Developer Reference for Intel® oneAPI Math Kernel Library - C

Developer Reference

Contents

?spgst

Reduces a real symmetric-definite generalized eigenvalue problem to the standard form using packed storage.

Syntax

lapack_int
LAPACKE_sspgst
(
int
matrix_layout
,
lapack_int
itype
,
char
uplo
,
lapack_int
n
,
float
*
ap
,
const
float
*
bp
);
lapack_int
LAPACKE_dspgst
(
int
matrix_layout
,
lapack_int
itype
,
char
uplo
,
lapack_int
n
,
double
*
ap
,
const
double
*
bp
);
Include Files
  • mkl.h
Description
The routine reduces real symmetric-definite generalized eigenproblems
A
*
x
 =
λ
*
B
*
x
,
A
*
B
*
x
 =
λ
*
x
, or
B
*
A
*
x
 =
λ
*
x
to the standard form
C
*
y
 =
λ
*
y
, using packed matrix storage. Here
A
 is a real symmetric matrix, and
B
 is a real symmetric positive-definite matrix. Before calling this routine, call
?pptrf
 to compute the Cholesky factorization:
B
 =
U
T
*U
 or
B
 =
L*L
T
.
Input Parameters
matrix_layout
Specifies whether matrix storage layout is row major (
LAPACK_ROW_MAJOR
) or column major (
LAPACK_COL_MAJOR
).
itype
Must be 1 or 2 or 3.
If
itype
 = 1
, the generalized eigenproblem is
A*z
 =
lambda
*B*z
for
uplo
 =
'U'
:
C
 = inv(
U
T
)*
A
*inv(
U
)
,
z
 = inv(
U
)*
y
;
for
uplo
 =
'L'
:
C
 = inv(
L
)*
A
*inv(
L
T
)
,
z
 = inv(
L
T
)*
y
.
If
itype
 = 2
, the generalized eigenproblem is
A
*
B
*
z
 =
lambda
*
z
for
uplo
 =
'U'
:
C
 =
U
*
A
*
U
T
,
z
 = inv(
U
)*
y
;
for
uplo
 =
'L'
:
C
 =
L
T
*
A
*
L
,
z
 = inv(
L
T
)*
y
.
If
itype
 = 3
, the generalized eigenproblem is
B*A
*
z
 =
lambda
*
z
for
uplo
 =
'U'
:
C
 =
U
*
A
*
U
T
,
z
 =
U
T
*
y
;
for
uplo
 =
'L'
:
C
 =
L
T
*
A
*
L
,
z
 =
L
*
y
.
uplo
Must be
'U'
 or
'L'
.
If
uplo
 =
'U'
,
ap
 stores the packed upper triangle of
A
;
you must supply
B
 in the factored form
B
 =
U
T
*U
.
If
uplo
 =
'L'
,
ap
 stores the packed lower triangle of
A
;
you must supply
B
 in the factored form
B
 =
L*L
T
.
n
The order of the matrices
A
 and
B
 (
n
≥
 0
).
ap
,
bp
Arrays:
ap
 contains the packed upper or lower triangle of
A
.
The dimension of
ap
 must be at least max(1,
n
*(
n
+1)/2).
bp
 contains the packed Cholesky factor of
B
 (as returned by
?pptrf
 with the same
uplo
 value).
The dimension of
bp
 must be at least max(1,
n
*(
n
+1)/2).
Output Parameters
ap
The upper or lower triangle of
A
 is overwritten by the upper or lower triangle of
C
, as specified by the arguments
itype
 and
uplo
.
Return Values
This function returns a value
info
.
If
info
=0
, the execution is successful.
If
info
 =
-i
, the
i
-th parameter had an illegal value.
Application Notes
Forming the reduced matrix
C
 is a stable procedure. However, it involves implicit multiplication by
inv(
B
)
 (if
itype
 = 1
) or
B
 (if
itype
 = 2
 or 3). When the routine 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 approximate number of floating-point operations is
n
3
.

Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.