Eigenvalues and Eigenvectors of a Hermitian Matrix in Packed Storage (Expert Driver)

Calling Sequence

CALL ZHPEVX	(JOBZ, RANGE, UPLO, N, ZA, DVL, DVU, IL, IU, DABTOL, NFOUND, DW, ZZ, LDZ, ZWORK, DWORK2, IWORK3, IFAIL, INFO)
CALL CHPEVX	(JOBZ, RANGE, UPLO, N, CA, SVL, SVU, IL, IU, SABTOL, NFOUND, SW, CZ, LDZ, CWORK, SWORK2, IWORK3, IFAIL, INFO)
void zhpevx	(char jobz, char range, char uplo, int n, doublecomplex *za, double dvl, double dvu, int il, int iu, double dabtol, int *nfound, double *dw, doublecomplex *zz, int ldz, int *ifail, int *info)
void chpevx	(char jobz, char range, char uplo, int n, complex *ca, float svl, float svu, int il, int iu, float sabtol, int *nfound, float *sw, complex *cz, int ldz, int *ifail, int *info)

Arguments

JOBZ	Indicates whether to compute eigenvalues only or to compute both eigenvalues and eigenvectors. The legal values for JOBZ are listed below. Any values not listed below are illegal.
	'N' or 'n'	Compute eigenvalues only.
	'V' or 'v'	Compute both eigenvalues and eigenvectors.
RANGE	Indicates whether to compute all eigenvalues or some subset. Legal values for RANGE are shown below. Any values not shown below are illegal.
	'A' or 'a'	All eigenvalues will be computed.
	'I' or 'i'	The ILth through IUth eigenvalues will be computed; see the descriptions of theIL and IU arguments below.
	'V' or 'v'	All eigenvalues in the half-open interval (VL,VU] will be computed; see the descriptions of the VL and VU arguments below.
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	On entry, the upper or lower triangle of the matrix A. The dimension of xA is (N × N + N) / 2. On exit, A is overwritten.
xVL, xVU	If RANGE = 'V' or 'v' then only eigenvalues in the half-open interval (VL,VU] will be computed. Otherwise not used.
IL, IU	If RANGE = 'I' or 'i' then only the ILth through IUth eigenvalues will be computed. Otherwise not used. 1 IL IU N.
xABTOL	The absolute error tolerance for the eigenvalues. An approximate eigenvalue is considered to have converged when it lies in an interval [a,b] of width less than or equal to ABTOL + × max(|a|, |b|) where is machine precision. If ABTOL 0 then × ||T||1 will be used in its place, where ||T||1 is the 1-norm of the tridiagonal matrix obtained by reducing A to tridiagonal form.
NFOUND	On exit, the total number of eigenvalues found. 0 NFOUND N. If RANGE = 'A' or 'a' then NFOUND = N. If RANGE = 'I' or 'i' then NFOUND = IU - IL + 1.
xW	On exit, W(1:NFOUND) contains the computed eigenvalues in ascending order.
xZ	On exit, if JOBZ = 'V' or 'v' then the first NFOUND columns of Z contain the orthonormal eigenvectors corresponding to the selected eigenvalues. If an eigenvector fails to converge then the corresponding column of Z contains an approximation to that eigenvector and its index is returned in IFAIL. If JOBZ = 'N' or 'n' then Z is not used. Note that Z must have at least NFOUND columns. If RANGE = 'V' or 'v' then NFOUND may not be known in advance and an upper bound must be used. For A(LDZ,K) where NFOUND > K, the eigenvectors will be stored into memory with unpredictable results.
LDZ	Leading dimension of the array Z as specified in a dimension or type statement. LDZ 1. If JOBZ = 'V' or 'v' then LDZ max(1, N).
xWORK	Scratch array with a dimension of 2 × N.
xWORK2	Scratch array with a dimension of 7 × N.
IWORK3	Scratch array with a dimension of 5 × N.
IFAIL	On exit, an N-element array that indicates which eigenvectors that failed to converge. If JOBZ = 'V' or 'v' and INFO = 0 then all N elements contain zero. If INFO > 0 then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N' or 'n' then IFAIL is not used.
INFO	On exit:
	INFO = 0	Subroutine completed normally.
	INFO < 0	The ith argument, where i = |INFO|, had an illegal value.
	INFO > 0	There were i eigenvectors, where i = INFO, that failed to converge. Their indices are stored in IFAIL.

Sample Program

PROGRAM TEST

IMPLICIT NONE

C

DOUBLE PRECISION ZERO

INTEGER LDEVAL, LDEVEC, LDIWRK, LDWORK, LDWRK2

INTEGER LENGTA, N

PARAMETER (N = 3)

PARAMETER (LDEVAL = N)

PARAMETER (LDEVEC = N)

PARAMETER (LDIWRK = 5 * N)

PARAMETER (LDWORK = 2 * N)

PARAMETER (LDWRK2 = 7 * N)

PARAMETER (LENGTA = (N * N + N) / 2)

PARAMETER (ZERO = 0.0D0)

C

DOUBLE PRECISION EVALS(LDEVAL), TEMP, WORK2(LDWRK2)

COMPLEX*16 A(LENGTA), EVECS(LDEVEC,N), WORK(LDWORK)

INTEGER ICOL, IFAIL(N), INFO, IROW, ITEMP

INTEGER IWORK(LDIWRK), NFOUND

C

EXTERNAL ZHPEVX

INTRINSIC CONJG

C

C Initialize the array A to store the coefficient array A

C shown below.

C

C 4 4-4i 0

C A = 4+4i 4 4-4i

C 0 4+4i 4

C

c DATA A / (4.0D0,0.0D0), (4.0D0,-4.0D0), (4.0D0,0.0D0),

c $ (0.0D0,0.0D0), (4.0D0,-4.0D0), (4.0D0,0.0D0) /

DATA A / (4.0D0,8D8), (4.0D0,-4.0D0), (4.0D0,8D8),

$ (0.0D0,0.0D0), (4.0D0,-4.0D0), (4.0D0,8D8) /

C

C Print the initial value of A.

C

PRINT 1000

PRINT 1010, DBLE(A(1)), ZERO, A(2), A(4)

PRINT 1010, CONJG(A(2)), DBLE(A(3)), ZERO, A(5)

PRINT 1010, CONJG(A(4)), CONJG(A(5)), DBLE(A(6)), ZERO

C

C Compute the eigenvalues and eigenvectors.

C

CALL ZHPEVX ('VALUES AND VECTORS', 'ALL VALUES',

$ 'UPPER TRIANGLE OF A STORED', N, A, TEMP, TEMP,

$ ITEMP, ITEMP, 0.0D0, NFOUND, EVALS, EVECS,

$ LDEVEC, WORK, WORK2, IWORK, IFAIL, INFO)

IF (INFO .LT. 0) THEN

PRINT 1020, ABS(INFO)

STOP 1

ELSE IF (INFO .GT. 0) THEN

PRINT 1030, INFO

STOP 2

END IF

C

C Print the eigenvalues and eigenvectors.

C

PRINT 1040

DO 200, IROW = 1, N

PRINT 1050, EVALS(IROW), (EVECS(IROW,ICOL), ICOL = 1, N)

200 CONTINUE

C

1000 FORMAT (1X, 'A:')

1010 FORMAT (1X, 10(: 2X, '(', F5.1, ',', F5.1, ')'))

1020 FORMAT (1X, 'Illegal argument to ZHPEVX, argument #', I2)

1030 FORMAT (1X, 'Convergence failure, INFO = ', I2)

1040 FORMAT (/1X, 'Eigenvalue', 16X, 'Eigenvector**T')

1050 FORMAT (1X, F8.3, 6X, '[', 3('(', F4.1, ',', F4.1, ') '),

$ ']')

C

END

Sample Output

A:

( 4.0, 0.0) ( 4.0, -4.0) ( 0.0, 0.0)

( 4.0, 4.0) ( 4.0, 0.0) ( 4.0, -4.0)

( 0.0, 0.0) ( 4.0, 4.0) ( 4.0, 0.0)

Eigenvalue Eigenvector**T

-4.000 [( 0.0, 0.5) ( 0.0, 0.7) ( 0.0, 0.5) ]

4.000 [( 0.5,-0.5) ( 0.0, 0.0) (-0.5, 0.5) ]

12.000 [(-0.5, 0.0) ( 0.7, 0.0) (-0.5, 0.0) ]