NAG Library Routine Document
F01BVF
1 Purpose
F01BVF transforms the generalized symmetricdefinite eigenproblem
$Ax=\lambda {\mathbf{B}}x$ to the equivalent standard eigenproblem
$Cy=\lambda y$, where
$A$,
${\mathbf{B}}$ and
$C$ are symmetric band matrices and
${\mathbf{B}}$ is positive definite.
${\mathbf{B}}$ must have been decomposed by
F01BUF.
2 Specification
SUBROUTINE F01BVF ( 
N, MA1, MB1, M3, K, A, LDA, B, LDB, V, LDV, W, IFAIL) 
INTEGER 
N, MA1, MB1, M3, K, LDA, LDB, LDV, IFAIL 
REAL (KIND=nag_wp) 
A(LDA,N), B(LDB,N), V(LDV,M3), W(M3) 

3 Description
$A$ is a symmetric band matrix of order
$n$ and bandwidth
$2{m}_{A}+1$. The positive definite symmetric band matrix
$B$, of order
$n$ and bandwidth
$2{m}_{B}+1$, must have been previously decomposed by
F01BUF as
$ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}$. F01BVF applies
$U$,
$L$ and
$D$ to
$A$,
${m}_{A}$ rows at a time, restoring the band form of
$A$ at each stage by plane rotations. The parameter
$k$ defines the changeover point in the decomposition of
$B$ as used by
F01BUF and is also used as a changeover point in the transformations applied by this routine. For maximum efficiency,
$k$ should be chosen to be the multiple of
${m}_{A}$ nearest to
$n/2$. The resulting symmetric band matrix
$C$ is overwritten on
A. The eigenvalues of
$C$, and thus of the original problem, may be found using
F08HEF (DSBTRD) and
F08JFF (DSTERF). For selected eigenvalues, use
F08HEF (DSBTRD) and
F08JJF (DSTEBZ).
4 References
Crawford C R (1973) Reduction of a bandsymmetric generalized eigenvalue problem Comm. ACM 16 41–44
5 Parameters
 1: N – INTEGERInput
On entry: $n$, the order of the matrices $A$, $B$ and $C$.
 2: MA1 – INTEGERInput
On entry: ${m}_{A}+1$, where ${m}_{A}$ is the number of nonzero superdiagonals in $A$. Normally ${\mathbf{MA1}}\ll {\mathbf{N}}$.
 3: MB1 – INTEGERInput
On entry: ${m}_{B}+1$, where ${m}_{B}$ is the number of nonzero superdiagonals in $B$.
Constraint:
${\mathbf{MB1}}\le {\mathbf{MA1}}$.
 4: M3 – INTEGERInput
On entry: the value of $3{m}_{A}+{m}_{B}$.
 5: K – INTEGERInput
On entry:
$k$, the changeover point in the transformations. It must be the same as the value used by
F01BUF in the decomposition of
$B$.
Suggested value:
the optimum value is the multiple of ${m}_{A}$ nearest to $n/2$.
Constraint:
${\mathbf{MB1}}1\le {\mathbf{K}}\le {\mathbf{N}}$.
 6: A(LDA,N) – 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
$\left({m}_{A}+1\right)$th row of the array, and the
${m}_{A}$ superdiagonals within the band stored in the first
${m}_{A}$ rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if
$n=6$ and
${m}_{A}=2$, the storage scheme is
Elements in the top left corner of the array need not be set. 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,JMA1+1), J
A(IJ+MA1,J) = matrix (I,J)
10 CONTINUE
20 CONTINUE
On exit: is overwritten by the corresponding elements of $C$.
 7: LDA – INTEGERInput
On entry: the first dimension of the array
A as declared in the (sub)program from which F01BVF is called.
Constraint:
${\mathbf{LDA}}\ge {\mathbf{MA1}}$.
 8: B(LDB,N) – REAL (KIND=nag_wp) arrayInput/Output
On entry: the elements of the decomposition of matrix
$B$ as returned by
F01BUF.
On exit: the elements of
B will have been permuted.
 9: LDB – INTEGERInput
On entry: the first dimension of the array
B as declared in the (sub)program from which F01BVF is called.
Constraint:
${\mathbf{LDB}}\ge {\mathbf{MB1}}$.
 10: V(LDV,M3) – REAL (KIND=nag_wp) arrayWorkspace
 11: LDV – INTEGERInput
On entry: the first dimension of the array
V as declared in the (sub)program from which F01BVF is called.
Constraint:
${\mathbf{LDV}}\ge {m}_{A}+{m}_{B}$.
 12: W(M3) – REAL (KIND=nag_wp) arrayWorkspace
 13: 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:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{MB1}}>{\mathbf{MA1}}$. 
7 Accuracy
In general the computed system is exactly congruent to a problem $\left(A+E\right)x=\lambda \left(B+F\right)x$, where $\Vert E\Vert $ and $\Vert F\Vert $ are of the order of $\epsilon \kappa \left(B\right)\Vert A\Vert $ and $\epsilon \kappa \left(B\right)\Vert B\Vert $ respectively, where $\kappa \left(B\right)$ is the condition number of $B$ with respect to inversion and $\epsilon $ is the machine precision. This means that when $B$ is positive definite but not wellconditioned with respect to inversion, the method, which effectively involves the inversion of $B$, may lead to a severe loss of accuracy in wellconditioned eigenvalues.
The time taken by F01BVF is approximately proportional to ${n}^{2}{m}_{B}^{2}$ and the distance of $k$ from $n/2$, e.g., $k=n/4$ and $k=3n/4$ take $502\%$ longer.
When $B$ is positive definite and wellconditioned with respect to inversion, the generalized symmetric eigenproblem can be reduced to the standard symmetric problem $Py=\lambda y$ where $P={L}^{1}A{L}^{\mathrm{T}}$ and $B=L{L}^{\mathrm{T}}$, the Cholesky factorization.
When
$A$ and
$B$ are of band form, especially if the bandwidth is small compared with the order of the matrices, storage considerations may rule out the possibility of working with
$P$ since it will be a full matrix in general. However, for any factorization of the form
$B=S{S}^{\mathrm{T}}$, the generalized symmetric problem reduces to the standard form
and there does exist a factorization such that
${S}^{1}A{S}^{\mathrm{T}}$ is still of band form (see
Crawford (1973)). Writing
the standard form is
$Cy=\lambda y$ and the bandwidth of
$C$ is the maximum bandwidth of
$A$ and
$B$.
Each stage in the transformation consists of two phases. The first reduces a leading principal submatrix of $B$ to the identity matrix and this introduces nonzero elements outside the band of $A$. In the second, further transformations are applied which leave the reduced part of $B$ unaltered and drive the extra elements upwards and off the top left corner of $A$. Alternatively, $B$ may be reduced to the identity matrix starting at the bottom righthand corner and the extra elements introduced in $A$ can be driven downwards.
The advantage of the $ULD{L}^{\mathrm{T}}{U}^{\mathrm{T}}$ decomposition of $B$ is that no extra elements have to be pushed over the whole length of $A$. If $k$ is taken as approximately $n/2$, the shifting is limited to halfway. At each stage the size of the triangular bumps produced in $A$ depends on the number of rows and columns of $B$ which are eliminated in the first phase and on the bandwidth of $B$. The number of rows and columns over which these triangles are moved at each step in the second phase is equal to the bandwidth of $A$.
In this routine,
A is defined as being at least as wide as
$B$ and must be filled out with zeros if necessary as it is overwritten with
$C$. The number of rows and columns of
$B$ which are effectively eliminated at each stage is
${m}_{A}$.
9 Example
This example finds the three smallest eigenvalues of
$Ax=\lambda Bx$, where
9.1 Program Text
Program Text (f01bvfe.f90)
9.2 Program Data
Program Data (f01bvfe.d)
9.3 Program Results
Program Results (f01bvfe.r)