LAPACK 3.12.0 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

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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 principal minor of order i of B is not positive. 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 REAL SROUNDUP_LWORK
328 EXTERNAL ilaenv, lsame, sroundup_lwork
329* ..
330* .. External Subroutines ..
331 EXTERNAL spotrf, ssyevx, ssygst, strmm, strsm, xerbla
332* ..
333* .. Intrinsic Functions ..
334 INTRINSIC max, min
335* ..
336* .. Executable Statements ..
337*
338* Test the input parameters.
339*
340 upper = lsame( uplo, 'U' )
341 wantz = lsame( jobz, 'V' )
342 alleig = lsame( range, 'A' )
343 valeig = lsame( range, 'V' )
344 indeig = lsame( range, 'I' )
345 lquery = ( lwork.EQ.-1 )
346*
347 info = 0
348 IF( itype.LT.1 .OR. itype.GT.3 ) THEN
349 info = -1
350 ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
351 info = -2
352 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
353 info = -3
354 ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
355 info = -4
356 ELSE IF( n.LT.0 ) THEN
357 info = -5
358 ELSE IF( lda.LT.max( 1, n ) ) THEN
359 info = -7
360 ELSE IF( ldb.LT.max( 1, n ) ) THEN
361 info = -9
362 ELSE
363 IF( valeig ) THEN
364 IF( n.GT.0 .AND. vu.LE.vl )
365 \$ info = -11
366 ELSE IF( indeig ) THEN
367 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
368 info = -12
369 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
370 info = -13
371 END IF
372 END IF
373 END IF
374 IF (info.EQ.0) THEN
375 IF (ldz.LT.1 .OR. (wantz .AND. ldz.LT.n)) THEN
376 info = -18
377 END IF
378 END IF
379*
380 IF( info.EQ.0 ) THEN
381 lwkmin = max( 1, 8*n )
382 nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
383 lwkopt = max( lwkmin, ( nb + 3 )*n )
384 work( 1 ) = sroundup_lwork(lwkopt)
385*
386 IF( lwork.LT.lwkmin .AND. .NOT.lquery ) THEN
387 info = -20
388 END IF
389 END IF
390*
391 IF( info.NE.0 ) THEN
392 CALL xerbla( 'SSYGVX', -info )
393 RETURN
394 ELSE IF( lquery ) THEN
395 RETURN
396 END IF
397*
398* Quick return if possible
399*
400 m = 0
401 IF( n.EQ.0 ) THEN
402 RETURN
403 END IF
404*
405* Form a Cholesky factorization of B.
406*
407 CALL spotrf( uplo, n, b, ldb, info )
408 IF( info.NE.0 ) THEN
409 info = n + info
410 RETURN
411 END IF
412*
413* Transform problem to standard eigenvalue problem and solve.
414*
415 CALL ssygst( itype, uplo, n, a, lda, b, ldb, info )
416 CALL ssyevx( jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol,
417 \$ m, w, z, ldz, work, lwork, iwork, ifail, info )
418*
419 IF( wantz ) THEN
420*
421* Backtransform eigenvectors to the original problem.
422*
423 IF( info.GT.0 )
424 \$ m = info - 1
425 IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
426*
427* For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
428* backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
429*
430 IF( upper ) THEN
431 trans = 'N'
432 ELSE
433 trans = 'T'
434 END IF
435*
436 CALL strsm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
437 \$ ldb, z, ldz )
438*
439 ELSE IF( itype.EQ.3 ) THEN
440*
441* For B*A*x=(lambda)*x;
442* backtransform eigenvectors: x = L*y or U**T*y
443*
444 IF( upper ) THEN
445 trans = 'T'
446 ELSE
447 trans = 'N'
448 END IF
449*
450 CALL strmm( 'Left', uplo, trans, 'Non-unit', n, m, one, b,
451 \$ ldb, z, ldz )
452 END IF
453 END IF
454*
455* Set WORK(1) to optimal workspace size.
456*
457 work( 1 ) = sroundup_lwork(lwkopt)
458*
459 RETURN
460*
461* End of SSYGVX
462*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
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 ssygst(itype, uplo, n, a, lda, b, ldb, info)
SSYGST
Definition ssygst.f:127
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
subroutine spotrf(uplo, n, a, lda, info)
SPOTRF
Definition spotrf.f:107
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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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