Annotation of rpl/lapack/lapack/dstein.f, revision 1.8
1.8 ! bertrand 1: *> \brief \b DSTEIN
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DSTEIN + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dstein.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dstein.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dstein.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
! 22: * IWORK, IFAIL, INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER INFO, LDZ, M, N
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
! 29: * $ IWORK( * )
! 30: * DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
! 31: * ..
! 32: *
! 33: *
! 34: *> \par Purpose:
! 35: * =============
! 36: *>
! 37: *> \verbatim
! 38: *>
! 39: *> DSTEIN computes the eigenvectors of a real symmetric tridiagonal
! 40: *> matrix T corresponding to specified eigenvalues, using inverse
! 41: *> iteration.
! 42: *>
! 43: *> The maximum number of iterations allowed for each eigenvector is
! 44: *> specified by an internal parameter MAXITS (currently set to 5).
! 45: *> \endverbatim
! 46: *
! 47: * Arguments:
! 48: * ==========
! 49: *
! 50: *> \param[in] N
! 51: *> \verbatim
! 52: *> N is INTEGER
! 53: *> The order of the matrix. N >= 0.
! 54: *> \endverbatim
! 55: *>
! 56: *> \param[in] D
! 57: *> \verbatim
! 58: *> D is DOUBLE PRECISION array, dimension (N)
! 59: *> The n diagonal elements of the tridiagonal matrix T.
! 60: *> \endverbatim
! 61: *>
! 62: *> \param[in] E
! 63: *> \verbatim
! 64: *> E is DOUBLE PRECISION array, dimension (N-1)
! 65: *> The (n-1) subdiagonal elements of the tridiagonal matrix
! 66: *> T, in elements 1 to N-1.
! 67: *> \endverbatim
! 68: *>
! 69: *> \param[in] M
! 70: *> \verbatim
! 71: *> M is INTEGER
! 72: *> The number of eigenvectors to be found. 0 <= M <= N.
! 73: *> \endverbatim
! 74: *>
! 75: *> \param[in] W
! 76: *> \verbatim
! 77: *> W is DOUBLE PRECISION array, dimension (N)
! 78: *> The first M elements of W contain the eigenvalues for
! 79: *> which eigenvectors are to be computed. The eigenvalues
! 80: *> should be grouped by split-off block and ordered from
! 81: *> smallest to largest within the block. ( The output array
! 82: *> W from DSTEBZ with ORDER = 'B' is expected here. )
! 83: *> \endverbatim
! 84: *>
! 85: *> \param[in] IBLOCK
! 86: *> \verbatim
! 87: *> IBLOCK is INTEGER array, dimension (N)
! 88: *> The submatrix indices associated with the corresponding
! 89: *> eigenvalues in W; IBLOCK(i)=1 if eigenvalue W(i) belongs to
! 90: *> the first submatrix from the top, =2 if W(i) belongs to
! 91: *> the second submatrix, etc. ( The output array IBLOCK
! 92: *> from DSTEBZ is expected here. )
! 93: *> \endverbatim
! 94: *>
! 95: *> \param[in] ISPLIT
! 96: *> \verbatim
! 97: *> ISPLIT is INTEGER array, dimension (N)
! 98: *> The splitting points, at which T breaks up into submatrices.
! 99: *> The first submatrix consists of rows/columns 1 to
! 100: *> ISPLIT( 1 ), the second of rows/columns ISPLIT( 1 )+1
! 101: *> through ISPLIT( 2 ), etc.
! 102: *> ( The output array ISPLIT from DSTEBZ is expected here. )
! 103: *> \endverbatim
! 104: *>
! 105: *> \param[out] Z
! 106: *> \verbatim
! 107: *> Z is DOUBLE PRECISION array, dimension (LDZ, M)
! 108: *> The computed eigenvectors. The eigenvector associated
! 109: *> with the eigenvalue W(i) is stored in the i-th column of
! 110: *> Z. Any vector which fails to converge is set to its current
! 111: *> iterate after MAXITS iterations.
! 112: *> \endverbatim
! 113: *>
! 114: *> \param[in] LDZ
! 115: *> \verbatim
! 116: *> LDZ is INTEGER
! 117: *> The leading dimension of the array Z. LDZ >= max(1,N).
! 118: *> \endverbatim
! 119: *>
! 120: *> \param[out] WORK
! 121: *> \verbatim
! 122: *> WORK is DOUBLE PRECISION array, dimension (5*N)
! 123: *> \endverbatim
! 124: *>
! 125: *> \param[out] IWORK
! 126: *> \verbatim
! 127: *> IWORK is INTEGER array, dimension (N)
! 128: *> \endverbatim
! 129: *>
! 130: *> \param[out] IFAIL
! 131: *> \verbatim
! 132: *> IFAIL is INTEGER array, dimension (M)
! 133: *> On normal exit, all elements of IFAIL are zero.
! 134: *> If one or more eigenvectors fail to converge after
! 135: *> MAXITS iterations, then their indices are stored in
! 136: *> array IFAIL.
! 137: *> \endverbatim
! 138: *>
! 139: *> \param[out] INFO
! 140: *> \verbatim
! 141: *> INFO is INTEGER
! 142: *> = 0: successful exit.
! 143: *> < 0: if INFO = -i, the i-th argument had an illegal value
! 144: *> > 0: if INFO = i, then i eigenvectors failed to converge
! 145: *> in MAXITS iterations. Their indices are stored in
! 146: *> array IFAIL.
! 147: *> \endverbatim
! 148: *
! 149: *> \par Internal Parameters:
! 150: * =========================
! 151: *>
! 152: *> \verbatim
! 153: *> MAXITS INTEGER, default = 5
! 154: *> The maximum number of iterations performed.
! 155: *>
! 156: *> EXTRA INTEGER, default = 2
! 157: *> The number of iterations performed after norm growth
! 158: *> criterion is satisfied, should be at least 1.
! 159: *> \endverbatim
! 160: *
! 161: * Authors:
! 162: * ========
! 163: *
! 164: *> \author Univ. of Tennessee
! 165: *> \author Univ. of California Berkeley
! 166: *> \author Univ. of Colorado Denver
! 167: *> \author NAG Ltd.
! 168: *
! 169: *> \date November 2011
! 170: *
! 171: *> \ingroup doubleOTHERcomputational
! 172: *
! 173: * =====================================================================
1.1 bertrand 174: SUBROUTINE DSTEIN( N, D, E, M, W, IBLOCK, ISPLIT, Z, LDZ, WORK,
175: $ IWORK, IFAIL, INFO )
176: *
1.8 ! bertrand 177: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 178: * -- LAPACK is a software package provided by Univ. of Tennessee, --
179: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.8 ! bertrand 180: * November 2011
1.1 bertrand 181: *
182: * .. Scalar Arguments ..
183: INTEGER INFO, LDZ, M, N
184: * ..
185: * .. Array Arguments ..
186: INTEGER IBLOCK( * ), IFAIL( * ), ISPLIT( * ),
187: $ IWORK( * )
188: DOUBLE PRECISION D( * ), E( * ), W( * ), WORK( * ), Z( LDZ, * )
189: * ..
190: *
191: * =====================================================================
192: *
193: * .. Parameters ..
194: DOUBLE PRECISION ZERO, ONE, TEN, ODM3, ODM1
195: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, TEN = 1.0D+1,
196: $ ODM3 = 1.0D-3, ODM1 = 1.0D-1 )
197: INTEGER MAXITS, EXTRA
198: PARAMETER ( MAXITS = 5, EXTRA = 2 )
199: * ..
200: * .. Local Scalars ..
201: INTEGER B1, BLKSIZ, BN, GPIND, I, IINFO, INDRV1,
202: $ INDRV2, INDRV3, INDRV4, INDRV5, ITS, J, J1,
203: $ JBLK, JMAX, NBLK, NRMCHK
204: DOUBLE PRECISION DTPCRT, EPS, EPS1, NRM, ONENRM, ORTOL, PERTOL,
205: $ SCL, SEP, TOL, XJ, XJM, ZTR
206: * ..
207: * .. Local Arrays ..
208: INTEGER ISEED( 4 )
209: * ..
210: * .. External Functions ..
211: INTEGER IDAMAX
212: DOUBLE PRECISION DASUM, DDOT, DLAMCH, DNRM2
213: EXTERNAL IDAMAX, DASUM, DDOT, DLAMCH, DNRM2
214: * ..
215: * .. External Subroutines ..
216: EXTERNAL DAXPY, DCOPY, DLAGTF, DLAGTS, DLARNV, DSCAL,
217: $ XERBLA
218: * ..
219: * .. Intrinsic Functions ..
220: INTRINSIC ABS, MAX, SQRT
221: * ..
222: * .. Executable Statements ..
223: *
224: * Test the input parameters.
225: *
226: INFO = 0
227: DO 10 I = 1, M
228: IFAIL( I ) = 0
229: 10 CONTINUE
230: *
231: IF( N.LT.0 ) THEN
232: INFO = -1
233: ELSE IF( M.LT.0 .OR. M.GT.N ) THEN
234: INFO = -4
235: ELSE IF( LDZ.LT.MAX( 1, N ) ) THEN
236: INFO = -9
237: ELSE
238: DO 20 J = 2, M
239: IF( IBLOCK( J ).LT.IBLOCK( J-1 ) ) THEN
240: INFO = -6
241: GO TO 30
242: END IF
243: IF( IBLOCK( J ).EQ.IBLOCK( J-1 ) .AND. W( J ).LT.W( J-1 ) )
244: $ THEN
245: INFO = -5
246: GO TO 30
247: END IF
248: 20 CONTINUE
249: 30 CONTINUE
250: END IF
251: *
252: IF( INFO.NE.0 ) THEN
253: CALL XERBLA( 'DSTEIN', -INFO )
254: RETURN
255: END IF
256: *
257: * Quick return if possible
258: *
259: IF( N.EQ.0 .OR. M.EQ.0 ) THEN
260: RETURN
261: ELSE IF( N.EQ.1 ) THEN
262: Z( 1, 1 ) = ONE
263: RETURN
264: END IF
265: *
266: * Get machine constants.
267: *
268: EPS = DLAMCH( 'Precision' )
269: *
270: * Initialize seed for random number generator DLARNV.
271: *
272: DO 40 I = 1, 4
273: ISEED( I ) = 1
274: 40 CONTINUE
275: *
276: * Initialize pointers.
277: *
278: INDRV1 = 0
279: INDRV2 = INDRV1 + N
280: INDRV3 = INDRV2 + N
281: INDRV4 = INDRV3 + N
282: INDRV5 = INDRV4 + N
283: *
284: * Compute eigenvectors of matrix blocks.
285: *
286: J1 = 1
287: DO 160 NBLK = 1, IBLOCK( M )
288: *
289: * Find starting and ending indices of block nblk.
290: *
291: IF( NBLK.EQ.1 ) THEN
292: B1 = 1
293: ELSE
294: B1 = ISPLIT( NBLK-1 ) + 1
295: END IF
296: BN = ISPLIT( NBLK )
297: BLKSIZ = BN - B1 + 1
298: IF( BLKSIZ.EQ.1 )
299: $ GO TO 60
300: GPIND = B1
301: *
302: * Compute reorthogonalization criterion and stopping criterion.
303: *
304: ONENRM = ABS( D( B1 ) ) + ABS( E( B1 ) )
305: ONENRM = MAX( ONENRM, ABS( D( BN ) )+ABS( E( BN-1 ) ) )
306: DO 50 I = B1 + 1, BN - 1
307: ONENRM = MAX( ONENRM, ABS( D( I ) )+ABS( E( I-1 ) )+
308: $ ABS( E( I ) ) )
309: 50 CONTINUE
310: ORTOL = ODM3*ONENRM
311: *
312: DTPCRT = SQRT( ODM1 / BLKSIZ )
313: *
314: * Loop through eigenvalues of block nblk.
315: *
316: 60 CONTINUE
317: JBLK = 0
318: DO 150 J = J1, M
319: IF( IBLOCK( J ).NE.NBLK ) THEN
320: J1 = J
321: GO TO 160
322: END IF
323: JBLK = JBLK + 1
324: XJ = W( J )
325: *
326: * Skip all the work if the block size is one.
327: *
328: IF( BLKSIZ.EQ.1 ) THEN
329: WORK( INDRV1+1 ) = ONE
330: GO TO 120
331: END IF
332: *
333: * If eigenvalues j and j-1 are too close, add a relatively
334: * small perturbation.
335: *
336: IF( JBLK.GT.1 ) THEN
337: EPS1 = ABS( EPS*XJ )
338: PERTOL = TEN*EPS1
339: SEP = XJ - XJM
340: IF( SEP.LT.PERTOL )
341: $ XJ = XJM + PERTOL
342: END IF
343: *
344: ITS = 0
345: NRMCHK = 0
346: *
347: * Get random starting vector.
348: *
349: CALL DLARNV( 2, ISEED, BLKSIZ, WORK( INDRV1+1 ) )
350: *
351: * Copy the matrix T so it won't be destroyed in factorization.
352: *
353: CALL DCOPY( BLKSIZ, D( B1 ), 1, WORK( INDRV4+1 ), 1 )
354: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV2+2 ), 1 )
355: CALL DCOPY( BLKSIZ-1, E( B1 ), 1, WORK( INDRV3+1 ), 1 )
356: *
357: * Compute LU factors with partial pivoting ( PT = LU )
358: *
359: TOL = ZERO
360: CALL DLAGTF( BLKSIZ, WORK( INDRV4+1 ), XJ, WORK( INDRV2+2 ),
361: $ WORK( INDRV3+1 ), TOL, WORK( INDRV5+1 ), IWORK,
362: $ IINFO )
363: *
364: * Update iteration count.
365: *
366: 70 CONTINUE
367: ITS = ITS + 1
368: IF( ITS.GT.MAXITS )
369: $ GO TO 100
370: *
371: * Normalize and scale the righthand side vector Pb.
372: *
373: SCL = BLKSIZ*ONENRM*MAX( EPS,
374: $ ABS( WORK( INDRV4+BLKSIZ ) ) ) /
375: $ DASUM( BLKSIZ, WORK( INDRV1+1 ), 1 )
376: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
377: *
378: * Solve the system LU = Pb.
379: *
380: CALL DLAGTS( -1, BLKSIZ, WORK( INDRV4+1 ), WORK( INDRV2+2 ),
381: $ WORK( INDRV3+1 ), WORK( INDRV5+1 ), IWORK,
382: $ WORK( INDRV1+1 ), TOL, IINFO )
383: *
384: * Reorthogonalize by modified Gram-Schmidt if eigenvalues are
385: * close enough.
386: *
387: IF( JBLK.EQ.1 )
388: $ GO TO 90
389: IF( ABS( XJ-XJM ).GT.ORTOL )
390: $ GPIND = J
391: IF( GPIND.NE.J ) THEN
392: DO 80 I = GPIND, J - 1
393: ZTR = -DDOT( BLKSIZ, WORK( INDRV1+1 ), 1, Z( B1, I ),
394: $ 1 )
395: CALL DAXPY( BLKSIZ, ZTR, Z( B1, I ), 1,
396: $ WORK( INDRV1+1 ), 1 )
397: 80 CONTINUE
398: END IF
399: *
400: * Check the infinity norm of the iterate.
401: *
402: 90 CONTINUE
403: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
404: NRM = ABS( WORK( INDRV1+JMAX ) )
405: *
406: * Continue for additional iterations after norm reaches
407: * stopping criterion.
408: *
409: IF( NRM.LT.DTPCRT )
410: $ GO TO 70
411: NRMCHK = NRMCHK + 1
412: IF( NRMCHK.LT.EXTRA+1 )
413: $ GO TO 70
414: *
415: GO TO 110
416: *
417: * If stopping criterion was not satisfied, update info and
418: * store eigenvector number in array ifail.
419: *
420: 100 CONTINUE
421: INFO = INFO + 1
422: IFAIL( INFO ) = J
423: *
424: * Accept iterate as jth eigenvector.
425: *
426: 110 CONTINUE
427: SCL = ONE / DNRM2( BLKSIZ, WORK( INDRV1+1 ), 1 )
428: JMAX = IDAMAX( BLKSIZ, WORK( INDRV1+1 ), 1 )
429: IF( WORK( INDRV1+JMAX ).LT.ZERO )
430: $ SCL = -SCL
431: CALL DSCAL( BLKSIZ, SCL, WORK( INDRV1+1 ), 1 )
432: 120 CONTINUE
433: DO 130 I = 1, N
434: Z( I, J ) = ZERO
435: 130 CONTINUE
436: DO 140 I = 1, BLKSIZ
437: Z( B1+I-1, J ) = WORK( INDRV1+I )
438: 140 CONTINUE
439: *
440: * Save the shift to check eigenvalue spacing at next
441: * iteration.
442: *
443: XJM = XJ
444: *
445: 150 CONTINUE
446: 160 CONTINUE
447: *
448: RETURN
449: *
450: * End of DSTEIN
451: *
452: END
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