdiagonals $k$ (and the number of subdiagonals) of the band matrix $A$.
Constraint:
${\mathbf{KD}}\ge 0$.
 4: $\mathrm{NRHS}$ – INTEGERInput

On entry: the number of righthand sides $r$, i.e., the number of columns of the matrix $B$.
Constraint:
${\mathbf{NRHS}}\ge 0$.
 5: $\mathrm{AB}\left({\mathbf{LDAB}},*\right)$ – 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 ${\mathbf{AB}}\left(k+1+ij,j\right)\text{ for}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,jk\right)\le i\le j$;
 if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+ij,j\right)\text{ for}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+k\right)\text{.}$
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: $\mathrm{LDAB}$ – 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: $\mathrm{B}\left({\mathbf{LDB}},*\right)$ – 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 righthand sides $B$.
On exit: if ${\mathbf{IFAIL}}={\mathbf{0}}$ or ${\mathbf{N}+{\mathbf{1}}}$, the $n$ by $r$ solution matrix $X$.
 8: $\mathrm{LDB}$ – 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: $\mathrm{RCOND}$ – 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 ${\mathbf{RCOND}}=1/\left({\Vert A\Vert}_{1}{\Vert {A}^{1}\Vert}_{1}\right)$.
 10: $\mathrm{ERRBND}$ – 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
$\hat{x}$, such that
${\Vert \hat{x}x\Vert}_{1}/{\Vert x\Vert}_{1}\le {\mathbf{ERRBND}}$, where
$\hat{x}$ 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, then
ERRBND is returned as unity.
 11: $\mathrm{IFAIL}$ – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction 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 parameter, 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).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}<0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{IFAIL}}\ne 999$

If ${\mathbf{IFAIL}}=i$, the $i$th argument had an illegal value.
 ${\mathbf{IFAIL}}>0\hspace{0.17em}\text{and}\hspace{0.17em}{\mathbf{IFAIL}}\le {\mathbf{N}}$

If ${\mathbf{IFAIL}}=i$, the leading minor of order $i$ of $A$ is not positive definite. The factorization could not be completed, and the solution has not been computed.
 ${\mathbf{IFAIL}}={\mathbf{N}}+1$

RCOND is less than
machine precision, so that the matrix
$A$ is numerically singular. A solution to the equations
$AX=B$ has nevertheless been computed.
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
7 Accuracy
The computed solution for a single righthand side,
$\hat{x}$, satisfies an equation of the form
where
and
$\epsilon $ is the
machine precision. An approximate error bound for the computed solution is given by
where
$\kappa \left(A\right)={\Vert {A}^{1}\Vert}_{1}{\Vert A\Vert}_{1}$, the condition number of
$A$ with respect to the solution of the linear equations. F04BFF uses the approximation
${\Vert E\Vert}_{1}=\epsilon {\Vert A\Vert}_{1}$ to estimate
ERRBND. See Section 4.4 of
Anderson et al. (1999)
for further details.
8 Parallelism 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 implementationspecific information.
The integer allocatable memory required is
N, and the real allocatable memory required is
$3\times {\mathbf{N}}$. Allocation failed before the solution could be computed.
The band storage scheme for the array
AB is illustrated by the following example, when
$n=6$,
$k=2$, and
${\mathbf{UPLO}}=\text{'U'}$:
Similarly, if
${\mathbf{UPLO}}=\text{'L'}$ the format of
AB is as follows:
Array elements marked $*$ need not be set and are not referenced by the routine.
Assuming that $n\gg k$, the total number of floatingpoint 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.
10 Example
This example solves the equations
where
$A$ is the symmetric positive definite band matrix
An estimate of the condition number of $A$ and an approximate error bound for the computed solutions are also printed.
10.1 Program Text
Program Text (f04bffe.f90)
10.2 Program Data
Program Data (f04bffe.d)
10.3 Program Results
Program Results (f04bffe.r)