LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ dsygvx()

 subroutine dsygvx ( integer itype, character jobz, character range, character uplo, integer n, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision vl, double precision vu, integer il, integer iu, double precision abstol, integer m, double precision, dimension( * ) w, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer, dimension( * ) iwork, integer, dimension( * ) ifail, integer info )

DSYGVX

Purpose:
``` DSYGVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*DLAMCH('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 DOUBLE PRECISION array, dimension (N) On normal exit, the first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DSYTRD 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: DPOTRF or DSYEVX returned an error code: <= N: if INFO = i, DSYEVX 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 dsygvx.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 DOUBLE PRECISION ABSTOL, VL, VU
306* ..
307* .. Array Arguments ..
308 INTEGER IFAIL( * ), IWORK( * )
309 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), W( * ), WORK( * ),
310 \$ Z( LDZ, * )
311* ..
312*
313* =====================================================================
314*
315* .. Parameters ..
316 DOUBLE PRECISION ONE
317 parameter( one = 1.0d+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 lsame, ilaenv
328* ..
329* .. External Subroutines ..
330 EXTERNAL dpotrf, dsyevx, dsygst, dtrmm, dtrsm, 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, 'DSYTRD', 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( 'DSYGVX', -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 dpotrf( 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 dsygst( itype, uplo, n, a, lda, b, ldb, info )
415 CALL dsyevx( 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 dtrsm( '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 dtrmm( '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 DSYGVX
461*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
subroutine dsyevx(jobz, range, uplo, n, a, lda, vl, vu, il, iu, abstol, m, w, z, ldz, work, lwork, iwork, ifail, info)
DSYEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices
Definition dsyevx.f:253
subroutine dsygst(itype, uplo, n, a, lda, b, ldb, info)
DSYGST
Definition dsygst.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 dpotrf(uplo, n, a, lda, info)
DPOTRF
Definition dpotrf.f:107
subroutine dtrmm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
DTRMM
Definition dtrmm.f:177
subroutine dtrsm(side, uplo, transa, diag, m, n, alpha, a, lda, b, ldb)
DTRSM
Definition dtrsm.f:181
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