LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ dspgvx()

subroutine dspgvx ( integer  ITYPE,
character  JOBZ,
character  RANGE,
character  UPLO,
integer  N,
double precision, dimension( * )  AP,
double precision, dimension( * )  BP,
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, dimension( * )  IWORK,
integer, dimension( * )  IFAIL,
integer  INFO 
)

DSPGVX

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Purpose:
 DSPGVX 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 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 A to tridiagonal form.

          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 (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:  DPPTRF or DSPEVX returned an error code:
             <= N:  if INFO = i, DSPEVX 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.
Date
June 2016
Contributors:
Mark Fahey, Department of Mathematics, Univ. of Kentucky, USA

Definition at line 274 of file dspgvx.f.

274 *
275 * -- LAPACK driver routine (version 3.7.0) --
276 * -- LAPACK is a software package provided by Univ. of Tennessee, --
277 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
278 * June 2016
279 *
280 * .. Scalar Arguments ..
281  CHARACTER jobz, range, uplo
282  INTEGER il, info, itype, iu, ldz, m, n
283  DOUBLE PRECISION abstol, vl, vu
284 * ..
285 * .. Array Arguments ..
286  INTEGER ifail( * ), iwork( * )
287  DOUBLE PRECISION ap( * ), bp( * ), w( * ), work( * ),
288  $ z( ldz, * )
289 * ..
290 *
291 * =====================================================================
292 *
293 * .. Local Scalars ..
294  LOGICAL alleig, indeig, upper, valeig, wantz
295  CHARACTER trans
296  INTEGER j
297 * ..
298 * .. External Functions ..
299  LOGICAL lsame
300  EXTERNAL lsame
301 * ..
302 * .. External Subroutines ..
303  EXTERNAL dpptrf, dspevx, dspgst, dtpmv, dtpsv, xerbla
304 * ..
305 * .. Intrinsic Functions ..
306  INTRINSIC min
307 * ..
308 * .. Executable Statements ..
309 *
310 * Test the input parameters.
311 *
312  upper = lsame( uplo, 'U' )
313  wantz = lsame( jobz, 'V' )
314  alleig = lsame( range, 'A' )
315  valeig = lsame( range, 'V' )
316  indeig = lsame( range, 'I' )
317 *
318  info = 0
319  IF( itype.LT.1 .OR. itype.GT.3 ) THEN
320  info = -1
321  ELSE IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
322  info = -2
323  ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
324  info = -3
325  ELSE IF( .NOT.( upper .OR. lsame( uplo, 'L' ) ) ) THEN
326  info = -4
327  ELSE IF( n.LT.0 ) THEN
328  info = -5
329  ELSE
330  IF( valeig ) THEN
331  IF( n.GT.0 .AND. vu.LE.vl ) THEN
332  info = -9
333  END IF
334  ELSE IF( indeig ) THEN
335  IF( il.LT.1 ) THEN
336  info = -10
337  ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
338  info = -11
339  END IF
340  END IF
341  END IF
342  IF( info.EQ.0 ) THEN
343  IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
344  info = -16
345  END IF
346  END IF
347 *
348  IF( info.NE.0 ) THEN
349  CALL xerbla( 'DSPGVX', -info )
350  RETURN
351  END IF
352 *
353 * Quick return if possible
354 *
355  m = 0
356  IF( n.EQ.0 )
357  $ RETURN
358 *
359 * Form a Cholesky factorization of B.
360 *
361  CALL dpptrf( uplo, n, bp, info )
362  IF( info.NE.0 ) THEN
363  info = n + info
364  RETURN
365  END IF
366 *
367 * Transform problem to standard eigenvalue problem and solve.
368 *
369  CALL dspgst( itype, uplo, n, ap, bp, info )
370  CALL dspevx( jobz, range, uplo, n, ap, vl, vu, il, iu, abstol, m,
371  $ w, z, ldz, work, iwork, ifail, info )
372 *
373  IF( wantz ) THEN
374 *
375 * Backtransform eigenvectors to the original problem.
376 *
377  IF( info.GT.0 )
378  $ m = info - 1
379  IF( itype.EQ.1 .OR. itype.EQ.2 ) THEN
380 *
381 * For A*x=(lambda)*B*x and A*B*x=(lambda)*x;
382 * backtransform eigenvectors: x = inv(L)**T*y or inv(U)*y
383 *
384  IF( upper ) THEN
385  trans = 'N'
386  ELSE
387  trans = 'T'
388  END IF
389 *
390  DO 10 j = 1, m
391  CALL dtpsv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
392  $ 1 )
393  10 CONTINUE
394 *
395  ELSE IF( itype.EQ.3 ) THEN
396 *
397 * For B*A*x=(lambda)*x;
398 * backtransform eigenvectors: x = L*y or U**T*y
399 *
400  IF( upper ) THEN
401  trans = 'T'
402  ELSE
403  trans = 'N'
404  END IF
405 *
406  DO 20 j = 1, m
407  CALL dtpmv( uplo, trans, 'Non-unit', n, bp, z( 1, j ),
408  $ 1 )
409  20 CONTINUE
410  END IF
411  END IF
412 *
413  RETURN
414 *
415 * End of DSPGVX
416 *
subroutine dspgst(ITYPE, UPLO, N, AP, BP, INFO)
DSPGST
Definition: dspgst.f:115
subroutine dtpsv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPSV
Definition: dtpsv.f:146
subroutine dpptrf(UPLO, N, AP, INFO)
DPPTRF
Definition: dpptrf.f:121
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
subroutine dtpmv(UPLO, TRANS, DIAG, N, AP, X, INCX)
DTPMV
Definition: dtpmv.f:144
subroutine dspevx(JOBZ, RANGE, UPLO, N, AP, VL, VU, IL, IU, ABSTOL, M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO)
DSPEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matric...
Definition: dspevx.f:236
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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