LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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◆ ssyevr()

 subroutine ssyevr ( character JOBZ, character RANGE, character UPLO, integer N, real, dimension( lda, * ) A, integer LDA, real VL, real VU, integer IL, integer IU, real ABSTOL, integer M, real, dimension( * ) W, real, dimension( ldz, * ) Z, integer LDZ, integer, dimension( * ) ISUPPZ, real, dimension( * ) WORK, integer LWORK, integer, dimension( * ) IWORK, integer LIWORK, integer INFO )

SSYEVR computes the eigenvalues and, optionally, the left and/or right eigenvectors for SY matrices

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

Purpose:
``` SSYEVR computes selected eigenvalues and, optionally, eigenvectors
of a real symmetric matrix A.  Eigenvalues and eigenvectors can be
selected by specifying either a range of values or a range of
indices for the desired eigenvalues.

SSYEVR first reduces the matrix A to tridiagonal form T with a call
to SSYTRD.  Then, whenever possible, SSYEVR calls SSTEMR to compute
the eigenspectrum using Relatively Robust Representations.  SSTEMR
computes eigenvalues by the dqds algorithm, while orthogonal
eigenvectors are computed from various "good" L D L^T representations
(also known as Relatively Robust Representations). Gram-Schmidt
orthogonalization is avoided as far as possible. More specifically,
the various steps of the algorithm are as follows.

For each unreduced block (submatrix) of T,
(a) Compute T - sigma I  = L D L^T, so that L and D
define all the wanted eigenvalues to high relative accuracy.
This means that small relative changes in the entries of D and L
cause only small relative changes in the eigenvalues and
eigenvectors. The standard (unfactored) representation of the
tridiagonal matrix T does not have this property in general.
(b) Compute the eigenvalues to suitable accuracy.
If the eigenvectors are desired, the algorithm attains full
accuracy of the computed eigenvalues only right before
the corresponding vectors have to be computed, see steps c) and d).
(c) For each cluster of close eigenvalues, select a new
shift close to the cluster, find a new factorization, and refine
the shifted eigenvalues to suitable accuracy.
(d) For each eigenvalue with a large enough relative separation compute
the corresponding eigenvector by forming a rank revealing twisted
factorization. Go back to (c) for any clusters that remain.

The desired accuracy of the output can be specified by the input
parameter ABSTOL.

For more details, see SSTEMR's documentation and:
- Inderjit S. Dhillon and Beresford N. Parlett: "Multiple representations
to compute orthogonal eigenvectors of symmetric tridiagonal matrices,"
Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.
- Inderjit Dhillon and Beresford Parlett: "Orthogonal Eigenvectors and
Relative Gaps," SIAM Journal on Matrix Analysis and Applications, Vol. 25,
2004.  Also LAPACK Working Note 154.
- Inderjit Dhillon: "A new O(n^2) algorithm for the symmetric
tridiagonal eigenvalue/eigenvector problem",
Computer Science Division Technical Report No. UCB/CSD-97-971,
UC Berkeley, May 1997.

Note 1 : SSYEVR calls SSTEMR when the full spectrum is requested
on machines which conform to the ieee-754 floating point standard.
SSYEVR calls SSTEBZ and SSTEIN on non-ieee machines and
when partial spectrum requests are made.

Normal execution of SSTEMR may create NaNs and infinities and
hence may abort due to a floating point exception in environments
which do not handle NaNs and infinities in the ieee standard default
manner.```
Parameters
 [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. For RANGE = 'V' or 'I' and IU - IL < N - 1, SSTEBZ and SSTEIN are called``` [in] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [in] N ``` N is INTEGER The order of the matrix A. 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] 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. See "Computing Small Singular Values of Bidiagonal Matrices with Guaranteed High Relative Accuracy," by Demmel and Kahan, LAPACK Working Note #3. If high relative accuracy is important, set ABSTOL to SLAMCH( 'Safe minimum' ). Doing so will guarantee that eigenvalues are computed to high relative accuracy when possible in future releases. The current code does not make any guarantees about high relative accuracy, but future releases will. See J. Barlow and J. Demmel, "Computing Accurate Eigensystems of Scaled Diagonally Dominant Matrices", LAPACK Working Note #7, for a discussion of which matrices define their eigenvalues to high relative accuracy.``` [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) The first M elements contain the selected eigenvalues in ascending order.``` [out] Z ``` Z is REAL array, dimension (LDZ, max(1,M)) 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). If JOBZ = 'N', then Z is not referenced. 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. Supplying N columns is always safe.``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. LDZ >= 1, and if JOBZ = 'V', LDZ >= max(1,N).``` [out] ISUPPZ ``` ISUPPZ is INTEGER array, dimension ( 2*max(1,M) ) The support of the eigenvectors in Z, i.e., the indices indicating the nonzero elements in Z. The i-th eigenvector is nonzero only in elements ISUPPZ( 2*i-1 ) through ISUPPZ( 2*i ). This is an output of SSTEMR (tridiagonal matrix). The support of the eigenvectors of A is typically 1:N because of the orthogonal transformations applied by SORMTR. Implemented only for RANGE = 'A' or 'I' and IU - IL = N - 1``` [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 dimension of the array WORK. LWORK >= max(1,26*N). For optimal efficiency, LWORK >= (NB+6)*N, where NB is the max of the blocksize for SSYTRD and SORMTR returned by ILAENV. If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.``` [out] IWORK ``` IWORK is INTEGER array, dimension (MAX(1,LIWORK)) On exit, if INFO = 0, IWORK(1) returns the optimal LWORK.``` [in] LIWORK ``` LIWORK is INTEGER The dimension of the array IWORK. LIWORK >= max(1,10*N). If LIWORK = -1, then a workspace query is assumed; the routine only calculates the optimal sizes of the WORK and IWORK arrays, returns these values as the first entries of the WORK and IWORK arrays, and no error message related to LWORK or LIWORK is issued by XERBLA.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: Internal error```
Contributors:
Inderjit Dhillon, IBM Almaden, USA
Osni Marques, LBNL/NERSC, USA
Ken Stanley, Computer Science Division, University of California at Berkeley, USA
Jason Riedy, Computer Science Division, University of California at Berkeley, USA

Definition at line 333 of file ssyevr.f.

336*
337* -- LAPACK driver routine --
338* -- LAPACK is a software package provided by Univ. of Tennessee, --
339* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
340*
341* .. Scalar Arguments ..
342 CHARACTER JOBZ, RANGE, UPLO
343 INTEGER IL, INFO, IU, LDA, LDZ, LIWORK, LWORK, M, N
344 REAL ABSTOL, VL, VU
345* ..
346* .. Array Arguments ..
347 INTEGER ISUPPZ( * ), IWORK( * )
348 REAL A( LDA, * ), W( * ), WORK( * ), Z( LDZ, * )
349* ..
350*
351* =====================================================================
352*
353* .. Parameters ..
354 REAL ZERO, ONE, TWO
355 parameter( zero = 0.0e+0, one = 1.0e+0, two = 2.0e+0 )
356* ..
357* .. Local Scalars ..
358 LOGICAL ALLEIG, INDEIG, LOWER, LQUERY, TEST, VALEIG,
359 \$ WANTZ, TRYRAC
360 CHARACTER ORDER
361 INTEGER I, IEEEOK, IINFO, IMAX, INDD, INDDD, INDE,
362 \$ INDEE, INDIBL, INDIFL, INDISP, INDIWO, INDTAU,
363 \$ INDWK, INDWKN, ISCALE, J, JJ, LIWMIN,
364 \$ LLWORK, LLWRKN, LWKOPT, LWMIN, NB, NSPLIT
365 REAL ABSTLL, ANRM, BIGNUM, EPS, RMAX, RMIN, SAFMIN,
366 \$ SIGMA, SMLNUM, TMP1, VLL, VUU
367* ..
368* .. External Functions ..
369 LOGICAL LSAME
370 INTEGER ILAENV
371 REAL SLAMCH, SLANSY
372 EXTERNAL lsame, ilaenv, slamch, slansy
373* ..
374* .. External Subroutines ..
375 EXTERNAL scopy, sormtr, sscal, sstebz, sstemr, sstein,
377* ..
378* .. Intrinsic Functions ..
379 INTRINSIC max, min, sqrt
380* ..
381* .. Executable Statements ..
382*
383* Test the input parameters.
384*
385 ieeeok = ilaenv( 10, 'SSYEVR', 'N', 1, 2, 3, 4 )
386*
387 lower = lsame( uplo, 'L' )
388 wantz = lsame( jobz, 'V' )
389 alleig = lsame( range, 'A' )
390 valeig = lsame( range, 'V' )
391 indeig = lsame( range, 'I' )
392*
393 lquery = ( ( lwork.EQ.-1 ) .OR. ( liwork.EQ.-1 ) )
394*
395 lwmin = max( 1, 26*n )
396 liwmin = max( 1, 10*n )
397*
398 info = 0
399 IF( .NOT.( wantz .OR. lsame( jobz, 'N' ) ) ) THEN
400 info = -1
401 ELSE IF( .NOT.( alleig .OR. valeig .OR. indeig ) ) THEN
402 info = -2
403 ELSE IF( .NOT.( lower .OR. lsame( uplo, 'U' ) ) ) THEN
404 info = -3
405 ELSE IF( n.LT.0 ) THEN
406 info = -4
407 ELSE IF( lda.LT.max( 1, n ) ) THEN
408 info = -6
409 ELSE
410 IF( valeig ) THEN
411 IF( n.GT.0 .AND. vu.LE.vl )
412 \$ info = -8
413 ELSE IF( indeig ) THEN
414 IF( il.LT.1 .OR. il.GT.max( 1, n ) ) THEN
415 info = -9
416 ELSE IF( iu.LT.min( n, il ) .OR. iu.GT.n ) THEN
417 info = -10
418 END IF
419 END IF
420 END IF
421 IF( info.EQ.0 ) THEN
422 IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.n ) ) THEN
423 info = -15
424 END IF
425 END IF
426*
427 IF( info.EQ.0 ) THEN
428 nb = ilaenv( 1, 'SSYTRD', uplo, n, -1, -1, -1 )
429 nb = max( nb, ilaenv( 1, 'SORMTR', uplo, n, -1, -1, -1 ) )
430 lwkopt = max( ( nb+1 )*n, lwmin )
431 work( 1 ) = lwkopt
432 iwork( 1 ) = liwmin
433*
434 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
435 info = -18
436 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
437 info = -20
438 END IF
439 END IF
440*
441 IF( info.NE.0 ) THEN
442 CALL xerbla( 'SSYEVR', -info )
443 RETURN
444 ELSE IF( lquery ) THEN
445 RETURN
446 END IF
447*
448* Quick return if possible
449*
450 m = 0
451 IF( n.EQ.0 ) THEN
452 work( 1 ) = 1
453 RETURN
454 END IF
455*
456 IF( n.EQ.1 ) THEN
457 work( 1 ) = 26
458 IF( alleig .OR. indeig ) THEN
459 m = 1
460 w( 1 ) = a( 1, 1 )
461 ELSE
462 IF( vl.LT.a( 1, 1 ) .AND. vu.GE.a( 1, 1 ) ) THEN
463 m = 1
464 w( 1 ) = a( 1, 1 )
465 END IF
466 END IF
467 IF( wantz ) THEN
468 z( 1, 1 ) = one
469 isuppz( 1 ) = 1
470 isuppz( 2 ) = 1
471 END IF
472 RETURN
473 END IF
474*
475* Get machine constants.
476*
477 safmin = slamch( 'Safe minimum' )
478 eps = slamch( 'Precision' )
479 smlnum = safmin / eps
480 bignum = one / smlnum
481 rmin = sqrt( smlnum )
482 rmax = min( sqrt( bignum ), one / sqrt( sqrt( safmin ) ) )
483*
484* Scale matrix to allowable range, if necessary.
485*
486 iscale = 0
487 abstll = abstol
488 IF (valeig) THEN
489 vll = vl
490 vuu = vu
491 END IF
492 anrm = slansy( 'M', uplo, n, a, lda, work )
493 IF( anrm.GT.zero .AND. anrm.LT.rmin ) THEN
494 iscale = 1
495 sigma = rmin / anrm
496 ELSE IF( anrm.GT.rmax ) THEN
497 iscale = 1
498 sigma = rmax / anrm
499 END IF
500 IF( iscale.EQ.1 ) THEN
501 IF( lower ) THEN
502 DO 10 j = 1, n
503 CALL sscal( n-j+1, sigma, a( j, j ), 1 )
504 10 CONTINUE
505 ELSE
506 DO 20 j = 1, n
507 CALL sscal( j, sigma, a( 1, j ), 1 )
508 20 CONTINUE
509 END IF
510 IF( abstol.GT.0 )
511 \$ abstll = abstol*sigma
512 IF( valeig ) THEN
513 vll = vl*sigma
514 vuu = vu*sigma
515 END IF
516 END IF
517
518* Initialize indices into workspaces. Note: The IWORK indices are
519* used only if SSTERF or SSTEMR fail.
520
521* WORK(INDTAU:INDTAU+N-1) stores the scalar factors of the
522* elementary reflectors used in SSYTRD.
523 indtau = 1
524* WORK(INDD:INDD+N-1) stores the tridiagonal's diagonal entries.
525 indd = indtau + n
526* WORK(INDE:INDE+N-1) stores the off-diagonal entries of the
527* tridiagonal matrix from SSYTRD.
528 inde = indd + n
529* WORK(INDDD:INDDD+N-1) is a copy of the diagonal entries over
530* -written by SSTEMR (the SSTERF path copies the diagonal to W).
531 inddd = inde + n
532* WORK(INDEE:INDEE+N-1) is a copy of the off-diagonal entries over
533* -written while computing the eigenvalues in SSTERF and SSTEMR.
534 indee = inddd + n
535* INDWK is the starting offset of the left-over workspace, and
536* LLWORK is the remaining workspace size.
537 indwk = indee + n
538 llwork = lwork - indwk + 1
539
540* IWORK(INDIBL:INDIBL+M-1) corresponds to IBLOCK in SSTEBZ and
541* stores the block indices of each of the M<=N eigenvalues.
542 indibl = 1
543* IWORK(INDISP:INDISP+NSPLIT-1) corresponds to ISPLIT in SSTEBZ and
544* stores the starting and finishing indices of each block.
545 indisp = indibl + n
546* IWORK(INDIFL:INDIFL+N-1) stores the indices of eigenvectors
547* that corresponding to eigenvectors that fail to converge in
548* SSTEIN. This information is discarded; if any fail, the driver
549* returns INFO > 0.
550 indifl = indisp + n
551* INDIWO is the offset of the remaining integer workspace.
552 indiwo = indifl + n
553
554*
555* Call SSYTRD to reduce symmetric matrix to tridiagonal form.
556*
557 CALL ssytrd( uplo, n, a, lda, work( indd ), work( inde ),
558 \$ work( indtau ), work( indwk ), llwork, iinfo )
559*
560* If all eigenvalues are desired
561* then call SSTERF or SSTEMR and SORMTR.
562*
563 test = .false.
564 IF( indeig ) THEN
565 IF( il.EQ.1 .AND. iu.EQ.n ) THEN
566 test = .true.
567 END IF
568 END IF
569 IF( ( alleig.OR.test ) .AND. ( ieeeok.EQ.1 ) ) THEN
570 IF( .NOT.wantz ) THEN
571 CALL scopy( n, work( indd ), 1, w, 1 )
572 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
573 CALL ssterf( n, w, work( indee ), info )
574 ELSE
575 CALL scopy( n-1, work( inde ), 1, work( indee ), 1 )
576 CALL scopy( n, work( indd ), 1, work( inddd ), 1 )
577*
578 IF (abstol .LE. two*n*eps) THEN
579 tryrac = .true.
580 ELSE
581 tryrac = .false.
582 END IF
583 CALL sstemr( jobz, 'A', n, work( inddd ), work( indee ),
584 \$ vl, vu, il, iu, m, w, z, ldz, n, isuppz,
585 \$ tryrac, work( indwk ), lwork, iwork, liwork,
586 \$ info )
587*
588*
589*
590* Apply orthogonal matrix used in reduction to tridiagonal
591* form to eigenvectors returned by SSTEMR.
592*
593 IF( wantz .AND. info.EQ.0 ) THEN
594 indwkn = inde
595 llwrkn = lwork - indwkn + 1
596 CALL sormtr( 'L', uplo, 'N', n, m, a, lda,
597 \$ work( indtau ), z, ldz, work( indwkn ),
598 \$ llwrkn, iinfo )
599 END IF
600 END IF
601*
602*
603 IF( info.EQ.0 ) THEN
604* Everything worked. Skip SSTEBZ/SSTEIN. IWORK(:) are
605* undefined.
606 m = n
607 GO TO 30
608 END IF
609 info = 0
610 END IF
611*
612* Otherwise, call SSTEBZ and, if eigenvectors are desired, SSTEIN.
613* Also call SSTEBZ and SSTEIN if SSTEMR fails.
614*
615 IF( wantz ) THEN
616 order = 'B'
617 ELSE
618 order = 'E'
619 END IF
620
621 CALL sstebz( range, order, n, vll, vuu, il, iu, abstll,
622 \$ work( indd ), work( inde ), m, nsplit, w,
623 \$ iwork( indibl ), iwork( indisp ), work( indwk ),
624 \$ iwork( indiwo ), info )
625*
626 IF( wantz ) THEN
627 CALL sstein( n, work( indd ), work( inde ), m, w,
628 \$ iwork( indibl ), iwork( indisp ), z, ldz,
629 \$ work( indwk ), iwork( indiwo ), iwork( indifl ),
630 \$ info )
631*
632* Apply orthogonal matrix used in reduction to tridiagonal
633* form to eigenvectors returned by SSTEIN.
634*
635 indwkn = inde
636 llwrkn = lwork - indwkn + 1
637 CALL sormtr( 'L', uplo, 'N', n, m, a, lda, work( indtau ), z,
638 \$ ldz, work( indwkn ), llwrkn, iinfo )
639 END IF
640*
641* If matrix was scaled, then rescale eigenvalues appropriately.
642*
643* Jump here if SSTEMR/SSTEIN succeeded.
644 30 CONTINUE
645 IF( iscale.EQ.1 ) THEN
646 IF( info.EQ.0 ) THEN
647 imax = m
648 ELSE
649 imax = info - 1
650 END IF
651 CALL sscal( imax, one / sigma, w, 1 )
652 END IF
653*
654* If eigenvalues are not in order, then sort them, along with
655* eigenvectors. Note: We do not sort the IFAIL portion of IWORK.
656* It may not be initialized (if SSTEMR/SSTEIN succeeded), and we do
657* not return this detailed information to the user.
658*
659 IF( wantz ) THEN
660 DO 50 j = 1, m - 1
661 i = 0
662 tmp1 = w( j )
663 DO 40 jj = j + 1, m
664 IF( w( jj ).LT.tmp1 ) THEN
665 i = jj
666 tmp1 = w( jj )
667 END IF
668 40 CONTINUE
669*
670 IF( i.NE.0 ) THEN
671 w( i ) = w( j )
672 w( j ) = tmp1
673 CALL sswap( n, z( 1, i ), 1, z( 1, j ), 1 )
674 END IF
675 50 CONTINUE
676 END IF
677*
678* Set WORK(1) to optimal workspace size.
679*
680 work( 1 ) = lwkopt
681 iwork( 1 ) = liwmin
682*
683 RETURN
684*
685* End of SSYEVR
686*
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 ssterf(N, D, E, INFO)
SSTERF
Definition: ssterf.f:86
subroutine sstebz(RANGE, ORDER, N, VL, VU, IL, IU, ABSTOL, D, E, M, NSPLIT, W, IBLOCK, ISPLIT, WORK, IWORK, INFO)
SSTEBZ
Definition: sstebz.f:273
subroutine sormtr(SIDE, UPLO, TRANS, M, N, A, LDA, TAU, C, LDC, WORK, LWORK, INFO)
SORMTR
Definition: sormtr.f:172
subroutine sstemr(JOBZ, RANGE, N, D, E, VL, VU, IL, IU, M, W, Z, LDZ, NZC, ISUPPZ, TRYRAC, WORK, LWORK, IWORK, LIWORK, INFO)
SSTEMR
Definition: sstemr.f:321
subroutine sstein(N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK, IWORK, IFAIL, INFO)
SSTEIN
Definition: sstein.f:174
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
subroutine ssytrd(UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO)
SSYTRD
Definition: ssytrd.f:192
subroutine sswap(N, SX, INCX, SY, INCY)
SSWAP
Definition: sswap.f:82
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine sscal(N, SA, SX, INCX)
SSCAL
Definition: sscal.f:79
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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