itype	Problem	uplo	$B$	$C$	$z$
$1$	$A z = λ B z$	'U' 'L'	$U^{T} U$ $L L^{T}$	$U^{- T} A U^{- 1}$ $L^{- 1} A L^{- T}$	$U^{- 1} y$ $L^{- T} y$
$2$	$A B z = λ z$	'U' 'L'	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{- 1} y$ $L^{- T} y$
$3$	$B A z = λ z$	'U' 'L'	$U^{T} U$ $L L^{T}$	$U A U^{T}$ $L^{T} A L$	$U^{T} y$ $L y$

References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

Parameters

Compulsory Input Parameters

1: $itype$ – int64int32nag_int scalar

Indicates how the standard form is computed.

$itype = 1$

if $uplo ='U'$ , $C = U^{- T} A U^{- 1}$ ;
if $uplo ='L'$ , $C = L^{- 1} A L^{- T}$ .

$itype = 2$ or $3$

if $uplo ='U'$ , $C = U A U^{T}$ ;
if $uplo ='L'$ , $C = L^{T} A L$ .

Constraint:

itype = 1

2

3

2: $uplo$ – string (length ≥ 1)

Indicates whether the upper or lower triangular part of

A

is stored and how

B

has been factorized.

$uplo ='U'$: The upper triangular part of $A$ is stored and $B = U^{T} U$ .
$uplo ='L'$: The lower triangular part of $A$ is stored and $B = L L^{T}$ .

Constraint:

uplo ='U'

'L'

3: $a (lda :)$ – double array

The first dimension of the array a must be at least

\max (1, n)

The second dimension of the array a must be at least

\max (1, n)

The

n

n

symmetric matrix

A

If $uplo ='U'$ , the upper triangular part of $a$ must be stored and the elements of the array below the diagonal are not referenced.
If $uplo ='L'$ , the lower triangular part of $a$ must be stored and the elements of the array above the diagonal are not referenced.

4: $b (ldb :)$ – double array

The first dimension of the array b must be at least

\max (1, n)

The second dimension of the array b must be at least

\max (1, n)

The Cholesky factor of

B

as specified by uplo and returned by nag_lapack_dpotrf (f07fd).

Optional Input Parameters

1: $n$ – int64int32nag_int scalar: Default: the first dimension of the arrays a, b and the second dimension of the arrays a, b.
$n$ , the order of the matrices $A$ and $B$ .

Constraint: $n \geq 0$ .

Output Parameters

1: $a (lda :)$ – double array: The first dimension of the array a will be $\max (1, n)$ .
The second dimension of the array a will be $\max (1, n)$ .

The upper or lower triangle of a stores the corresponding upper or lower triangle of $C$ as specified by itype and uplo.
2: $info$ – int64int32nag_int scalar: $info = 0$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

$info = - i$: If $info = - i$ , parameter $i$ had an illegal value on entry. The parameters are numbered as follows:
1: itype, 2: uplo, 3: n, 4: a, 5: lda, 6: b, 7: ldb, 8: info.
It is possible that info refers to a parameter that is omitted from the MATLAB interface. This usually indicates that an error in one of the other input parameters has caused an incorrect value to be inferred.

Accuracy

Forming the reduced matrix

C

is a stable procedure. However it involves implicit multiplication by

B^{- 1}

(if

itype = 1

) or

B

(if

itype = 2

3

). When nag_lapack_dsygst (f08se) 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. See the document for nag_lapack_dsygv (f08sa) for further details.

Further Comments

The total number of floating-point operations is approximately

n^{3}

The complex analogue of this function is nag_lapack_zhegst (f08ss).

Example

This example computes all the eigenvalues of

A z = λ B z

, where

A = (\begin{array}{r} 0.24 & 0.39 & 0.42 & - 0.16 \\ 0.39 & - 0.11 & 0.79 & 0.63 \\ 0.42 & 0.79 & - 0.25 & 0.48 \\ - 0.16 & 0.63 & 0.48 & - 0.03 \end{array}) and B = (\begin{array}{r} 4.16 & - 3.12 & 0.56 & - 0.10 \\ - 3.12 & 5.03 & - 0.83 & 1.09 \\ 0.56 & - 0.83 & 0.76 & 0.34 \\ - 0.10 & 1.09 & 0.34 & 1.18 \end{array}) .

Here

B

is symmetric positive definite and must first be factorized by nag_lapack_dpotrf (f07fd). The program calls nag_lapack_dsygst (f08se) to reduce the problem to the standard form

C y = λ y

; then nag_lapack_dsytrd (f08fe) to reduce

C

to tridiagonal form, and nag_lapack_dsterf (f08jf) to compute the eigenvalues.

Open in the MATLAB editor: f08se_example

function f08se_example


fprintf('f08se example results\n\n');

% Sove Az = lambda Bz
  % A and B are symmetric, B is positive definite:
a = [ 0.24,  0.39,  0.42, -0.16;
      0.39, -0.11,  0.79,  0.63;
      0.42,  0.79, -0.25,  0.48;
     -0.16,  0.63,  0.48, -0.03];
b = [ 4.16  -3.12   0.56  -0.10;
     -3.12   5.03  -0.83   1.09;
      0.56  -0.83   0.76   0.34;
     -0.10   1.09   0.34   1.18];

% Factorize B
uplo = 'L';
[bfac, info] = f07fd(uplo, b);

% Reduce problem to standard form Cy = lambda*y
itype = int64(1);
[c, info] = f08se( ...
		   itype, uplo, a, bfac);

% Find eigenvalues lambda
jobz = 'No Vectors';
[~, w, info] = f08fa( ...
		      jobz, uplo, c);

disp('Eigenvalues:');
disp(w);

f08se example results

Eigenvalues:
   -2.2254
   -0.4548
    0.1001
    1.1270

PDF version (NAG web site, 64-bit version, 64-bit version)

Chapter Contents

Chapter Introduction

NAG Toolbox