Equilibration Scale Factors for a Symmetric Positive Definite Matrix in Packed Storage

Calling Sequence

CALL DPPEQU	(UPLO, N, DA, DSCALE, DSCOND, DAMAX, INFO)
CALL SPPEQU	(UPLO, N, SA, SSCALE, SSCOND, SAMAX, INFO)
CALL ZPPEQU	(UPLO, N, ZA, DSCALE, DSCOND, DAMAX, INFO)
CALL CPPEQU	(UPLO, N, CA, SSCALE, SSCOND, SAMAX, INFO)
void dppequ	(char uplo, int n, double *da, double *dscale, double *scond, double *damax, int *info)
void sppequ	(char uplo, int n, float *sa, float *sscale, float *scond, float *samax, int *info)
void zppequ	(char uplo, int n, doublecomplex *za, double *dscale, double *scond, double *damax, int *info)
void cppequ	(char uplo, int n, complex *ca, float *sscale, float *scond, float *samax, int *info)

Arguments

UPLO	Indicates whether xA contains the upper or lower triangle of the matrix. The legal values for UPLO are listed below. Any values not listed below are illegal.
	'U' or 'u'	xA contains the upper triangle.
	'L' or 'l'	xA contains the lower triangle.
N	Order of the matrix A. N 0.
xA	Upper or lower triangle of the matrix A. The dimension of xA is (N × N + N) / 2.
xSCALE	On exit, the N scale factors for A.
xSCOND	On exit, the ratio of the smallest SCALE(i) to the largest SCALE(i). If SCOND 0.1 and AMAX is not close to overflow or underflow, it is not worth scaling by SCALE.
xAMAX	On exit, the absolute value of the largest element of A. If AMAX is very close to overflow or underflow, A should be scaled regardless of the value of SCOND.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	The ith diagonal entry, where i = INFO, is non-positive.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

INTEGER LDA, LDSCAL, N

PARAMETER (N = 4)

PARAMETER (LDA = (N * (N + 1)) / 2)

PARAMETER (LDSCAL = N)

C

DOUBLE PRECISION A(LDA), AMAX, SCALE(N), SCOND

INTEGER ICOL, INFO, IROW

C

EXTERNAL DPPEQU

INTRINSIC ABS

C

C Initialize the array A to store in symmetric form the upper

C triangle of the 4x4 symmetric positive definite matrix A

C shown below.

C

C 4096 -2 0 0

C A = -2 256 -2 0

C 0 -2 16 -2

C 0 0 -2 1

C

DATA A / 4.096D3, -2.0D0, 2.56D2, 0.0D0, -2.0D0, 1.6D1,

$ 0.0D0, 0.0D0, -2.0D0, 1.0D0 /

C

C Print the initial values of the arrays.

C

PRINT 1000

PRINT 1010, A(1), A(2), A(4), A(7)

PRINT 1010, A(2), A(3), A(5), A(8)

PRINT 1010, A(4), A(5), A(6), A(9)

PRINT 1010, A(7), A(8), A(9), A(10)

C

C Compute the scale factors to use to equilibrate A.

C

CALL DPPEQU ('UPPER TRIANGLE OF A STORED', N, A, SCALE,

$ SCOND, AMAX, INFO)

IF (INFO .NE. 0) THEN

PRINT 1030, INFO

STOP 1

END IF

C

C Apply the scale factors to A then print the result.

C

DO 210, ICOL = 1, N

DO 200, IROW = 1, ICOL

A(IROW + (ICOL - 1) * ICOL / 2) =

$ A(IROW + (ICOL - 1) * ICOL / 2) * SCALE(IROW) *

$ SCALE(ICOL)

200 CONTINUE

210 CONTINUE

PRINT 1040

PRINT 1050, SCALE

PRINT 1060

PRINT 1010, A(1), A(2), A(4), A(7)

PRINT 1010, A(2), A(3), A(5), A(8)

PRINT 1010, A(4), A(5), A(6), A(9)

PRINT 1010, A(7), A(8), A(9), A(10)

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (4(3X, F10.5))

1020 FORMAT (/1X, 'A in symmetric form: (* in unused elements)')

1030 FORMAT (1X, 'Error computing scale factors for A, INFO =',

$ 1X, I5)

1040 FORMAT (/1X, 'Scale factors:')

1050 FORMAT (3X, F8.6)

1060 FORMAT (/1X, 'Equilibrated A:')

C

END

Sample Output

A:
4096.00000 -2.00000 0.00000 0.00000
-2.00000 256.00000 -2.00000 0.00000
0.00000 -2.00000 16.00000 -2.00000
0.00000 0.00000 -2.00000 1.00000
Scale factors:
0.015625
0.062500
0.250000
1.000000
Equilibrated A:
1.00000 -0.00195 0.00000 0.00000
-0.00195 1.00000 -0.03125 0.00000
0.00000 -0.03125 1.00000 -0.50000
0.00000 0.00000 -0.50000 1.00000