Annotation of rpl/lapack/lapack/dstevx.f, revision 1.8
1.8 ! bertrand 1: *> \brief <b> DSTEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for OTHER matrices</b>
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DSTEVX + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstevx.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstevx.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstevx.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
! 22: * M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * CHARACTER JOBZ, RANGE
! 26: * INTEGER IL, INFO, IU, LDZ, M, N
! 27: * DOUBLE PRECISION ABSTOL, VL, VU
! 28: * ..
! 29: * .. Array Arguments ..
! 30: * INTEGER IFAIL( * ), IWORK( * )
! 31: * DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
! 32: * ..
! 33: *
! 34: *
! 35: *> \par Purpose:
! 36: * =============
! 37: *>
! 38: *> \verbatim
! 39: *>
! 40: *> DSTEVX computes selected eigenvalues and, optionally, eigenvectors
! 41: *> of a real symmetric tridiagonal matrix A. Eigenvalues and
! 42: *> eigenvectors can be selected by specifying either a range of values
! 43: *> or a range of indices for the desired eigenvalues.
! 44: *> \endverbatim
! 45: *
! 46: * Arguments:
! 47: * ==========
! 48: *
! 49: *> \param[in] JOBZ
! 50: *> \verbatim
! 51: *> JOBZ is CHARACTER*1
! 52: *> = 'N': Compute eigenvalues only;
! 53: *> = 'V': Compute eigenvalues and eigenvectors.
! 54: *> \endverbatim
! 55: *>
! 56: *> \param[in] RANGE
! 57: *> \verbatim
! 58: *> RANGE is CHARACTER*1
! 59: *> = 'A': all eigenvalues will be found.
! 60: *> = 'V': all eigenvalues in the half-open interval (VL,VU]
! 61: *> will be found.
! 62: *> = 'I': the IL-th through IU-th eigenvalues will be found.
! 63: *> \endverbatim
! 64: *>
! 65: *> \param[in] N
! 66: *> \verbatim
! 67: *> N is INTEGER
! 68: *> The order of the matrix. N >= 0.
! 69: *> \endverbatim
! 70: *>
! 71: *> \param[in,out] D
! 72: *> \verbatim
! 73: *> D is DOUBLE PRECISION array, dimension (N)
! 74: *> On entry, the n diagonal elements of the tridiagonal matrix
! 75: *> A.
! 76: *> On exit, D may be multiplied by a constant factor chosen
! 77: *> to avoid over/underflow in computing the eigenvalues.
! 78: *> \endverbatim
! 79: *>
! 80: *> \param[in,out] E
! 81: *> \verbatim
! 82: *> E is DOUBLE PRECISION array, dimension (max(1,N-1))
! 83: *> On entry, the (n-1) subdiagonal elements of the tridiagonal
! 84: *> matrix A in elements 1 to N-1 of E.
! 85: *> On exit, E may be multiplied by a constant factor chosen
! 86: *> to avoid over/underflow in computing the eigenvalues.
! 87: *> \endverbatim
! 88: *>
! 89: *> \param[in] VL
! 90: *> \verbatim
! 91: *> VL is DOUBLE PRECISION
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[in] VU
! 95: *> \verbatim
! 96: *> VU is DOUBLE PRECISION
! 97: *> If RANGE='V', the lower and upper bounds of the interval to
! 98: *> be searched for eigenvalues. VL < VU.
! 99: *> Not referenced if RANGE = 'A' or 'I'.
! 100: *> \endverbatim
! 101: *>
! 102: *> \param[in] IL
! 103: *> \verbatim
! 104: *> IL is INTEGER
! 105: *> \endverbatim
! 106: *>
! 107: *> \param[in] IU
! 108: *> \verbatim
! 109: *> IU is INTEGER
! 110: *> If RANGE='I', the indices (in ascending order) of the
! 111: *> smallest and largest eigenvalues to be returned.
! 112: *> 1 <= IL <= IU <= N, if N > 0; IL = 1 and IU = 0 if N = 0.
! 113: *> Not referenced if RANGE = 'A' or 'V'.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[in] ABSTOL
! 117: *> \verbatim
! 118: *> ABSTOL is DOUBLE PRECISION
! 119: *> The absolute error tolerance for the eigenvalues.
! 120: *> An approximate eigenvalue is accepted as converged
! 121: *> when it is determined to lie in an interval [a,b]
! 122: *> of width less than or equal to
! 123: *>
! 124: *> ABSTOL + EPS * max( |a|,|b| ) ,
! 125: *>
! 126: *> where EPS is the machine precision. If ABSTOL is less
! 127: *> than or equal to zero, then EPS*|T| will be used in
! 128: *> its place, where |T| is the 1-norm of the tridiagonal
! 129: *> matrix.
! 130: *>
! 131: *> Eigenvalues will be computed most accurately when ABSTOL is
! 132: *> set to twice the underflow threshold 2*DLAMCH('S'), not zero.
! 133: *> If this routine returns with INFO>0, indicating that some
! 134: *> eigenvectors did not converge, try setting ABSTOL to
! 135: *> 2*DLAMCH('S').
! 136: *>
! 137: *> See "Computing Small Singular Values of Bidiagonal Matrices
! 138: *> with Guaranteed High Relative Accuracy," by Demmel and
! 139: *> Kahan, LAPACK Working Note #3.
! 140: *> \endverbatim
! 141: *>
! 142: *> \param[out] M
! 143: *> \verbatim
! 144: *> M is INTEGER
! 145: *> The total number of eigenvalues found. 0 <= M <= N.
! 146: *> If RANGE = 'A', M = N, and if RANGE = 'I', M = IU-IL+1.
! 147: *> \endverbatim
! 148: *>
! 149: *> \param[out] W
! 150: *> \verbatim
! 151: *> W is DOUBLE PRECISION array, dimension (N)
! 152: *> The first M elements contain the selected eigenvalues in
! 153: *> ascending order.
! 154: *> \endverbatim
! 155: *>
! 156: *> \param[out] Z
! 157: *> \verbatim
! 158: *> Z is DOUBLE PRECISION array, dimension (LDZ, max(1,M) )
! 159: *> If JOBZ = 'V', then if INFO = 0, the first M columns of Z
! 160: *> contain the orthonormal eigenvectors of the matrix A
! 161: *> corresponding to the selected eigenvalues, with the i-th
! 162: *> column of Z holding the eigenvector associated with W(i).
! 163: *> If an eigenvector fails to converge (INFO > 0), then that
! 164: *> column of Z contains the latest approximation to the
! 165: *> eigenvector, and the index of the eigenvector is returned
! 166: *> in IFAIL. If JOBZ = 'N', then Z is not referenced.
! 167: *> Note: the user must ensure that at least max(1,M) columns are
! 168: *> supplied in the array Z; if RANGE = 'V', the exact value of M
! 169: *> is not known in advance and an upper bound must be used.
! 170: *> \endverbatim
! 171: *>
! 172: *> \param[in] LDZ
! 173: *> \verbatim
! 174: *> LDZ is INTEGER
! 175: *> The leading dimension of the array Z. LDZ >= 1, and if
! 176: *> JOBZ = 'V', LDZ >= max(1,N).
! 177: *> \endverbatim
! 178: *>
! 179: *> \param[out] WORK
! 180: *> \verbatim
! 181: *> WORK is DOUBLE PRECISION array, dimension (5*N)
! 182: *> \endverbatim
! 183: *>
! 184: *> \param[out] IWORK
! 185: *> \verbatim
! 186: *> IWORK is INTEGER array, dimension (5*N)
! 187: *> \endverbatim
! 188: *>
! 189: *> \param[out] IFAIL
! 190: *> \verbatim
! 191: *> IFAIL is INTEGER array, dimension (N)
! 192: *> If JOBZ = 'V', then if INFO = 0, the first M elements of
! 193: *> IFAIL are zero. If INFO > 0, then IFAIL contains the
! 194: *> indices of the eigenvectors that failed to converge.
! 195: *> If JOBZ = 'N', then IFAIL is not referenced.
! 196: *> \endverbatim
! 197: *>
! 198: *> \param[out] INFO
! 199: *> \verbatim
! 200: *> INFO is INTEGER
! 201: *> = 0: successful exit
! 202: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 203: *> > 0: if INFO = i, then i eigenvectors failed to converge.
! 204: *> Their indices are stored in array IFAIL.
! 205: *> \endverbatim
! 206: *
! 207: * Authors:
! 208: * ========
! 209: *
! 210: *> \author Univ. of Tennessee
! 211: *> \author Univ. of California Berkeley
! 212: *> \author Univ. of Colorado Denver
! 213: *> \author NAG Ltd.
! 214: *
! 215: *> \date November 2011
! 216: *
! 217: *> \ingroup doubleOTHEReigen
! 218: *
! 219: * =====================================================================
1.1 bertrand 220: SUBROUTINE DSTEVX( JOBZ, RANGE, N, D, E, VL, VU, IL, IU, ABSTOL,
221: $ M, W, Z, LDZ, WORK, IWORK, IFAIL, INFO )
222: *
1.8 ! bertrand 223: * -- LAPACK driver routine (version 3.4.0) --
1.1 bertrand 224: * -- LAPACK is a software package provided by Univ. of Tennessee, --
225: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 226: * November 2011
1.1 bertrand 227: *
228: * .. Scalar Arguments ..
229: CHARACTER JOBZ, RANGE
230: INTEGER IL, INFO, IU, LDZ, M, N
231: DOUBLE PRECISION ABSTOL, VL, VU
232: * ..
233: * .. Array Arguments ..
234: INTEGER IFAIL( * ), IWORK( * )
235: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
236: * ..
237: *
238: * =====================================================================
239: *
240: * .. Parameters ..
241: DOUBLE PRECISION ZERO, ONE
242: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
243: * ..
244: * .. Local Scalars ..
245: LOGICAL ALLEIG, INDEIG, TEST, VALEIG, WANTZ
246: CHARACTER ORDER
247: INTEGER I, IMAX, INDIBL, INDISP, INDIWO, INDWRK,
248: $ ISCALE, ITMP1, J, JJ, NSPLIT
249: DOUBLE PRECISION BIGNUM, EPS, RMAX, RMIN, SAFMIN, SIGMA, SMLNUM,
250: $ TMP1, TNRM, VLL, VUU
251: * ..
252: * .. External Functions ..
253: LOGICAL LSAME
254: DOUBLE PRECISION DLAMCH, DLANST
255: EXTERNAL LSAME, DLAMCH, DLANST
256: * ..
257: * .. External Subroutines ..
258: EXTERNAL DCOPY, DSCAL, DSTEBZ, DSTEIN, DSTEQR, DSTERF,
259: $ DSWAP, XERBLA
260: * ..
261: * .. Intrinsic Functions ..
262: INTRINSIC MAX, MIN, SQRT
263: * ..
264: * .. Executable Statements ..
265: *
266: * Test the input parameters.
267: *
268: WANTZ = LSAME( JOBZ, 'V' )
269: ALLEIG = LSAME( RANGE, 'A' )
270: VALEIG = LSAME( RANGE, 'V' )
271: INDEIG = LSAME( RANGE, 'I' )
272: *
273: INFO = 0
274: IF( .NOT.( WANTZ .OR. LSAME( JOBZ, 'N' ) ) ) THEN
275: INFO = -1
276: ELSE IF( .NOT.( ALLEIG .OR. VALEIG .OR. INDEIG ) ) THEN
277: INFO = -2
278: ELSE IF( N.LT.0 ) THEN
279: INFO = -3
280: ELSE
281: IF( VALEIG ) THEN
282: IF( N.GT.0 .AND. VU.LE.VL )
283: $ INFO = -7
284: ELSE IF( INDEIG ) THEN
285: IF( IL.LT.1 .OR. IL.GT.MAX( 1, N ) ) THEN
286: INFO = -8
287: ELSE IF( IU.LT.MIN( N, IL ) .OR. IU.GT.N ) THEN
288: INFO = -9
289: END IF
290: END IF
291: END IF
292: IF( INFO.EQ.0 ) THEN
293: IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.N ) )
294: $ INFO = -14
295: END IF
296: *
297: IF( INFO.NE.0 ) THEN
298: CALL XERBLA( 'DSTEVX', -INFO )
299: RETURN
300: END IF
301: *
302: * Quick return if possible
303: *
304: M = 0
305: IF( N.EQ.0 )
306: $ RETURN
307: *
308: IF( N.EQ.1 ) THEN
309: IF( ALLEIG .OR. INDEIG ) THEN
310: M = 1
311: W( 1 ) = D( 1 )
312: ELSE
313: IF( VL.LT.D( 1 ) .AND. VU.GE.D( 1 ) ) THEN
314: M = 1
315: W( 1 ) = D( 1 )
316: END IF
317: END IF
318: IF( WANTZ )
319: $ Z( 1, 1 ) = ONE
320: RETURN
321: END IF
322: *
323: * Get machine constants.
324: *
325: SAFMIN = DLAMCH( 'Safe minimum' )
326: EPS = DLAMCH( 'Precision' )
327: SMLNUM = SAFMIN / EPS
328: BIGNUM = ONE / SMLNUM
329: RMIN = SQRT( SMLNUM )
330: RMAX = MIN( SQRT( BIGNUM ), ONE / SQRT( SQRT( SAFMIN ) ) )
331: *
332: * Scale matrix to allowable range, if necessary.
333: *
334: ISCALE = 0
335: IF( VALEIG ) THEN
336: VLL = VL
337: VUU = VU
338: ELSE
339: VLL = ZERO
340: VUU = ZERO
341: END IF
342: TNRM = DLANST( 'M', N, D, E )
343: IF( TNRM.GT.ZERO .AND. TNRM.LT.RMIN ) THEN
344: ISCALE = 1
345: SIGMA = RMIN / TNRM
346: ELSE IF( TNRM.GT.RMAX ) THEN
347: ISCALE = 1
348: SIGMA = RMAX / TNRM
349: END IF
350: IF( ISCALE.EQ.1 ) THEN
351: CALL DSCAL( N, SIGMA, D, 1 )
352: CALL DSCAL( N-1, SIGMA, E( 1 ), 1 )
353: IF( VALEIG ) THEN
354: VLL = VL*SIGMA
355: VUU = VU*SIGMA
356: END IF
357: END IF
358: *
359: * If all eigenvalues are desired and ABSTOL is less than zero, then
360: * call DSTERF or SSTEQR. If this fails for some eigenvalue, then
361: * try DSTEBZ.
362: *
363: TEST = .FALSE.
364: IF( INDEIG ) THEN
365: IF( IL.EQ.1 .AND. IU.EQ.N ) THEN
366: TEST = .TRUE.
367: END IF
368: END IF
369: IF( ( ALLEIG .OR. TEST ) .AND. ( ABSTOL.LE.ZERO ) ) THEN
370: CALL DCOPY( N, D, 1, W, 1 )
371: CALL DCOPY( N-1, E( 1 ), 1, WORK( 1 ), 1 )
372: INDWRK = N + 1
373: IF( .NOT.WANTZ ) THEN
374: CALL DSTERF( N, W, WORK, INFO )
375: ELSE
376: CALL DSTEQR( 'I', N, W, WORK, Z, LDZ, WORK( INDWRK ), INFO )
377: IF( INFO.EQ.0 ) THEN
378: DO 10 I = 1, N
379: IFAIL( I ) = 0
380: 10 CONTINUE
381: END IF
382: END IF
383: IF( INFO.EQ.0 ) THEN
384: M = N
385: GO TO 20
386: END IF
387: INFO = 0
388: END IF
389: *
390: * Otherwise, call DSTEBZ and, if eigenvectors are desired, SSTEIN.
391: *
392: IF( WANTZ ) THEN
393: ORDER = 'B'
394: ELSE
395: ORDER = 'E'
396: END IF
397: INDWRK = 1
398: INDIBL = 1
399: INDISP = INDIBL + N
400: INDIWO = INDISP + N
401: CALL DSTEBZ( RANGE, ORDER, N, VLL, VUU, IL, IU, ABSTOL, D, E, M,
402: $ NSPLIT, W, IWORK( INDIBL ), IWORK( INDISP ),
403: $ WORK( INDWRK ), IWORK( INDIWO ), INFO )
404: *
405: IF( WANTZ ) THEN
406: CALL DSTEIN( N, D, E, M, W, IWORK( INDIBL ), IWORK( INDISP ),
407: $ Z, LDZ, WORK( INDWRK ), IWORK( INDIWO ), IFAIL,
408: $ INFO )
409: END IF
410: *
411: * If matrix was scaled, then rescale eigenvalues appropriately.
412: *
413: 20 CONTINUE
414: IF( ISCALE.EQ.1 ) THEN
415: IF( INFO.EQ.0 ) THEN
416: IMAX = M
417: ELSE
418: IMAX = INFO - 1
419: END IF
420: CALL DSCAL( IMAX, ONE / SIGMA, W, 1 )
421: END IF
422: *
423: * If eigenvalues are not in order, then sort them, along with
424: * eigenvectors.
425: *
426: IF( WANTZ ) THEN
427: DO 40 J = 1, M - 1
428: I = 0
429: TMP1 = W( J )
430: DO 30 JJ = J + 1, M
431: IF( W( JJ ).LT.TMP1 ) THEN
432: I = JJ
433: TMP1 = W( JJ )
434: END IF
435: 30 CONTINUE
436: *
437: IF( I.NE.0 ) THEN
438: ITMP1 = IWORK( INDIBL+I-1 )
439: W( I ) = W( J )
440: IWORK( INDIBL+I-1 ) = IWORK( INDIBL+J-1 )
441: W( J ) = TMP1
442: IWORK( INDIBL+J-1 ) = ITMP1
443: CALL DSWAP( N, Z( 1, I ), 1, Z( 1, J ), 1 )
444: IF( INFO.NE.0 ) THEN
445: ITMP1 = IFAIL( I )
446: IFAIL( I ) = IFAIL( J )
447: IFAIL( J ) = ITMP1
448: END IF
449: END IF
450: 40 CONTINUE
451: END IF
452: *
453: RETURN
454: *
455: * End of DSTEVX
456: *
457: END
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