F08 Chapter Contents
F08 Chapter Introduction
NAG Library Manual

NAG Library Routine DocumentF08UEF (DSBGST)

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

F08UEF (DSBGST) reduces a real symmetric-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$ and $B$ are band matrices, $A$ is a real symmetric matrix, and $B$ has been factorized by F08UFF (DPBSTF).

2  Specification

 SUBROUTINE F08UEF ( VECT, UPLO, N, KA, KB, AB, LDAB, BB, LDBB, X, LDX, WORK, INFO)
 INTEGER N, KA, KB, LDAB, LDBB, LDX, INFO REAL (KIND=nag_wp) AB(LDAB,*), BB(LDBB,*), X(LDX,*), WORK(2*N) CHARACTER(1) VECT, UPLO
The routine may be called by its LAPACK name dsbgst.

3  Description

To reduce the real symmetric-definite generalized eigenproblem $Az=\lambda Bz$ to the standard form $Cy=\lambda y$, where $A$, $B$ and $C$ are banded, F08UEF (DSBGST) must be preceded by a call to F08UFF (DPBSTF) which computes the split Cholesky factorization of the positive definite matrix $B$: $B={S}^{\mathrm{T}}S$. The split Cholesky factorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.
This routine overwrites $A$ with $C={X}^{\mathrm{T}}AX$, where $X={S}^{-1}Q$ and $Q$ is a orthogonal matrix chosen (implicitly) to preserve the bandwidth of $A$. The routine also has an option to allow the accumulation of $X$, and then, if $z$ is an eigenvector of $C$, $Xz$ is an eigenvector of the original system.

4  References

Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44
Kaufman L (1984) Banded eigenvalue solvers on vector machines ACM Trans. Math. Software 10 73–86

5  Parameters

1:     VECT – CHARACTER(1)Input
On entry: indicates whether $X$ is to be returned.
${\mathbf{VECT}}=\text{'N'}$
$X$ is not returned.
${\mathbf{VECT}}=\text{'V'}$
$X$ is returned.
Constraint: ${\mathbf{VECT}}=\text{'N'}$ or $\text{'V'}$.
2:     UPLO – CHARACTER(1)Input
On entry: indicates whether the upper or lower triangular part of $A$ is stored.
${\mathbf{UPLO}}=\text{'U'}$
The upper triangular part of $A$ is stored.
${\mathbf{UPLO}}=\text{'L'}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{UPLO}}=\text{'U'}$ or $\text{'L'}$.
3:     N – INTEGERInput
On entry: $n$, the order of the matrices $A$ and $B$.
Constraint: ${\mathbf{N}}\ge 0$.
4:     KA – INTEGERInput
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the number of superdiagonals, ${k}_{a}$, of the matrix $A$.
If ${\mathbf{UPLO}}=\text{'L'}$, the number of subdiagonals, ${k}_{a}$, of the matrix $A$.
Constraint: ${\mathbf{KA}}\ge 0$.
5:     KB – INTEGERInput
On entry: if ${\mathbf{UPLO}}=\text{'U'}$, the number of superdiagonals, ${k}_{b}$, of the matrix $B$.
If ${\mathbf{UPLO}}=\text{'L'}$, the number of subdiagonals, ${k}_{b}$, of the matrix $B$.
Constraint: ${\mathbf{KA}}\ge {\mathbf{KB}}\ge 0$.
6:     AB(LDAB,$*$) – 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 upper or lower triangle of the $n$ by $n$ symmetric band matrix $A$.
The matrix is stored in rows $1$ to ${k}_{a}+1$, 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 ${\mathbf{AB}}\left({k}_{a}+1+i-j,j\right)\text{​ for ​}\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,j-{k}_{a}\right)\le i\le j$;
• if ${\mathbf{UPLO}}=\text{'L'}$, the elements of the lower triangle of $A$ within the band must be stored with element ${A}_{ij}$ in ${\mathbf{AB}}\left(1+i-j,j\right)\text{​ for ​}j\le i\le \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(n,j+{k}_{a}\right)\text{.}$
On exit: the upper or lower triangle of AB is overwritten by the corresponding upper or lower triangle of $C$ as specified by UPLO.
7:     LDAB – INTEGERInput
On entry: the first dimension of the array AB as declared in the (sub)program from which F08UEF (DSBGST) is called.
Constraint: ${\mathbf{LDAB}}\ge {\mathbf{KA}}+1$.
8:     BB(LDBB,$*$) – REAL (KIND=nag_wp) arrayInput
Note: the second dimension of the array BB must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$.
On entry: the banded split Cholesky factor of $B$ as specified by UPLO, N and KB and returned by F08UFF (DPBSTF).
9:     LDBB – INTEGERInput
On entry: the first dimension of the array BB as declared in the (sub)program from which F08UEF (DSBGST) is called.
Constraint: ${\mathbf{LDBB}}\ge {\mathbf{KB}}+1$.
10:   X(LDX,$*$) – REAL (KIND=nag_wp) arrayOutput
Note: the second dimension of the array X must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$ if ${\mathbf{VECT}}=\text{'V'}$ and at least $1$ if ${\mathbf{VECT}}=\text{'N'}$.
On exit: the $n$ by $n$ matrix $X={S}^{-1}Q$, if ${\mathbf{VECT}}=\text{'V'}$.
If ${\mathbf{VECT}}=\text{'N'}$, X is not referenced.
11:   LDX – INTEGERInput
On entry: the first dimension of the array X as declared in the (sub)program from which F08UEF (DSBGST) is called.
Constraints:
• if ${\mathbf{VECT}}=\text{'V'}$, ${\mathbf{LDX}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{N}}\right)$;
• if ${\mathbf{VECT}}=\text{'N'}$, ${\mathbf{LDX}}\ge 1$.
12:   WORK($2×{\mathbf{N}}$) – REAL (KIND=nag_wp) arrayWorkspace
13:   INFO – INTEGEROutput
On exit: ${\mathbf{INFO}}=0$ unless the routine detects an error (see Section 6).

6  Error 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.

7  Accuracy

Forming the reduced matrix $C$ is a stable procedure. However it involves implicit multiplication by ${B}^{-1}$. When F08UEF (DSBGST) is used as a step in the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of accuracy if $B$ is ill-conditioned with respect to inversion.

The total number of floating point operations is approximately $6{n}^{2}{k}_{B}$, when ${\mathbf{VECT}}=\text{'N'}$, assuming $n\gg {k}_{A},{k}_{B}$; there are an additional $\left(3/2\right){n}^{3}\left({k}_{B}/{k}_{A}\right)$ operations when ${\mathbf{VECT}}=\text{'V'}$.
The complex analogue of this routine is F08USF (ZHBGST).

9  Example

This example computes all the eigenvalues of $Az=\lambda Bz$, where
 $A = 0.24 0.39 0.42 0.00 0.39 -0.11 0.79 0.63 0.42 0.79 -0.25 0.48 0.00 0.63 0.48 -0.03 and B= 2.07 0.95 0.00 0.00 0.95 1.69 -0.29 0.00 0.00 -0.29 0.65 -0.33 0.00 0.00 -0.33 1.17 .$
Here $A$ is symmetric, $B$ is symmetric positive definite, and $A$ and $B$ are treated as band matrices. $B$ must first be factorized by F08UFF (DPBSTF). The program calls F08UEF (DSBGST) to reduce the problem to the standard form $Cy=\lambda y$, then F08HEF (DSBTRD) to reduce $C$ to tridiagonal form, and F08JFF (DSTERF) to compute the eigenvalues.

9.1  Program Text

Program Text (f08uefe.f90)

9.2  Program Data

Program Data (f08uefe.d)

9.3  Program Results

Program Results (f08uefe.r)