hide long namesshow long names
hide short namesshow short names
Integer type:  int32  int64  nag_int  show int32  show int32  show int64  show int64  show nag_int  show nag_int

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_matop_real_symm_posdef_geneig (f01bv)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_matop_real_symm_posdef_geneig (f01bv) transforms the generalized symmetric-definite eigenproblem Ax=λbx to the equivalent standard eigenproblem Cy=λy, where A, b and C are symmetric band matrices and b is positive definite. b must have been decomposed by nag_matop_real_symm_posdef_fac (f01bu).


[a, b, ifail] = f01bv(k, a, b, 'n', n, 'ma1', ma1, 'mb1', mb1)
[a, b, ifail] = nag_matop_real_symm_posdef_geneig(k, a, b, 'n', n, 'ma1', ma1, 'mb1', mb1)
Note: the interface to this routine has changed since earlier releases of the toolbox:
At Mark 22: ma1 and mb1 were made optional


A is a symmetric band matrix of order n and bandwidth 2mA+1. The positive definite symmetric band matrix B, of order n and bandwidth 2mB+1, must have been previously decomposed by nag_matop_real_symm_posdef_fac (f01bu) as ULDLTUT. nag_matop_real_symm_posdef_geneig (f01bv) applies U, L and D to A, mA rows at a time, restoring the band form of A at each stage by plane rotations. The argument k defines the change-over point in the decomposition of B as used by nag_matop_real_symm_posdef_fac (f01bu) and is also used as a change-over point in the transformations applied by this function. For maximum efficiency, k should be chosen to be the multiple of mA 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 nag_lapack_dsbtrd (f08he) and nag_lapack_dsterf (f08jf). For selected eigenvalues, use nag_lapack_dsbtrd (f08he) and nag_lapack_dstebz (f08jj).


Crawford C R (1973) Reduction of a band-symmetric generalized eigenvalue problem Comm. ACM 16 41–44


Compulsory Input Parameters

1:     k int64int32nag_int scalar
Suggested value: the optimum value is the multiple of mA nearest to n/2.
k, the change-over point in the transformations. It must be the same as the value used by nag_matop_real_symm_posdef_fac (f01bu) in the decomposition of B.
Constraint: mb1-1kn.
2:     aldan – double array
lda, the first dimension of the array, must satisfy the constraint ldama1.
The upper triangle of the n by n symmetric band matrix A, with the diagonal of the matrix stored in the mA+1th row of the array, and the mA superdiagonals within the band stored in the first mA rows of the array. Each column of the matrix is stored in the corresponding column of the array. For example, if n=6 and mA=2, the storage scheme is
* * a13 a24 a35 a46 * a12 a23 a34 a45 a56 a11 a22 a33 a44 a55 a66  
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:
 for j=1:n
   for i=max(1,j-ma1+1):j
     a(i-j+ma1,j) = matrix(i,j);
3:     bldbn – double array
ldb, the first dimension of the array, must satisfy the constraint ldbmb1.
The elements of the decomposition of matrix B as returned by nag_matop_real_symm_posdef_fac (f01bu).

Optional Input Parameters

1:     n int64int32nag_int scalar
Default: the second dimension of the arrays a, b. (An error is raised if these dimensions are not equal.)
n, the order of the matrices A, B and C.
2:     ma1 int64int32nag_int scalar
Default: the first dimension of the array a.
mA+1, where mA is the number of nonzero superdiagonals in A. Normally ma1n.
3:     mb1 int64int32nag_int scalar
Default: the first dimension of the array b.
mB+1, where mB is the number of nonzero superdiagonals in B.
Constraint: mb1ma1.

Output Parameters

1:     aldan – double array
Stores the corresponding elements of C.
2:     bldbn – double array
The elements of b will have been permuted.
3:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,mb1>ma1.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


In general the computed system is exactly congruent to a problem A+Ex=λB+Fx, where E and F are of the order of εκBA and εκBB respectively, where κB is the condition number of B with respect to inversion and ε is the machine precision. This means that when B is positive definite but not well-conditioned with respect to inversion, the method, which effectively involves the inversion of B, may lead to a severe loss of accuracy in well-conditioned eigenvalues.

Further Comments

The time taken by nag_matop_real_symm_posdef_geneig (f01bv) is approximately proportional to n2mB2 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 well-conditioned with respect to inversion, the generalized symmetric eigenproblem can be reduced to the standard symmetric problem Py=λy where P=L-1AL-T and B=LLT, 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=SST, the generalized symmetric problem reduces to the standard form
and there does exist a factorization such that S-1AS-T is still of band form (see Crawford (1973)). Writing
C=S-1AS-T  and  y=STx  
the standard form is Cy=λy 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 sub-matrix 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 right-hand corner and the extra elements introduced in A can be driven downwards.
The advantage of the ULDLTUT 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 function, 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 mA.


This example finds the three smallest eigenvalues of Ax=λBx, where
A= 11 12 12 12 13 13 13 14 14 14 15 15 15 16 16 16 17 17 17 18 18 18 19 19 19  
B= 101 22 22 102 23 23 103 24 24 104 25 25 105 26 26 106 27 27 107 28 28 108 29 29 109 .  
function f01bv_example

fprintf('f01bv example results\n\n');

% The matrices A and B
n = 9;
kd = int64(1);
a = [  0,  12,  13,  14,  15,  16,  17,  18,  19;
      11,  12,  13,  14,  15,  16,  17,  18,  19];
b = [  0,  22,  23,  24,  25,  26,  27,  28,  29;
     101, 102, 103, 104, 105, 106, 107, 108, 109];

% Decompose B
k = int64(4);
[b, ifail] = f01bu(k, b);

% Use decomposition to transform Ax = lambda Bx to Cy = lambda y
[c, b, ifail] = f01bv(k, a, b);

% Reduce C to tridiagonal form
vect = 'No vectors';
uplo = 'Upper';
q = zeros(n, n);
[~, d, e, ~, info] = f08he( ...
                            vect, uplo, kd, c, q);

% Compute eigenvalues from tridiagonal form
m1 = int64(1);
m2 = int64(3);
range = 'Indexed';
order = 'Blocked';
vl = 0;
vu = 0;
abstol = 0;
[m, nsplit, w, iblock, isplit, info] = ...
f08jj( ...
       range, order, vl, vu, m1, m2, abstol, d, e);

disp('Selected eigenvalues');

f01bv example results

Selected eigenvalues
   -0.2643   -0.1530   -0.0418

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

© The Numerical Algorithms Group Ltd, Oxford, UK. 2009–2015