 LAPACK  3.10.0 LAPACK: Linear Algebra PACKage

## ◆ sspgvx()

 subroutine sspgvx ( integer ITYPE, character JOBZ, character RANGE, character UPLO, integer N, real, dimension( * ) AP, real, dimension( * ) BP, 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, dimension( * ) IWORK, integer, dimension( * ) IFAIL, integer INFO )

SSPGVX

Purpose:
``` SSPGVX 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, stored in packed storage, 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] AP ``` AP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix A, packed columnwise in a linear array. The j-th column of A is stored in the array AP as follows: if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; if UPLO = 'L', AP(i + (j-1)*(2*n-j)/2) = A(i,j) for j<=i<=n. On exit, the contents of AP are destroyed.``` [in,out] BP ``` BP is REAL array, dimension (N*(N+1)/2) On entry, the upper or lower triangle of the symmetric matrix B, packed columnwise in a linear array. The j-th column of B is stored in the array BP as follows: if UPLO = 'U', BP(i + (j-1)*j/2) = B(i,j) for 1<=i<=j; if UPLO = 'L', BP(i + (j-1)*(2*n-j)/2) = B(i,j) for j<=i<=n. On exit, the triangular factor U or L from the Cholesky factorization B = U**T*U or B = L*L**T, in the same storage format as B.``` [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 A to tridiagonal form. Eigenvalues will be computed most accurately when ABSTOL is set to twice the underflow threshold 2*SLAMCH('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 (8*N)` [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: SPPTRF or SSPEVX returned an error code: <= N: if INFO = i, SSPEVX 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 269 of file sspgvx.f.

272 *
273 * -- LAPACK driver routine --
274 * -- LAPACK is a software package provided by Univ. of Tennessee, --
275 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
276 *
277 * .. Scalar Arguments ..
278  CHARACTER JOBZ, RANGE, UPLO
279  INTEGER IL, INFO, ITYPE, IU, LDZ, M, N
280  REAL ABSTOL, VL, VU
281 * ..
282 * .. Array Arguments ..
283  INTEGER IFAIL( * ), IWORK( * )
284  REAL AP( * ), BP( * ), W( * ), WORK( * ),
285  \$ Z( LDZ, * )
286 * ..
287 *
288 * =====================================================================
289 *
290 * .. Local Scalars ..
291  LOGICAL ALLEIG, INDEIG, UPPER, VALEIG, WANTZ
292  CHARACTER TRANS
293  INTEGER J
294 * ..
295 * .. External Functions ..
296  LOGICAL LSAME
297  EXTERNAL lsame
298 * ..
299 * .. External Subroutines ..
300  EXTERNAL spptrf, sspevx, sspgst, stpmv, stpsv, xerbla
301 * ..
302 * .. Intrinsic Functions ..
303  INTRINSIC min
304 * ..
305 * .. Executable Statements ..
306 *
307 * Test the input parameters.
308 *
309  upper = lsame( uplo, 'U' )
310  wantz = lsame( jobz, 'V' )
311  alleig = lsame( range, 'A' )
312  valeig = lsame( range, 'V' )
313  indeig = lsame( range, 'I' )
314 *
315  info = 0
316  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
317  info = -1
318  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
319  info = -2
320  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
321  info = -3
322  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
323  info = -4
324  ELSE IF( n.LT.0 ) THEN
325  info = -5
326  ELSE
327  IF( valeig ) THEN
328  IF( n.GT.0 .AND. vu.LE.vl ) THEN
329  info = -9
330  END IF
331  ELSE IF( indeig ) THEN
332  IF( il.LT.1 ) THEN
333  info = -10
334  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
335  info = -11
336  END IF
337  END IF
338  END IF
339  IF( info.EQ.0 ) THEN
340  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
341  info = -16
342  END IF
343  END IF
344 *
345  IF( info.NE.0 ) THEN
346  CALL xerbla( 'SSPGVX', -info )
347  RETURN
348  END IF
349 *
350 * Quick return if possible
351 *
352  m = 0
353  IF( n.EQ.0 )
354  \$ RETURN
355 *
356 * Form a Cholesky factorization of B.
357 *
358  CALL spptrf( uplo, n, bp, info )
359  IF( info.NE.0 ) THEN
360  info = n + info
361  RETURN
362  END IF
363 *
364 * Transform problem to standard eigenvalue problem and solve.
365 *
366  CALL sspgst( itype, uplo, n, ap, bp, info )
367  CALL sspevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
368  \$ w, z, ldz, work, iwork, ifail, info )
369 *
370  IF( wantz ) THEN
371 *
372 * Backtransform eigenvectors to the original problem.
373 *
374  IF( info.GT.0 )
375  \$ m = info - 1
376  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
377 *
378 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
379 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
380 *
381  IF( upper ) THEN
382  trans = 'N'
383  ELSE
384  trans = 'T'
385  END IF
386 *
387  DO 10 j = 1, m
388  CALL stpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
389  \$ 1 )
390  10 CONTINUE
391 *
392  ELSE IF( itype.EQ.3 ) THEN
393 *
394 * For B*A*x=(lambda)*x;
395 * backtransform eigenvectors: x = L*y or U**T*y
396 *
397  IF( upper ) THEN
398  trans = 'T'
399  ELSE
400  trans = 'N'
401  END IF
402 *
403  DO 20 j = 1, m
404  CALL stpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
405  \$ 1 )
406  20 CONTINUE
407  END IF
408  END IF
409 *
410  RETURN
411 *
412 * End of SSPGVX
413 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine sspgst(ITYPE, UPLO, N, AP, BP, INFO)
SSPGST
Definition: sspgst.f:113
subroutine spptrf(UPLO, N, AP, INFO)
SPPTRF
Definition: spptrf.f:119
subroutine sspevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrice...
Definition: sspevx.f:234
subroutine stpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPMV
Definition: stpmv.f:142
subroutine stpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
STPSV
Definition: stpsv.f:144
Here is the call graph for this function:
Here is the caller graph for this function: