NAG Library Routine Document
f01buf (real_symm_posdef_fac)
1
Purpose
f01buf performs a $ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}$ decomposition of a real symmetric positive definite band matrix.
2
Specification
Fortran Interface
Integer, Intent (In)  ::  n, m1, k, lda  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (Inout)  ::  a(lda,n)  Real (Kind=nag_wp), Intent (Out)  ::  w(m1) 

C Header Interface
#include nagmk26.h
void 
f01buf_ (const Integer *n, const Integer *m1, const Integer *k, double a[], const Integer *lda, double w[], Integer *ifail) 

3
Description
The symmetric positive definite matrix $A$, of order $n$ and bandwidth $2m+1$, is divided into the leading principal submatrix of order $k$ and its complement, where $m\le k\le n$. A $UD{U}^{\mathrm{T}}$ decomposition of the latter and an $LD{L}^{\mathrm{T}}$ decomposition of the former are obtained by means of a sequence of elementary transformations, where $U$ is unit upper triangular, $L$ is unit lower triangular and $D$ is diagonal. Thus if $k=n$, an $LD{L}^{\mathrm{T}}$ decomposition of $A$ is obtained.
This routine is specifically designed to precede
f01bvf for the transformation of the symmetricdefinite eigenproblem
$Ax=\lambda Bx$ by the method of Crawford where
$A$ and
$B$ are of band form. In this context,
$k$ is chosen to be close to
$n/2$ and the decomposition is applied to the matrix
$B$.
4
References
Wilkinson J H (1965) The Algebraic Eigenvalue Problem Oxford University Press, Oxford
Wilkinson J H and Reinsch C (1971) Handbook for Automatic Computation II, Linear Algebra Springer–Verlag
5
Arguments
 1: $\mathbf{n}$ – IntegerInput

On entry: $n$, the order of the matrix $A$.
 2: $\mathbf{m1}$ – IntegerInput

On entry: $m+1$, where $m$ is the number of nonzero superdiagonals in $A$. Normally ${\mathbf{m1}}\ll {\mathbf{n}}$.
 3: $\mathbf{k}$ – IntegerInput

On entry: $k$, the changeover point in the decomposition.
Constraint:
${\mathbf{m1}}1\le {\mathbf{k}}\le {\mathbf{n}}$.
 4: $\mathbf{a}\left({\mathbf{lda}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the upper triangle of the
$n$ by
$n$ symmetric band matrix
$A$, with the diagonal of the matrix stored in the
$\left(m+1\right)$th row of the array, and the
$m$ superdiagonals within the band stored in the first
$m$ rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if
$n=6$ and
$m=2$, the storage scheme is
Elements in the top left corner of the array are not used. The following code assigns the matrix elements within the band to the correct elements of the array:
Do 20 j = 1, n
Do 10 i = max(1,jm1+1), j
a(ij+m1,j) = matrix(i,j)
10 Continue
20 Continue
On exit: $A$ is overwritten by the corresponding elements of $L$, $D$ and $U$.
 5: $\mathbf{lda}$ – IntegerInput

On entry: the first dimension of the array
a as declared in the (sub)program from which
f01buf is called.
Constraint:
${\mathbf{lda}}\ge {\mathbf{m1}}$.
 6: $\mathbf{w}\left({\mathbf{m1}}\right)$ – Real (Kind=nag_wp) arrayWorkspace

 7: $\mathbf{ifail}$ – 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).
6
Error 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:
 ${\mathbf{ifail}}=1$

On entry,  ${\mathbf{k}}<{\mathbf{m1}}1$ or ${\mathbf{k}}>{\mathbf{n}}$. 
 ${\mathbf{ifail}}=2$
 ${\mathbf{ifail}}=3$

The matrix
$A$ is not positive definite, perhaps as a result of rounding errors, giving an element of
$D$ which is zero or negative.
${\mathbf{ifail}}={\mathbf{3}}$ when the failure occurs in the leading principal submatrix of order
k and
${\mathbf{ifail}}={\mathbf{2}}$ when it occurs in the complement.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
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.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The Cholesky decomposition of a positive definite matrix is known for its remarkable numerical stability (see
Wilkinson (1965)). The computed
$U$,
$L$ and
$D$ satisfy the relation
$ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}=A+E$ where the
$2$norms of
$A$ and
$E$ are related by
$\Vert E\Vert \le c{\left(m+1\right)}^{2}\epsilon \Vert A\Vert $ where
$c$ is a constant of order unity and
$\epsilon $ is the
machine precision. In practice, the error is usually appreciably smaller than this.
8
Parallelism and Performance
f01buf is not threaded in any implementation.
The time taken by f01buf is approximately proportional to $n{m}^{2}+3nm$.
This routine is specifically designed for use as the first stage in the solution of the generalized symmetric eigenproblem $Ax=\lambda Bx$ by Crawford's method which preserves band form in the transformation to a similar standard problem. In this context, for maximum efficiency, $k$ should be chosen as the multiple of $m$ nearest to $n/2$.
The matrix $U$ is such that ${U}^{1}A{U}^{\mathrm{T}}$ is diagonal in its last $nk$ rows and columns, $L$ is such that ${L}^{1}{U}^{1}A{U}^{\mathrm{T}}{L}^{\mathrm{T}}=D$ and $D$ is diagonal. To find $U$, $L$ and $D$ where $A=ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}$ requires $nm\left(m+3\right)/2m\left(m+1\right)\left(m+2\right)/3$ multiplications and divisions which, is independent of $k$.
10
Example
This example finds a
$ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}$ decomposition of the real symmetric positive definite matrix
10.1
Program Text
Program Text (f01bufe.f90)
10.2
Program Data
Program Data (f01bufe.d)
10.3
Program Results
Program Results (f01bufe.r)