LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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

Download SSYGVX + dependencies [TGZ] [ZIP] [TXT]

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.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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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