 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ ssygvx()

 subroutine ssygvx ( integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real, dimension( ldb, * ) B, integer LDB, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

SSYGVX

Purpose:
``` SSYGVX computes selected eigenvalues, and optionally, eigenvectors
of a real generalized symmetric-definite eigenproblem, of the form
A*x=(lambda)*B*x,  A*Bx=(lambda)*x,  or B*A*x=(lambda)*x.  Here A
and B are assumed to be symmetric and B is also positive definite.
Eigenvalues and eigenvectors can be selected by specifying either a
range of values or a range of indices for the desired eigenvalues.```
Parameters
 [in] ITYPE ``` ITYPE is INTEGER Specifies the problem type to be solved: = 1: A*x = (lambda)*B*x = 2: A*B*x = (lambda)*x = 3: B*A*x = (lambda)*x``` [in] JOBZ ``` JOBZ is CHARACTER*1 = 'N': Compute eigenvalues only; = 'V': Compute eigenvalues and eigenvectors.``` [in] RANGE ``` RANGE is CHARACTER*1 = 'A': all eigenvalues will be found. = 'V': all eigenvalues in the half-open interval (VL,VU] will be found. = 'I': the IL-th through IU-th eigenvalues will be found.``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A and B are stored; = 'L': Lower triangle of A and B are stored.``` [in] N ``` N is INTEGER The order of the matrix pencil (A,B). N >= 0.``` [in,out] A ``` A is REAL array, dimension (LDA, N) On entry, the symmetric matrix A. If UPLO = 'U', the leading N-by-N upper triangular part of A contains the upper triangular part of the matrix A. If UPLO = 'L', the leading N-by-N lower triangular part of A contains the lower triangular part of the matrix A. On exit, the lower triangle (if UPLO='L') or the upper triangle (if UPLO='U') of A, including the diagonal, is destroyed.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in,out] B ``` B is REAL array, dimension (LDB, N) On entry, the symmetric matrix B. If UPLO = 'U', the leading N-by-N upper triangular part of B contains the upper triangular part of the matrix B. If UPLO = 'L', the leading N-by-N lower triangular part of B contains the lower triangular part of the matrix B. On exit, if INFO <= N, the part of B containing the matrix is overwritten by the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in] VL ``` VL is REAL If RANGE='V', the lower bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] VU ``` VU is REAL If RANGE='V', the upper bound of the interval to be searched for eigenvalues. VL < VU. Not referenced if RANGE = 'A' or 'I'.``` [in] IL ``` IL is INTEGER If RANGE='I', the index of the smallest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] IU ``` IU is INTEGER If RANGE='I', the index of the largest eigenvalue to be returned. 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0. Not referenced if RANGE = 'A' or 'V'.``` [in] ABSTOL ``` ABSTOL is REAL The absolute error tolerance for the eigenvalues. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width less than or equal to ABSTOL + EPS * max( |a|,|b| ) , where EPS is the machine precision. If ABSTOL is less than or equal to zero, then EPS*|T| will be used in its place, where |T| is the 1-norm of the tridiagonal matrix obtained by reducing C to tridiagonal form, where C is the symmetric matrix of the standard symmetric problem to which the generalized problem is transformed. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*DLAMCH('S'), not zero. If this routine returns with INFO>0, indicating that some eigenvectors did not converge, try setting ABSTOL to 2*SLAMCH('S').``` [out] M ``` M is INTEGER The total number of eigenvalues found. 0 <= M <= N. If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.``` [out] W ``` W is REAL array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is REAL array, dimension (LDZ, max(1,M)) If JOBZ = 'N', then Z is not referenced. If JOBZ = 'V', then if INFO = 0, the first M columns of Z contain the orthonormal eigenvectors of the matrix A corresponding to the selected eigenvalues, with the i-th column of Z holding the eigenvector associated with W(i). The eigenvectors are normalized as follows: if ITYPE = 1 or 2, Z**T*B*Z = I; if ITYPE = 3, Z**T*inv(B)*Z = I. If an eigenvector fails to converge, then that column of Z contains the latest approximation to the eigenvector, and the index of the eigenvector is returned in IFAIL. Note: the user must ensure that at least max(1,M) columns are supplied in the array Z; if RANGE = 'V', the exact value of M is not known in advance and an upper bound must be used.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).``` [out] WORK ``` WORK is REAL array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.``` [in] LWORK ``` LWORK is INTEGER The length of the array WORK. LWORK >= max(1,8*N). For optimal efficiency, LWORK >= (NB+3)*N, where NB is the blocksize for SSYTRD returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.``` [out] IWORK ` IWORK is INTEGER array, dimension (5*N)` [out] IFAIL ``` IFAIL is INTEGER array, dimension (N) If JOBZ = 'V', then if INFO = 0, the first M elements of IFAIL are zero. If INFO > 0, then IFAIL contains the indices of the eigenvectors that failed to converge. If JOBZ = 'N', then IFAIL is not referenced.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: SPOTRF or SSYEVX returned an error code: <= N: if INFO = i, SSYEVX failed to converge; i eigenvectors failed to converge. Their indices are stored in array IFAIL. > N: if INFO = N + i, for 1 <= i <= N, then the leading minor of order i of B is not positive definite. The factorization of B could not be completed and no eigenvalues or eigenvectors were computed.```
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 294 of file ssygvx.f.

297 *
298 * -- LAPACK driver routine --
299 * -- LAPACK is a software package provided by Univ. of Tennessee, --
300 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
301 *
302 * .. Scalar Arguments ..
303  CHARACTER JOBZ, RANGE, UPLO
304  INTEGER IL, INFO, ITYPE, IU, LDA, LDB, LDZ, LWORK, M, N
305  REAL ABSTOL, VL, VU
306 * ..
307 * .. Array Arguments ..
308  INTEGER IFAIL( * ), IWORK( * )
309  REAL A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
310  \$ Z( LDZ, * )
311 * ..
312 *
313 * =====================================================================
314 *
315 * .. Parameters ..
316  REAL ONE
317  parameter( one = 1.0e+0 )
318 * ..
319 * .. Local Scalars ..
320  LOGICAL ALLEIG, INDEIG, LQUERY, UPPER, VALEIG, WANTZ
321  CHARACTER TRANS
322  INTEGER LWKMIN, LWKOPT, NB
323 * ..
324 * .. External Functions ..
325  LOGICAL LSAME
326  INTEGER ILAENV
327  EXTERNAL ilaenv, lsame
328 * ..
329 * .. External Subroutines ..
330  EXTERNAL spotrf, ssyevx, ssygst, strmm, strsm, xerbla
331 * ..
332 * .. Intrinsic Functions ..
333  INTRINSIC max, min
334 * ..
335 * .. Executable Statements ..
336 *
337 * Test the input parameters.
338 *
339  upper = lsame( uplo, 'U' )
340  wantz = lsame( jobz, 'V' )
341  alleig = lsame( range, 'A' )
342  valeig = lsame( range, 'V' )
343  indeig = lsame( range, 'I' )
344  lquery = ( lwork.EQ.-1 )
345 *
346  info = 0
347  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
348  info = -1
349  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
350  info = -2
351  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
352  info = -3
353  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
354  info = -4
355  ELSE IF( n.LT.0 ) THEN
356  info = -5
357  ELSE IF( lda.LT.max( 1, n ) ) THEN
358  info = -7
359  ELSE IF( ldb.LT.max( 1, n ) ) THEN
360  info = -9
361  ELSE
362  IF( valeig ) THEN
363  IF( n.GT.0 .AND. vu.LE.vl )
364  \$ info = -11
365  ELSE IF( indeig ) THEN
366  IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
367  info = -12
368  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
369  info = -13
370  END IF
371  END IF
372  END IF
373  IF (info.EQ.0) THEN
374  IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
375  info = -18
376  END IF
377  END IF
378 *
379  IF( info.EQ.0 ) THEN
380  lwkmin = max( 1, 8*n )
381  nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
382  lwkopt = max( lwkmin, ( nb + 3 )*n )
383  work( 1 ) = lwkopt
384 *
385  IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
386  info = -20
387  END IF
388  END IF
389 *
390  IF( info.NE.0 ) THEN
391  CALL xerbla( 'SSYGVX', -info )
392  RETURN
393  ELSE IF( lquery ) THEN
394  RETURN
395  END IF
396 *
397 * Quick return if possible
398 *
399  m = 0
400  IF( n.EQ.0 ) THEN
401  RETURN
402  END IF
403 *
404 * Form a Cholesky factorization of B.
405 *
406  CALL spotrf( uplo, n, b, ldb, info )
407  IF( info.NE.0 ) THEN
408  info = n + info
409  RETURN
410  END IF
411 *
412 * Transform problem to standard eigenvalue problem and solve.
413 *
414  CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
415  CALL ssyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
416  \$ m, w, z, ldz, work, lwork, iwork, ifail, info )
417 *
418  IF( wantz ) THEN
419 *
420 * Backtransform eigenvectors to the original problem.
421 *
422  IF( info.GT.0 )
423  \$ m = info - 1
424  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
425 *
426 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
427 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
428 *
429  IF( upper ) THEN
430  trans = 'N'
431  ELSE
432  trans = 'T'
433  END IF
434 *
435  CALL strsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
436  \$ ldb, z, ldz )
437 *
438  ELSE IF( itype.EQ.3 ) THEN
439 *
440 * For B*A*x=(lambda)*x;
441 * backtransform eigenvectors: x = L*y or U**T*y
442 *
443  IF( upper ) THEN
444  trans = 'T'
445  ELSE
446  trans = 'N'
447  END IF
448 *
449  CALL strmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
450  \$ ldb, z, ldz )
451  END IF
452  END IF
453 *
454 * Set WORK(1) to optimal workspace size.
455 *
456  work( 1 ) = lwkopt
457 *
458  RETURN
459 *
460 * End of SSYGVX
461 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine spotrf(UPLO, N, A, LDA, INFO)
SPOTRF
Definition: spotrf.f:107
subroutine ssygst(ITYPE, UPLO, N, A, LDA, B, LDB, INFO)
SSYGST
Definition: ssygst.f:127
subroutine ssyevx(JOBZ, RANGE, UPLO, N, A, LDA, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, LWORK, IWORK, IFAIL, INFO)
SSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
Definition: ssyevx.f:253
subroutine strmm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRMM
Definition: strmm.f:177
subroutine strsm(SIDE, UPLO, TRANSA, DIAG, M, N, ALPHA, A, LDA, B, LDB)
STRSM
Definition: strsm.f:181
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