1: *> \brief \b ZGEJSV
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
22: * M, N, A, LDA, SVA, U, LDU, V, LDV,
23: * CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
24: *
25: * .. Scalar Arguments ..
26: * IMPLICIT NONE
27: * INTEGER INFO, LDA, LDU, LDV, LWORK, M, N
28: * ..
29: * .. Array Arguments ..
30: * DOUBLE COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ), CWORK( LWORK )
31: * DOUBLE PRECISION SVA( N ), RWORK( LRWORK )
32: * INTEGER IWORK( * )
33: * CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
34: * ..
35: *
36: *
37: *> \par Purpose:
38: * =============
39: *>
40: *> \verbatim
41: *>
42: * ZGEJSV computes the singular value decomposition (SVD) of a real M-by-N
43: * matrix [A], where M >= N. The SVD of [A] is written as
44: *
45: * [A] = [U] * [SIGMA] * [V]^*,
46: *
47: * where [SIGMA] is an N-by-N (M-by-N) matrix which is zero except for its N
48: * diagonal elements, [U] is an M-by-N (or M-by-M) orthonormal matrix, and
49: * [V] is an N-by-N orthogonal matrix. The diagonal elements of [SIGMA] are
50: * the singular values of [A]. The columns of [U] and [V] are the left and
51: * the right singular vectors of [A], respectively. The matrices [U] and [V]
52: * are computed and stored in the arrays U and V, respectively. The diagonal
53: * of [SIGMA] is computed and stored in the array SVA.
54: *
55: * Arguments:
56: * ==========
57: *>
58: *> \param[in] JOBA
59: *> \verbatim
60: *> JOBA is CHARACTER*1
61: *> Specifies the level of accuracy:
62: *> = 'C': This option works well (high relative accuracy) if A = B * D,
63: *> with well-conditioned B and arbitrary diagonal matrix D.
64: *> The accuracy cannot be spoiled by COLUMN scaling. The
65: *> accuracy of the computed output depends on the condition of
66: *> B, and the procedure aims at the best theoretical accuracy.
67: *> The relative error max_{i=1:N}|d sigma_i| / sigma_i is
68: *> bounded by f(M,N)*epsilon* cond(B), independent of D.
69: *> The input matrix is preprocessed with the QRF with column
70: *> pivoting. This initial preprocessing and preconditioning by
71: *> a rank revealing QR factorization is common for all values of
72: *> JOBA. Additional actions are specified as follows:
73: *> = 'E': Computation as with 'C' with an additional estimate of the
74: *> condition number of B. It provides a realistic error bound.
75: *> = 'F': If A = D1 * C * D2 with ill-conditioned diagonal scalings
76: *> D1, D2, and well-conditioned matrix C, this option gives
77: *> higher accuracy than the 'C' option. If the structure of the
78: *> input matrix is not known, and relative accuracy is
79: *> desirable, then this option is advisable. The input matrix A
80: *> is preprocessed with QR factorization with FULL (row and
81: *> column) pivoting.
82: *> = 'G' Computation as with 'F' with an additional estimate of the
83: *> condition number of B, where A=D*B. If A has heavily weighted
84: *> rows, then using this condition number gives too pessimistic
85: *> error bound.
86: *> = 'A': Small singular values are the noise and the matrix is treated
87: *> as numerically rank defficient. The error in the computed
88: *> singular values is bounded by f(m,n)*epsilon*||A||.
89: *> The computed SVD A = U * S * V^* restores A up to
90: *> f(m,n)*epsilon*||A||.
91: *> This gives the procedure the licence to discard (set to zero)
92: *> all singular values below N*epsilon*||A||.
93: *> = 'R': Similar as in 'A'. Rank revealing property of the initial
94: *> QR factorization is used do reveal (using triangular factor)
95: *> a gap sigma_{r+1} < epsilon * sigma_r in which case the
96: *> numerical RANK is declared to be r. The SVD is computed with
97: *> absolute error bounds, but more accurately than with 'A'.
98: *> \endverbatim
99: *>
100: *> \param[in] JOBU
101: *> \verbatim
102: *> JOBU is CHARACTER*1
103: *> Specifies whether to compute the columns of U:
104: *> = 'U': N columns of U are returned in the array U.
105: *> = 'F': full set of M left sing. vectors is returned in the array U.
106: *> = 'W': U may be used as workspace of length M*N. See the description
107: *> of U.
108: *> = 'N': U is not computed.
109: *> \endverbatim
110: *>
111: *> \param[in] JOBV
112: *> \verbatim
113: *> JOBV is CHARACTER*1
114: *> Specifies whether to compute the matrix V:
115: *> = 'V': N columns of V are returned in the array V; Jacobi rotations
116: *> are not explicitly accumulated.
117: *> = 'J': N columns of V are returned in the array V, but they are
118: *> computed as the product of Jacobi rotations. This option is
119: *> allowed only if JOBU .NE. 'N', i.e. in computing the full SVD.
120: *> = 'W': V may be used as workspace of length N*N. See the description
121: *> of V.
122: *> = 'N': V is not computed.
123: *> \endverbatim
124: *>
125: *> \param[in] JOBR
126: *> \verbatim
127: *> JOBR is CHARACTER*1
128: *> Specifies the RANGE for the singular values. Issues the licence to
129: *> set to zero small positive singular values if they are outside
130: *> specified range. If A .NE. 0 is scaled so that the largest singular
131: *> value of c*A is around SQRT(BIG), BIG=DLAMCH('O'), then JOBR issues
132: *> the licence to kill columns of A whose norm in c*A is less than
133: *> SQRT(SFMIN) (for JOBR.EQ.'R'), or less than SMALL=SFMIN/EPSLN,
134: *> where SFMIN=DLAMCH('S'), EPSLN=DLAMCH('E').
135: *> = 'N': Do not kill small columns of c*A. This option assumes that
136: *> BLAS and QR factorizations and triangular solvers are
137: *> implemented to work in that range. If the condition of A
138: *> is greater than BIG, use ZGESVJ.
139: *> = 'R': RESTRICTED range for sigma(c*A) is [SQRT(SFMIN), SQRT(BIG)]
140: *> (roughly, as described above). This option is recommended.
141: *> ===========================
142: *> For computing the singular values in the FULL range [SFMIN,BIG]
143: *> use ZGESVJ.
144: *> \endverbatim
145: *>
146: *> \param[in] JOBT
147: *> \verbatim
148: *> JOBT is CHARACTER*1
149: *> If the matrix is square then the procedure may determine to use
150: *> transposed A if A^* seems to be better with respect to convergence.
151: *> If the matrix is not square, JOBT is ignored. This is subject to
152: *> changes in the future.
153: *> The decision is based on two values of entropy over the adjoint
154: *> orbit of A^* * A. See the descriptions of WORK(6) and WORK(7).
155: *> = 'T': transpose if entropy test indicates possibly faster
156: *> convergence of Jacobi process if A^* is taken as input. If A is
157: *> replaced with A^*, then the row pivoting is included automatically.
158: *> = 'N': do not speculate.
159: *> This option can be used to compute only the singular values, or the
160: *> full SVD (U, SIGMA and V). For only one set of singular vectors
161: *> (U or V), the caller should provide both U and V, as one of the
162: *> matrices is used as workspace if the matrix A is transposed.
163: *> The implementer can easily remove this constraint and make the
164: *> code more complicated. See the descriptions of U and V.
165: *> \endverbatim
166: *>
167: *> \param[in] JOBP
168: *> \verbatim
169: *> JOBP is CHARACTER*1
170: *> Issues the licence to introduce structured perturbations to drown
171: *> denormalized numbers. This licence should be active if the
172: *> denormals are poorly implemented, causing slow computation,
173: *> especially in cases of fast convergence (!). For details see [1,2].
174: *> For the sake of simplicity, this perturbations are included only
175: *> when the full SVD or only the singular values are requested. The
176: *> implementer/user can easily add the perturbation for the cases of
177: *> computing one set of singular vectors.
178: *> = 'P': introduce perturbation
179: *> = 'N': do not perturb
180: *> \endverbatim
181: *>
182: *> \param[in] M
183: *> \verbatim
184: *> M is INTEGER
185: *> The number of rows of the input matrix A. M >= 0.
186: *> \endverbatim
187: *>
188: *> \param[in] N
189: *> \verbatim
190: *> N is INTEGER
191: *> The number of columns of the input matrix A. M >= N >= 0.
192: *> \endverbatim
193: *>
194: *> \param[in,out] A
195: *> \verbatim
196: *> A is DOUBLE COMPLEX array, dimension (LDA,N)
197: *> On entry, the M-by-N matrix A.
198: *> \endverbatim
199: *>
200: *> \param[in] LDA
201: *> \verbatim
202: *> LDA is INTEGER
203: *> The leading dimension of the array A. LDA >= max(1,M).
204: *> \endverbatim
205: *>
206: *> \param[out] SVA
207: *> \verbatim
208: *> SVA is DOUBLE PRECISION array, dimension (N)
209: *> On exit,
210: *> - For WORK(1)/WORK(2) = ONE: The singular values of A. During the
211: *> computation SVA contains Euclidean column norms of the
212: *> iterated matrices in the array A.
213: *> - For WORK(1) .NE. WORK(2): The singular values of A are
214: *> (WORK(1)/WORK(2)) * SVA(1:N). This factored form is used if
215: *> sigma_max(A) overflows or if small singular values have been
216: *> saved from underflow by scaling the input matrix A.
217: *> - If JOBR='R' then some of the singular values may be returned
218: *> as exact zeros obtained by "set to zero" because they are
219: *> below the numerical rank threshold or are denormalized numbers.
220: *> \endverbatim
221: *>
222: *> \param[out] U
223: *> \verbatim
224: *> U is DOUBLE COMPLEX array, dimension ( LDU, N )
225: *> If JOBU = 'U', then U contains on exit the M-by-N matrix of
226: *> the left singular vectors.
227: *> If JOBU = 'F', then U contains on exit the M-by-M matrix of
228: *> the left singular vectors, including an ONB
229: *> of the orthogonal complement of the Range(A).
230: *> If JOBU = 'W' .AND. (JOBV.EQ.'V' .AND. JOBT.EQ.'T' .AND. M.EQ.N),
231: *> then U is used as workspace if the procedure
232: *> replaces A with A^*. In that case, [V] is computed
233: *> in U as left singular vectors of A^* and then
234: *> copied back to the V array. This 'W' option is just
235: *> a reminder to the caller that in this case U is
236: *> reserved as workspace of length N*N.
237: *> If JOBU = 'N' U is not referenced.
238: *> \endverbatim
239: *>
240: *> \param[in] LDU
241: *> \verbatim
242: *> LDU is INTEGER
243: *> The leading dimension of the array U, LDU >= 1.
244: *> IF JOBU = 'U' or 'F' or 'W', then LDU >= M.
245: *> \endverbatim
246: *>
247: *> \param[out] V
248: *> \verbatim
249: *> V is DOUBLE COMPLEX array, dimension ( LDV, N )
250: *> If JOBV = 'V', 'J' then V contains on exit the N-by-N matrix of
251: *> the right singular vectors;
252: *> If JOBV = 'W', AND (JOBU.EQ.'U' AND JOBT.EQ.'T' AND M.EQ.N),
253: *> then V is used as workspace if the pprocedure
254: *> replaces A with A^*. In that case, [U] is computed
255: *> in V as right singular vectors of A^* and then
256: *> copied back to the U array. This 'W' option is just
257: *> a reminder to the caller that in this case V is
258: *> reserved as workspace of length N*N.
259: *> If JOBV = 'N' V is not referenced.
260: *> \endverbatim
261: *>
262: *> \param[in] LDV
263: *> \verbatim
264: *> LDV is INTEGER
265: *> The leading dimension of the array V, LDV >= 1.
266: *> If JOBV = 'V' or 'J' or 'W', then LDV >= N.
267: *> \endverbatim
268: *>
269: *> \param[out] CWORK
270: *> \verbatim
271: *> CWORK (workspace)
272: *> CWORK is DOUBLE COMPLEX array, dimension at least LWORK.
273: *> \endverbatim
274: *>
275: *> \param[in] LWORK
276: *> \verbatim
277: *> LWORK is INTEGER
278: *> Length of CWORK to confirm proper allocation of workspace.
279: *> LWORK depends on the job:
280: *>
281: *> 1. If only SIGMA is needed ( JOBU.EQ.'N', JOBV.EQ.'N' ) and
282: *> 1.1 .. no scaled condition estimate required (JOBE.EQ.'N'):
283: *> LWORK >= 2*N+1. This is the minimal requirement.
284: *> ->> For optimal performance (blocked code) the optimal value
285: *> is LWORK >= N + (N+1)*NB. Here NB is the optimal
286: *> block size for ZGEQP3 and ZGEQRF.
287: *> In general, optimal LWORK is computed as
288: *> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF)).
289: *> 1.2. .. an estimate of the scaled condition number of A is
290: *> required (JOBA='E', or 'G'). In this case, LWORK the minimal
291: *> requirement is LWORK >= N*N + 3*N.
292: *> ->> For optimal performance (blocked code) the optimal value
293: *> is LWORK >= max(N+(N+1)*NB, N*N+3*N).
294: *> In general, the optimal length LWORK is computed as
295: *> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(ZGEQRF),
296: *> N+N*N+LWORK(CPOCON)).
297: *>
298: *> 2. If SIGMA and the right singular vectors are needed (JOBV.EQ.'V'),
299: *> (JOBU.EQ.'N')
300: *> -> the minimal requirement is LWORK >= 3*N.
301: *> -> For optimal performance, LWORK >= max(N+(N+1)*NB, 3*N,2*N+N*NB),
302: *> where NB is the optimal block size for ZGEQP3, ZGEQRF, ZGELQ,
303: *> CUNMLQ. In general, the optimal length LWORK is computed as
304: *> LWORK >= max(N+LWORK(ZGEQP3), N+LWORK(CPOCON), N+LWORK(ZGESVJ),
305: *> N+LWORK(ZGELQF), 2*N+LWORK(ZGEQRF), N+LWORK(CUNMLQ)).
306: *>
307: *> 3. If SIGMA and the left singular vectors are needed
308: *> -> the minimal requirement is LWORK >= 3*N.
309: *> -> For optimal performance:
310: *> if JOBU.EQ.'U' :: LWORK >= max(3*N, N+(N+1)*NB, 2*N+N*NB),
311: *> where NB is the optimal block size for ZGEQP3, ZGEQRF, CUNMQR.
312: *> In general, the optimal length LWORK is computed as
313: *> LWORK >= max(N+LWORK(ZGEQP3),N+LWORK(CPOCON),
314: *> 2*N+LWORK(ZGEQRF), N+LWORK(CUNMQR)).
315: *>
316: *> 4. If the full SVD is needed: (JOBU.EQ.'U' or JOBU.EQ.'F') and
317: *> 4.1. if JOBV.EQ.'V'
318: *> the minimal requirement is LWORK >= 5*N+2*N*N.
319: *> 4.2. if JOBV.EQ.'J' the minimal requirement is
320: *> LWORK >= 4*N+N*N.
321: *> In both cases, the allocated CWORK can accomodate blocked runs
322: *> of ZGEQP3, ZGEQRF, ZGELQF, SUNMQR, CUNMLQ.
323: *> \endverbatim
324: *>
325: *> \param[out] RWORK
326: *> \verbatim
327: *> RWORK is DOUBLE PRECISION array, dimension at least LRWORK.
328: *> On exit,
329: *> RWORK(1) = Determines the scaling factor SCALE = RWORK(2) / RWORK(1)
330: *> such that SCALE*SVA(1:N) are the computed singular values
331: *> of A. (See the description of SVA().)
332: *> RWORK(2) = See the description of RWORK(1).
333: *> RWORK(3) = SCONDA is an estimate for the condition number of
334: *> column equilibrated A. (If JOBA .EQ. 'E' or 'G')
335: *> SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
336: *> It is computed using SPOCON. It holds
337: *> N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
338: *> where R is the triangular factor from the QRF of A.
339: *> However, if R is truncated and the numerical rank is
340: *> determined to be strictly smaller than N, SCONDA is
341: *> returned as -1, thus indicating that the smallest
342: *> singular values might be lost.
343: *>
344: *> If full SVD is needed, the following two condition numbers are
345: *> useful for the analysis of the algorithm. They are provied for
346: *> a developer/implementer who is familiar with the details of
347: *> the method.
348: *>
349: *> RWORK(4) = an estimate of the scaled condition number of the
350: *> triangular factor in the first QR factorization.
351: *> RWORK(5) = an estimate of the scaled condition number of the
352: *> triangular factor in the second QR factorization.
353: *> The following two parameters are computed if JOBT .EQ. 'T'.
354: *> They are provided for a developer/implementer who is familiar
355: *> with the details of the method.
356: *> RWORK(6) = the entropy of A^* * A :: this is the Shannon entropy
357: *> of diag(A^* * A) / Trace(A^* * A) taken as point in the
358: *> probability simplex.
359: *> RWORK(7) = the entropy of A * A^*. (See the description of RWORK(6).)
360: *> \endverbatim
361: *>
362: *> \param[in] LRWORK
363: *> \verbatim
364: *> LRWORK is INTEGER
365: *> Length of RWORK to confirm proper allocation of workspace.
366: *> LRWORK depends on the job:
367: *>
368: *> 1. If only singular values are requested i.e. if
369: *> LSAME(JOBU,'N') .AND. LSAME(JOBV,'N')
370: *> then:
371: *> 1.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
372: *> then LRWORK = max( 7, N + 2 * M ).
373: *> 1.2. Otherwise, LRWORK = max( 7, 2 * N ).
374: *> 2. If singular values with the right singular vectors are requested
375: *> i.e. if
376: *> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J')) .AND.
377: *> .NOT.(LSAME(JOBU,'U').OR.LSAME(JOBU,'F'))
378: *> then:
379: *> 2.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
380: *> then LRWORK = max( 7, N + 2 * M ).
381: *> 2.2. Otherwise, LRWORK = max( 7, 2 * N ).
382: *> 3. If singular values with the left singular vectors are requested, i.e. if
383: *> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
384: *> .NOT.(LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
385: *> then:
386: *> 3.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
387: *> then LRWORK = max( 7, N + 2 * M ).
388: *> 3.2. Otherwise, LRWORK = max( 7, 2 * N ).
389: *> 4. If singular values with both the left and the right singular vectors
390: *> are requested, i.e. if
391: *> (LSAME(JOBU,'U').OR.LSAME(JOBU,'F')) .AND.
392: *> (LSAME(JOBV,'V').OR.LSAME(JOBV,'J'))
393: *> then:
394: *> 4.1. If LSAME(JOBT,'T') .OR. LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'),
395: *> then LRWORK = max( 7, N + 2 * M ).
396: *> 4.2. Otherwise, LRWORK = max( 7, 2 * N ).
397: *> \endverbatim
398: *>
399: *> \param[out] IWORK
400: *> \verbatim
401: *> IWORK is INTEGER array, of dimension:
402: *> If LSAME(JOBA,'F') .OR. LSAME(JOBA,'G'), then
403: *> the dimension of IWORK is max( 3, 2 * N + M ).
404: *> Otherwise, the dimension of IWORK is
405: *> -> max( 3, 2*N ) for full SVD
406: *> -> max( 3, N ) for singular values only or singular
407: *> values with one set of singular vectors (left or right)
408: *> On exit,
409: *> IWORK(1) = the numerical rank determined after the initial
410: *> QR factorization with pivoting. See the descriptions
411: *> of JOBA and JOBR.
412: *> IWORK(2) = the number of the computed nonzero singular values
413: *> IWORK(3) = if nonzero, a warning message:
414: *> If IWORK(3).EQ.1 then some of the column norms of A
415: *> were denormalized floats. The requested high accuracy
416: *> is not warranted by the data.
417: *> \endverbatim
418: *>
419: *> \param[out] INFO
420: *> \verbatim
421: *> INFO is INTEGER
422: *> < 0 : if INFO = -i, then the i-th argument had an illegal value.
423: *> = 0 : successfull exit;
424: *> > 0 : ZGEJSV did not converge in the maximal allowed number
425: *> of sweeps. The computed values may be inaccurate.
426: *> \endverbatim
427: *
428: * Authors:
429: * ========
430: *
431: *> \author Univ. of Tennessee
432: *> \author Univ. of California Berkeley
433: *> \author Univ. of Colorado Denver
434: *> \author NAG Ltd.
435: *
436: *> \date November 2015
437: *
438: *> \ingroup complex16GEsing
439: *
440: *> \par Further Details:
441: * =====================
442: *>
443: *> \verbatim
444: *>
445: *> ZGEJSV implements a preconditioned Jacobi SVD algorithm. It uses ZGEQP3,
446: *> ZGEQRF, and ZGELQF as preprocessors and preconditioners. Optionally, an
447: *> additional row pivoting can be used as a preprocessor, which in some
448: *> cases results in much higher accuracy. An example is matrix A with the
449: *> structure A = D1 * C * D2, where D1, D2 are arbitrarily ill-conditioned
450: *> diagonal matrices and C is well-conditioned matrix. In that case, complete
451: *> pivoting in the first QR factorizations provides accuracy dependent on the
452: *> condition number of C, and independent of D1, D2. Such higher accuracy is
453: *> not completely understood theoretically, but it works well in practice.
454: *> Further, if A can be written as A = B*D, with well-conditioned B and some
455: *> diagonal D, then the high accuracy is guaranteed, both theoretically and
456: *> in software, independent of D. For more details see [1], [2].
457: *> The computational range for the singular values can be the full range
458: *> ( UNDERFLOW,OVERFLOW ), provided that the machine arithmetic and the BLAS
459: *> & LAPACK routines called by ZGEJSV are implemented to work in that range.
460: *> If that is not the case, then the restriction for safe computation with
461: *> the singular values in the range of normalized IEEE numbers is that the
462: *> spectral condition number kappa(A)=sigma_max(A)/sigma_min(A) does not
463: *> overflow. This code (ZGEJSV) is best used in this restricted range,
464: *> meaning that singular values of magnitude below ||A||_2 / DLAMCH('O') are
465: *> returned as zeros. See JOBR for details on this.
466: *> Further, this implementation is somewhat slower than the one described
467: *> in [1,2] due to replacement of some non-LAPACK components, and because
468: *> the choice of some tuning parameters in the iterative part (ZGESVJ) is
469: *> left to the implementer on a particular machine.
470: *> The rank revealing QR factorization (in this code: ZGEQP3) should be
471: *> implemented as in [3]. We have a new version of ZGEQP3 under development
472: *> that is more robust than the current one in LAPACK, with a cleaner cut in
473: *> rank defficient cases. It will be available in the SIGMA library [4].
474: *> If M is much larger than N, it is obvious that the inital QRF with
475: *> column pivoting can be preprocessed by the QRF without pivoting. That
476: *> well known trick is not used in ZGEJSV because in some cases heavy row
477: *> weighting can be treated with complete pivoting. The overhead in cases
478: *> M much larger than N is then only due to pivoting, but the benefits in
479: *> terms of accuracy have prevailed. The implementer/user can incorporate
480: *> this extra QRF step easily. The implementer can also improve data movement
481: *> (matrix transpose, matrix copy, matrix transposed copy) - this
482: *> implementation of ZGEJSV uses only the simplest, naive data movement.
483: *
484: *> \par Contributors:
485: * ==================
486: *>
487: *> Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
488: *
489: *> \par References:
490: * ================
491: *>
492: *> \verbatim
493: *>
494: * [1] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm I.
495: * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1322-1342.
496: * LAPACK Working note 169.
497: * [2] Z. Drmac and K. Veselic: New fast and accurate Jacobi SVD algorithm II.
498: * SIAM J. Matrix Anal. Appl. Vol. 35, No. 2 (2008), pp. 1343-1362.
499: * LAPACK Working note 170.
500: * [3] Z. Drmac and Z. Bujanovic: On the failure of rank-revealing QR
501: * factorization software - a case study.
502: * ACM Trans. math. Softw. Vol. 35, No 2 (2008), pp. 1-28.
503: * LAPACK Working note 176.
504: * [4] Z. Drmac: SIGMA - mathematical software library for accurate SVD, PSV,
505: * QSVD, (H,K)-SVD computations.
506: * Department of Mathematics, University of Zagreb, 2008.
507: *> \endverbatim
508: *
509: *> \par Bugs, examples and comments:
510: * =================================
511: *>
512: *> Please report all bugs and send interesting examples and/or comments to
513: *> drmac@math.hr. Thank you.
514: *>
515: * =====================================================================
516: SUBROUTINE ZGEJSV( JOBA, JOBU, JOBV, JOBR, JOBT, JOBP,
517: $ M, N, A, LDA, SVA, U, LDU, V, LDV,
518: $ CWORK, LWORK, RWORK, LRWORK, IWORK, INFO )
519: *
520: * -- LAPACK computational routine (version 3.6.0) --
521: * -- LAPACK is a software package provided by Univ. of Tennessee, --
522: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
523: * November 2015
524: *
525: * .. Scalar Arguments ..
526: IMPLICIT NONE
527: INTEGER INFO, LDA, LDU, LDV, LWORK, LRWORK, M, N
528: * ..
529: * .. Array Arguments ..
530: DOUBLE COMPLEX A( LDA, * ), U( LDU, * ), V( LDV, * ),
531: $ CWORK( LWORK )
532: DOUBLE PRECISION SVA( N ), RWORK( * )
533: INTEGER IWORK( * )
534: CHARACTER*1 JOBA, JOBP, JOBR, JOBT, JOBU, JOBV
535: * ..
536: *
537: * ===========================================================================
538: *
539: * .. Local Parameters ..
540: DOUBLE PRECISION ZERO, ONE
541: PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
542: DOUBLE COMPLEX CZERO, CONE
543: PARAMETER ( CZERO = ( 0.0D0, 0.0D0 ), CONE = ( 1.0D0, 0.0D0 ) )
544: * ..
545: * .. Local Scalars ..
546: DOUBLE COMPLEX CTEMP
547: DOUBLE PRECISION AAPP, AAQQ, AATMAX, AATMIN, BIG, BIG1,
548: $ COND_OK, CONDR1, CONDR2, ENTRA, ENTRAT, EPSLN,
549: $ MAXPRJ, SCALEM, SCONDA, SFMIN, SMALL, TEMP1,
550: $ USCAL1, USCAL2, XSC
551: INTEGER IERR, N1, NR, NUMRANK, p, q, WARNING
552: LOGICAL ALMORT, DEFR, ERREST, GOSCAL, JRACC, KILL, LSVEC,
553: $ L2ABER, L2KILL, L2PERT, L2RANK, L2TRAN,
554: $ NOSCAL, ROWPIV, RSVEC, TRANSP
555: * ..
556: * .. Intrinsic Functions ..
557: INTRINSIC ABS, DCMPLX, DCONJG, DLOG, DMAX1, DMIN1, DFLOAT,
558: $ MAX0, MIN0, NINT, DSQRT
559: * ..
560: * .. External Functions ..
561: DOUBLE PRECISION DLAMCH, DZNRM2
562: INTEGER IDAMAX
563: LOGICAL LSAME
564: EXTERNAL IDAMAX, LSAME, DLAMCH, DZNRM2
565: * ..
566: * .. External Subroutines ..
567: EXTERNAL ZCOPY, ZGELQF, ZGEQP3, ZGEQRF, ZLACPY, ZLASCL,
568: $ ZLASET, ZLASSQ, ZLASWP, ZUNGQR, ZUNMLQ,
569: $ ZUNMQR, ZPOCON, DSCAL, ZDSCAL, ZSWAP, ZTRSM, XERBLA
570: *
571: EXTERNAL ZGESVJ
572: * ..
573: *
574: * Test the input arguments
575: *
576:
577: LSVEC = LSAME( JOBU, 'U' ) .OR. LSAME( JOBU, 'F' )
578: JRACC = LSAME( JOBV, 'J' )
579: RSVEC = LSAME( JOBV, 'V' ) .OR. JRACC
580: ROWPIV = LSAME( JOBA, 'F' ) .OR. LSAME( JOBA, 'G' )
581: L2RANK = LSAME( JOBA, 'R' )
582: L2ABER = LSAME( JOBA, 'A' )
583: ERREST = LSAME( JOBA, 'E' ) .OR. LSAME( JOBA, 'G' )
584: L2TRAN = LSAME( JOBT, 'T' )
585: L2KILL = LSAME( JOBR, 'R' )
586: DEFR = LSAME( JOBR, 'N' )
587: L2PERT = LSAME( JOBP, 'P' )
588: *
589: IF ( .NOT.(ROWPIV .OR. L2RANK .OR. L2ABER .OR.
590: $ ERREST .OR. LSAME( JOBA, 'C' ) )) THEN
591: INFO = - 1
592: ELSE IF ( .NOT.( LSVEC .OR. LSAME( JOBU, 'N' ) .OR.
593: $ LSAME( JOBU, 'W' )) ) THEN
594: INFO = - 2
595: ELSE IF ( .NOT.( RSVEC .OR. LSAME( JOBV, 'N' ) .OR.
596: $ LSAME( JOBV, 'W' )) .OR. ( JRACC .AND. (.NOT.LSVEC) ) ) THEN
597: INFO = - 3
598: ELSE IF ( .NOT. ( L2KILL .OR. DEFR ) ) THEN
599: INFO = - 4
600: ELSE IF ( .NOT. ( L2TRAN .OR. LSAME( JOBT, 'N' ) ) ) THEN
601: INFO = - 5
602: ELSE IF ( .NOT. ( L2PERT .OR. LSAME( JOBP, 'N' ) ) ) THEN
603: INFO = - 6
604: ELSE IF ( M .LT. 0 ) THEN
605: INFO = - 7
606: ELSE IF ( ( N .LT. 0 ) .OR. ( N .GT. M ) ) THEN
607: INFO = - 8
608: ELSE IF ( LDA .LT. M ) THEN
609: INFO = - 10
610: ELSE IF ( LSVEC .AND. ( LDU .LT. M ) ) THEN
611: INFO = - 13
612: ELSE IF ( RSVEC .AND. ( LDV .LT. N ) ) THEN
613: INFO = - 15
614: ELSE IF ( (.NOT.(LSVEC .OR. RSVEC .OR. ERREST).AND.
615: $ (LWORK .LT. 2*N+1)) .OR.
616: $ (.NOT.(LSVEC .OR. RSVEC) .AND. ERREST .AND.
617: $ (LWORK .LT. N*N+3*N)) .OR.
618: $ (LSVEC .AND. (.NOT.RSVEC) .AND. (LWORK .LT. 3*N))
619: $ .OR.
620: $ (RSVEC .AND. (.NOT.LSVEC) .AND. (LWORK .LT. 3*N))
621: $ .OR.
622: $ (LSVEC .AND. RSVEC .AND. (.NOT.JRACC) .AND.
623: $ (LWORK.LT.5*N+2*N*N))
624: $ .OR. (LSVEC .AND. RSVEC .AND. JRACC .AND.
625: $ LWORK.LT.4*N+N*N))
626: $ THEN
627: INFO = - 17
628: ELSE IF ( LRWORK.LT. MAX0(N+2*M,7)) THEN
629: INFO = -19
630: ELSE
631: * #:)
632: INFO = 0
633: END IF
634: *
635: IF ( INFO .NE. 0 ) THEN
636: * #:(
637: CALL XERBLA( 'ZGEJSV', - INFO )
638: RETURN
639: END IF
640: *
641: * Quick return for void matrix (Y3K safe)
642: * #:)
643: IF ( ( M .EQ. 0 ) .OR. ( N .EQ. 0 ) ) RETURN
644: *
645: * Determine whether the matrix U should be M x N or M x M
646: *
647: IF ( LSVEC ) THEN
648: N1 = N
649: IF ( LSAME( JOBU, 'F' ) ) N1 = M
650: END IF
651: *
652: * Set numerical parameters
653: *
654: *! NOTE: Make sure DLAMCH() does not fail on the target architecture.
655: *
656: EPSLN = DLAMCH('Epsilon')
657: SFMIN = DLAMCH('SafeMinimum')
658: SMALL = SFMIN / EPSLN
659: BIG = DLAMCH('O')
660: * BIG = ONE / SFMIN
661: *
662: * Initialize SVA(1:N) = diag( ||A e_i||_2 )_1^N
663: *
664: *(!) If necessary, scale SVA() to protect the largest norm from
665: * overflow. It is possible that this scaling pushes the smallest
666: * column norm left from the underflow threshold (extreme case).
667: *
668: SCALEM = ONE / DSQRT(DFLOAT(M)*DFLOAT(N))
669: NOSCAL = .TRUE.
670: GOSCAL = .TRUE.
671: DO 1874 p = 1, N
672: AAPP = ZERO
673: AAQQ = ONE
674: CALL ZLASSQ( M, A(1,p), 1, AAPP, AAQQ )
675: IF ( AAPP .GT. BIG ) THEN
676: INFO = - 9
677: CALL XERBLA( 'ZGEJSV', -INFO )
678: RETURN
679: END IF
680: AAQQ = DSQRT(AAQQ)
681: IF ( ( AAPP .LT. (BIG / AAQQ) ) .AND. NOSCAL ) THEN
682: SVA(p) = AAPP * AAQQ
683: ELSE
684: NOSCAL = .FALSE.
685: SVA(p) = AAPP * ( AAQQ * SCALEM )
686: IF ( GOSCAL ) THEN
687: GOSCAL = .FALSE.
688: CALL DSCAL( p-1, SCALEM, SVA, 1 )
689: END IF
690: END IF
691: 1874 CONTINUE
692: *
693: IF ( NOSCAL ) SCALEM = ONE
694: *
695: AAPP = ZERO
696: AAQQ = BIG
697: DO 4781 p = 1, N
698: AAPP = DMAX1( AAPP, SVA(p) )
699: IF ( SVA(p) .NE. ZERO ) AAQQ = DMIN1( AAQQ, SVA(p) )
700: 4781 CONTINUE
701: *
702: * Quick return for zero M x N matrix
703: * #:)
704: IF ( AAPP .EQ. ZERO ) THEN
705: IF ( LSVEC ) CALL ZLASET( 'G', M, N1, CZERO, CONE, U, LDU )
706: IF ( RSVEC ) CALL ZLASET( 'G', N, N, CZERO, CONE, V, LDV )
707: RWORK(1) = ONE
708: RWORK(2) = ONE
709: IF ( ERREST ) RWORK(3) = ONE
710: IF ( LSVEC .AND. RSVEC ) THEN
711: RWORK(4) = ONE
712: RWORK(5) = ONE
713: END IF
714: IF ( L2TRAN ) THEN
715: RWORK(6) = ZERO
716: RWORK(7) = ZERO
717: END IF
718: IWORK(1) = 0
719: IWORK(2) = 0
720: IWORK(3) = 0
721: RETURN
722: END IF
723: *
724: * Issue warning if denormalized column norms detected. Override the
725: * high relative accuracy request. Issue licence to kill columns
726: * (set them to zero) whose norm is less than sigma_max / BIG (roughly).
727: * #:(
728: WARNING = 0
729: IF ( AAQQ .LE. SFMIN ) THEN
730: L2RANK = .TRUE.
731: L2KILL = .TRUE.
732: WARNING = 1
733: END IF
734: *
735: * Quick return for one-column matrix
736: * #:)
737: IF ( N .EQ. 1 ) THEN
738: *
739: IF ( LSVEC ) THEN
740: CALL ZLASCL( 'G',0,0,SVA(1),SCALEM, M,1,A(1,1),LDA,IERR )
741: CALL ZLACPY( 'A', M, 1, A, LDA, U, LDU )
742: * computing all M left singular vectors of the M x 1 matrix
743: IF ( N1 .NE. N ) THEN
744: CALL ZGEQRF( M, N, U,LDU, CWORK, CWORK(N+1),LWORK-N,IERR )
745: CALL ZUNGQR( M,N1,1, U,LDU,CWORK,CWORK(N+1),LWORK-N,IERR )
746: CALL ZCOPY( M, A(1,1), 1, U(1,1), 1 )
747: END IF
748: END IF
749: IF ( RSVEC ) THEN
750: V(1,1) = CONE
751: END IF
752: IF ( SVA(1) .LT. (BIG*SCALEM) ) THEN
753: SVA(1) = SVA(1) / SCALEM
754: SCALEM = ONE
755: END IF
756: RWORK(1) = ONE / SCALEM
757: RWORK(2) = ONE
758: IF ( SVA(1) .NE. ZERO ) THEN
759: IWORK(1) = 1
760: IF ( ( SVA(1) / SCALEM) .GE. SFMIN ) THEN
761: IWORK(2) = 1
762: ELSE
763: IWORK(2) = 0
764: END IF
765: ELSE
766: IWORK(1) = 0
767: IWORK(2) = 0
768: END IF
769: IWORK(3) = 0
770: IF ( ERREST ) RWORK(3) = ONE
771: IF ( LSVEC .AND. RSVEC ) THEN
772: RWORK(4) = ONE
773: RWORK(5) = ONE
774: END IF
775: IF ( L2TRAN ) THEN
776: RWORK(6) = ZERO
777: RWORK(7) = ZERO
778: END IF
779: RETURN
780: *
781: END IF
782: *
783: TRANSP = .FALSE.
784: L2TRAN = L2TRAN .AND. ( M .EQ. N )
785: *
786: AATMAX = -ONE
787: AATMIN = BIG
788: IF ( ROWPIV .OR. L2TRAN ) THEN
789: *
790: * Compute the row norms, needed to determine row pivoting sequence
791: * (in the case of heavily row weighted A, row pivoting is strongly
792: * advised) and to collect information needed to compare the
793: * structures of A * A^* and A^* * A (in the case L2TRAN.EQ..TRUE.).
794: *
795: IF ( L2TRAN ) THEN
796: DO 1950 p = 1, M
797: XSC = ZERO
798: TEMP1 = ONE
799: CALL ZLASSQ( N, A(p,1), LDA, XSC, TEMP1 )
800: * ZLASSQ gets both the ell_2 and the ell_infinity norm
801: * in one pass through the vector
802: RWORK(M+N+p) = XSC * SCALEM
803: RWORK(N+p) = XSC * (SCALEM*DSQRT(TEMP1))
804: AATMAX = DMAX1( AATMAX, RWORK(N+p) )
805: IF (RWORK(N+p) .NE. ZERO)
806: $ AATMIN = DMIN1(AATMIN,RWORK(N+p))
807: 1950 CONTINUE
808: ELSE
809: DO 1904 p = 1, M
810: RWORK(M+N+p) = SCALEM*ABS( A(p,IDAMAX(N,A(p,1),LDA)) )
811: AATMAX = DMAX1( AATMAX, RWORK(M+N+p) )
812: AATMIN = DMIN1( AATMIN, RWORK(M+N+p) )
813: 1904 CONTINUE
814: END IF
815: *
816: END IF
817: *
818: * For square matrix A try to determine whether A^* would be better
819: * input for the preconditioned Jacobi SVD, with faster convergence.
820: * The decision is based on an O(N) function of the vector of column
821: * and row norms of A, based on the Shannon entropy. This should give
822: * the right choice in most cases when the difference actually matters.
823: * It may fail and pick the slower converging side.
824: *
825: ENTRA = ZERO
826: ENTRAT = ZERO
827: IF ( L2TRAN ) THEN
828: *
829: XSC = ZERO
830: TEMP1 = ONE
831: CALL ZLASSQ( N, SVA, 1, XSC, TEMP1 )
832: TEMP1 = ONE / TEMP1
833: *
834: ENTRA = ZERO
835: DO 1113 p = 1, N
836: BIG1 = ( ( SVA(p) / XSC )**2 ) * TEMP1
837: IF ( BIG1 .NE. ZERO ) ENTRA = ENTRA + BIG1 * DLOG(BIG1)
838: 1113 CONTINUE
839: ENTRA = - ENTRA / DLOG(DFLOAT(N))
840: *
841: * Now, SVA().^2/Trace(A^* * A) is a point in the probability simplex.
842: * It is derived from the diagonal of A^* * A. Do the same with the
843: * diagonal of A * A^*, compute the entropy of the corresponding
844: * probability distribution. Note that A * A^* and A^* * A have the
845: * same trace.
846: *
847: ENTRAT = ZERO
848: DO 1114 p = N+1, N+M
849: BIG1 = ( ( RWORK(p) / XSC )**2 ) * TEMP1
850: IF ( BIG1 .NE. ZERO ) ENTRAT = ENTRAT + BIG1 * DLOG(BIG1)
851: 1114 CONTINUE
852: ENTRAT = - ENTRAT / DLOG(DFLOAT(M))
853: *
854: * Analyze the entropies and decide A or A^*. Smaller entropy
855: * usually means better input for the algorithm.
856: *
857: TRANSP = ( ENTRAT .LT. ENTRA )
858: TRANSP = .TRUE.
859: *
860: * If A^* is better than A, take the adjoint of A.
861: *
862: IF ( TRANSP ) THEN
863: * In an optimal implementation, this trivial transpose
864: * should be replaced with faster transpose.
865: DO 1115 p = 1, N - 1
866: A(p,p) = DCONJG(A(p,p))
867: DO 1116 q = p + 1, N
868: CTEMP = DCONJG(A(q,p))
869: A(q,p) = DCONJG(A(p,q))
870: A(p,q) = CTEMP
871: 1116 CONTINUE
872: 1115 CONTINUE
873: A(N,N) = DCONJG(A(N,N))
874: DO 1117 p = 1, N
875: RWORK(M+N+p) = SVA(p)
876: SVA(p) = RWORK(N+p)
877: * previously computed row 2-norms are now column 2-norms
878: * of the transposed matrix
879: 1117 CONTINUE
880: TEMP1 = AAPP
881: AAPP = AATMAX
882: AATMAX = TEMP1
883: TEMP1 = AAQQ
884: AAQQ = AATMIN
885: AATMIN = TEMP1
886: KILL = LSVEC
887: LSVEC = RSVEC
888: RSVEC = KILL
889: IF ( LSVEC ) N1 = N
890: *
891: ROWPIV = .TRUE.
892: END IF
893: *
894: END IF
895: * END IF L2TRAN
896: *
897: * Scale the matrix so that its maximal singular value remains less
898: * than SQRT(BIG) -- the matrix is scaled so that its maximal column
899: * has Euclidean norm equal to SQRT(BIG/N). The only reason to keep
900: * SQRT(BIG) instead of BIG is the fact that ZGEJSV uses LAPACK and
901: * BLAS routines that, in some implementations, are not capable of
902: * working in the full interval [SFMIN,BIG] and that they may provoke
903: * overflows in the intermediate results. If the singular values spread
904: * from SFMIN to BIG, then ZGESVJ will compute them. So, in that case,
905: * one should use ZGESVJ instead of ZGEJSV.
906: *
907: BIG1 = DSQRT( BIG )
908: TEMP1 = DSQRT( BIG / DFLOAT(N) )
909: *
910: CALL ZLASCL( 'G', 0, 0, AAPP, TEMP1, N, 1, SVA, N, IERR )
911: IF ( AAQQ .GT. (AAPP * SFMIN) ) THEN
912: AAQQ = ( AAQQ / AAPP ) * TEMP1
913: ELSE
914: AAQQ = ( AAQQ * TEMP1 ) / AAPP
915: END IF
916: TEMP1 = TEMP1 * SCALEM
917: CALL ZLASCL( 'G', 0, 0, AAPP, TEMP1, M, N, A, LDA, IERR )
918: *
919: * To undo scaling at the end of this procedure, multiply the
920: * computed singular values with USCAL2 / USCAL1.
921: *
922: USCAL1 = TEMP1
923: USCAL2 = AAPP
924: *
925: IF ( L2KILL ) THEN
926: * L2KILL enforces computation of nonzero singular values in
927: * the restricted range of condition number of the initial A,
928: * sigma_max(A) / sigma_min(A) approx. SQRT(BIG)/SQRT(SFMIN).
929: XSC = DSQRT( SFMIN )
930: ELSE
931: XSC = SMALL
932: *
933: * Now, if the condition number of A is too big,
934: * sigma_max(A) / sigma_min(A) .GT. SQRT(BIG/N) * EPSLN / SFMIN,
935: * as a precaution measure, the full SVD is computed using ZGESVJ
936: * with accumulated Jacobi rotations. This provides numerically
937: * more robust computation, at the cost of slightly increased run
938: * time. Depending on the concrete implementation of BLAS and LAPACK
939: * (i.e. how they behave in presence of extreme ill-conditioning) the
940: * implementor may decide to remove this switch.
941: IF ( ( AAQQ.LT.DSQRT(SFMIN) ) .AND. LSVEC .AND. RSVEC ) THEN
942: JRACC = .TRUE.
943: END IF
944: *
945: END IF
946: IF ( AAQQ .LT. XSC ) THEN
947: DO 700 p = 1, N
948: IF ( SVA(p) .LT. XSC ) THEN
949: CALL ZLASET( 'A', M, 1, CZERO, CZERO, A(1,p), LDA )
950: SVA(p) = ZERO
951: END IF
952: 700 CONTINUE
953: END IF
954: *
955: * Preconditioning using QR factorization with pivoting
956: *
957: IF ( ROWPIV ) THEN
958: * Optional row permutation (Bjoerck row pivoting):
959: * A result by Cox and Higham shows that the Bjoerck's
960: * row pivoting combined with standard column pivoting
961: * has similar effect as Powell-Reid complete pivoting.
962: * The ell-infinity norms of A are made nonincreasing.
963: DO 1952 p = 1, M - 1
964: q = IDAMAX( M-p+1, RWORK(M+N+p), 1 ) + p - 1
965: IWORK(2*N+p) = q
966: IF ( p .NE. q ) THEN
967: TEMP1 = RWORK(M+N+p)
968: RWORK(M+N+p) = RWORK(M+N+q)
969: RWORK(M+N+q) = TEMP1
970: END IF
971: 1952 CONTINUE
972: CALL ZLASWP( N, A, LDA, 1, M-1, IWORK(2*N+1), 1 )
973: END IF
974:
975: *
976: * End of the preparation phase (scaling, optional sorting and
977: * transposing, optional flushing of small columns).
978: *
979: * Preconditioning
980: *
981: * If the full SVD is needed, the right singular vectors are computed
982: * from a matrix equation, and for that we need theoretical analysis
983: * of the Businger-Golub pivoting. So we use ZGEQP3 as the first RR QRF.
984: * In all other cases the first RR QRF can be chosen by other criteria
985: * (eg speed by replacing global with restricted window pivoting, such
986: * as in xGEQPX from TOMS # 782). Good results will be obtained using
987: * xGEQPX with properly (!) chosen numerical parameters.
988: * Any improvement of ZGEQP3 improves overal performance of ZGEJSV.
989: *
990: * A * P1 = Q1 * [ R1^* 0]^*:
991: DO 1963 p = 1, N
992: * .. all columns are free columns
993: IWORK(p) = 0
994: 1963 CONTINUE
995: CALL ZGEQP3( M, N, A, LDA, IWORK, CWORK, CWORK(N+1), LWORK-N,
996: $ RWORK, IERR )
997: *
998: * The upper triangular matrix R1 from the first QRF is inspected for
999: * rank deficiency and possibilities for deflation, or possible
1000: * ill-conditioning. Depending on the user specified flag L2RANK,
1001: * the procedure explores possibilities to reduce the numerical
1002: * rank by inspecting the computed upper triangular factor. If
1003: * L2RANK or L2ABER are up, then ZGEJSV will compute the SVD of
1004: * A + dA, where ||dA|| <= f(M,N)*EPSLN.
1005: *
1006: NR = 1
1007: IF ( L2ABER ) THEN
1008: * Standard absolute error bound suffices. All sigma_i with
1009: * sigma_i < N*EPSLN*||A|| are flushed to zero. This is an
1010: * agressive enforcement of lower numerical rank by introducing a
1011: * backward error of the order of N*EPSLN*||A||.
1012: TEMP1 = DSQRT(DFLOAT(N))*EPSLN
1013: DO 3001 p = 2, N
1014: IF ( ABS(A(p,p)) .GE. (TEMP1*ABS(A(1,1))) ) THEN
1015: NR = NR + 1
1016: ELSE
1017: GO TO 3002
1018: END IF
1019: 3001 CONTINUE
1020: 3002 CONTINUE
1021: ELSE IF ( L2RANK ) THEN
1022: * .. similarly as above, only slightly more gentle (less agressive).
1023: * Sudden drop on the diagonal of R1 is used as the criterion for
1024: * close-to-rank-defficient.
1025: TEMP1 = DSQRT(SFMIN)
1026: DO 3401 p = 2, N
1027: IF ( ( ABS(A(p,p)) .LT. (EPSLN*ABS(A(p-1,p-1))) ) .OR.
1028: $ ( ABS(A(p,p)) .LT. SMALL ) .OR.
1029: $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3402
1030: NR = NR + 1
1031: 3401 CONTINUE
1032: 3402 CONTINUE
1033: *
1034: ELSE
1035: * The goal is high relative accuracy. However, if the matrix
1036: * has high scaled condition number the relative accuracy is in
1037: * general not feasible. Later on, a condition number estimator
1038: * will be deployed to estimate the scaled condition number.
1039: * Here we just remove the underflowed part of the triangular
1040: * factor. This prevents the situation in which the code is
1041: * working hard to get the accuracy not warranted by the data.
1042: TEMP1 = DSQRT(SFMIN)
1043: DO 3301 p = 2, N
1044: IF ( ( ABS(A(p,p)) .LT. SMALL ) .OR.
1045: $ ( L2KILL .AND. (ABS(A(p,p)) .LT. TEMP1) ) ) GO TO 3302
1046: NR = NR + 1
1047: 3301 CONTINUE
1048: 3302 CONTINUE
1049: *
1050: END IF
1051: *
1052: ALMORT = .FALSE.
1053: IF ( NR .EQ. N ) THEN
1054: MAXPRJ = ONE
1055: DO 3051 p = 2, N
1056: TEMP1 = ABS(A(p,p)) / SVA(IWORK(p))
1057: MAXPRJ = DMIN1( MAXPRJ, TEMP1 )
1058: 3051 CONTINUE
1059: IF ( MAXPRJ**2 .GE. ONE - DFLOAT(N)*EPSLN ) ALMORT = .TRUE.
1060: END IF
1061: *
1062: *
1063: SCONDA = - ONE
1064: CONDR1 = - ONE
1065: CONDR2 = - ONE
1066: *
1067: IF ( ERREST ) THEN
1068: IF ( N .EQ. NR ) THEN
1069: IF ( RSVEC ) THEN
1070: * .. V is available as workspace
1071: CALL ZLACPY( 'U', N, N, A, LDA, V, LDV )
1072: DO 3053 p = 1, N
1073: TEMP1 = SVA(IWORK(p))
1074: CALL ZDSCAL( p, ONE/TEMP1, V(1,p), 1 )
1075: 3053 CONTINUE
1076: CALL ZPOCON( 'U', N, V, LDV, ONE, TEMP1,
1077: $ CWORK(N+1), RWORK, IERR )
1078: *
1079: ELSE IF ( LSVEC ) THEN
1080: * .. U is available as workspace
1081: CALL ZLACPY( 'U', N, N, A, LDA, U, LDU )
1082: DO 3054 p = 1, N
1083: TEMP1 = SVA(IWORK(p))
1084: CALL ZDSCAL( p, ONE/TEMP1, U(1,p), 1 )
1085: 3054 CONTINUE
1086: CALL ZPOCON( 'U', N, U, LDU, ONE, TEMP1,
1087: $ CWORK(N+1), RWORK, IERR )
1088: ELSE
1089: CALL ZLACPY( 'U', N, N, A, LDA, CWORK(N+1), N )
1090: DO 3052 p = 1, N
1091: TEMP1 = SVA(IWORK(p))
1092: CALL ZDSCAL( p, ONE/TEMP1, CWORK(N+(p-1)*N+1), 1 )
1093: 3052 CONTINUE
1094: * .. the columns of R are scaled to have unit Euclidean lengths.
1095: CALL ZPOCON( 'U', N, CWORK(N+1), N, ONE, TEMP1,
1096: $ CWORK(N+N*N+1), RWORK, IERR )
1097: *
1098: END IF
1099: SCONDA = ONE / DSQRT(TEMP1)
1100: * SCONDA is an estimate of SQRT(||(R^* * R)^(-1)||_1).
1101: * N^(-1/4) * SCONDA <= ||R^(-1)||_2 <= N^(1/4) * SCONDA
1102: ELSE
1103: SCONDA = - ONE
1104: END IF
1105: END IF
1106: *
1107: L2PERT = L2PERT .AND. ( ABS( A(1,1)/A(NR,NR) ) .GT. DSQRT(BIG1) )
1108: * If there is no violent scaling, artificial perturbation is not needed.
1109: *
1110: * Phase 3:
1111: *
1112: IF ( .NOT. ( RSVEC .OR. LSVEC ) ) THEN
1113: *
1114: * Singular Values only
1115: *
1116: * .. transpose A(1:NR,1:N)
1117: DO 1946 p = 1, MIN0( N-1, NR )
1118: CALL ZCOPY( N-p, A(p,p+1), LDA, A(p+1,p), 1 )
1119: CALL ZLACGV( N-p+1, A(p,p), 1 )
1120: 1946 CONTINUE
1121: IF ( NR .EQ. N ) A(N,N) = DCONJG(A(N,N))
1122: *
1123: * The following two DO-loops introduce small relative perturbation
1124: * into the strict upper triangle of the lower triangular matrix.
1125: * Small entries below the main diagonal are also changed.
1126: * This modification is useful if the computing environment does not
1127: * provide/allow FLUSH TO ZERO underflow, for it prevents many
1128: * annoying denormalized numbers in case of strongly scaled matrices.
1129: * The perturbation is structured so that it does not introduce any
1130: * new perturbation of the singular values, and it does not destroy
1131: * the job done by the preconditioner.
1132: * The licence for this perturbation is in the variable L2PERT, which
1133: * should be .FALSE. if FLUSH TO ZERO underflow is active.
1134: *
1135: IF ( .NOT. ALMORT ) THEN
1136: *
1137: IF ( L2PERT ) THEN
1138: * XSC = SQRT(SMALL)
1139: XSC = EPSLN / DFLOAT(N)
1140: DO 4947 q = 1, NR
1141: CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
1142: DO 4949 p = 1, N
1143: IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
1144: $ .OR. ( p .LT. q ) )
1145: * $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
1146: $ A(p,q) = CTEMP
1147: 4949 CONTINUE
1148: 4947 CONTINUE
1149: ELSE
1150: CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, A(1,2),LDA )
1151: END IF
1152: *
1153: * .. second preconditioning using the QR factorization
1154: *
1155: CALL ZGEQRF( N,NR, A,LDA, CWORK, CWORK(N+1),LWORK-N, IERR )
1156: *
1157: * .. and transpose upper to lower triangular
1158: DO 1948 p = 1, NR - 1
1159: CALL ZCOPY( NR-p, A(p,p+1), LDA, A(p+1,p), 1 )
1160: CALL ZLACGV( NR-p+1, A(p,p), 1 )
1161: 1948 CONTINUE
1162: *
1163: END IF
1164: *
1165: * Row-cyclic Jacobi SVD algorithm with column pivoting
1166: *
1167: * .. again some perturbation (a "background noise") is added
1168: * to drown denormals
1169: IF ( L2PERT ) THEN
1170: * XSC = SQRT(SMALL)
1171: XSC = EPSLN / DFLOAT(N)
1172: DO 1947 q = 1, NR
1173: CTEMP = DCMPLX(XSC*ABS(A(q,q)),ZERO)
1174: DO 1949 p = 1, NR
1175: IF ( ( (p.GT.q) .AND. (ABS(A(p,q)).LE.TEMP1) )
1176: $ .OR. ( p .LT. q ) )
1177: * $ A(p,q) = TEMP1 * ( A(p,q) / ABS(A(p,q)) )
1178: $ A(p,q) = CTEMP
1179: 1949 CONTINUE
1180: 1947 CONTINUE
1181: ELSE
1182: CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, A(1,2), LDA )
1183: END IF
1184: *
1185: * .. and one-sided Jacobi rotations are started on a lower
1186: * triangular matrix (plus perturbation which is ignored in
1187: * the part which destroys triangular form (confusing?!))
1188: *
1189: CALL ZGESVJ( 'L', 'NoU', 'NoV', NR, NR, A, LDA, SVA,
1190: $ N, V, LDV, CWORK, LWORK, RWORK, LRWORK, INFO )
1191: *
1192: SCALEM = RWORK(1)
1193: NUMRANK = NINT(RWORK(2))
1194: *
1195: *
1196: ELSE IF ( RSVEC .AND. ( .NOT. LSVEC ) ) THEN
1197: *
1198: * -> Singular Values and Right Singular Vectors <-
1199: *
1200: IF ( ALMORT ) THEN
1201: *
1202: * .. in this case NR equals N
1203: DO 1998 p = 1, NR
1204: CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1205: CALL ZLACGV( N-p+1, V(p,p), 1 )
1206: 1998 CONTINUE
1207: CALL ZLASET( 'Upper', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
1208: *
1209: CALL ZGESVJ( 'L','U','N', N, NR, V,LDV, SVA, NR, A,LDA,
1210: $ CWORK, LWORK, RWORK, LRWORK, INFO )
1211: SCALEM = RWORK(1)
1212: NUMRANK = NINT(RWORK(2))
1213:
1214: ELSE
1215: *
1216: * .. two more QR factorizations ( one QRF is not enough, two require
1217: * accumulated product of Jacobi rotations, three are perfect )
1218: *
1219: CALL ZLASET( 'Lower', NR-1,NR-1, CZERO, CZERO, A(2,1), LDA )
1220: CALL ZGELQF( NR,N, A, LDA, CWORK, CWORK(N+1), LWORK-N, IERR)
1221: CALL ZLACPY( 'Lower', NR, NR, A, LDA, V, LDV )
1222: CALL ZLASET( 'Upper', NR-1,NR-1, CZERO, CZERO, V(1,2), LDV )
1223: CALL ZGEQRF( NR, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
1224: $ LWORK-2*N, IERR )
1225: DO 8998 p = 1, NR
1226: CALL ZCOPY( NR-p+1, V(p,p), LDV, V(p,p), 1 )
1227: CALL ZLACGV( NR-p+1, V(p,p), 1 )
1228: 8998 CONTINUE
1229: CALL ZLASET( 'Upper', NR-1, NR-1, ZERO, ZERO, V(1,2), LDV )
1230: *
1231: CALL ZGESVJ( 'Lower', 'U','N', NR, NR, V,LDV, SVA, NR, U,
1232: $ LDU, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
1233: SCALEM = RWORK(1)
1234: NUMRANK = NINT(RWORK(2))
1235: IF ( NR .LT. N ) THEN
1236: CALL ZLASET( 'A',N-NR, NR, CZERO,CZERO, V(NR+1,1), LDV )
1237: CALL ZLASET( 'A',NR, N-NR, CZERO,CZERO, V(1,NR+1), LDV )
1238: CALL ZLASET( 'A',N-NR,N-NR,CZERO,CONE, V(NR+1,NR+1),LDV )
1239: END IF
1240: *
1241: CALL ZUNMLQ( 'Left', 'C', N, N, NR, A, LDA, CWORK,
1242: $ V, LDV, CWORK(N+1), LWORK-N, IERR )
1243: *
1244: END IF
1245: *
1246: DO 8991 p = 1, N
1247: CALL ZCOPY( N, V(p,1), LDV, A(IWORK(p),1), LDA )
1248: 8991 CONTINUE
1249: CALL ZLACPY( 'All', N, N, A, LDA, V, LDV )
1250: *
1251: IF ( TRANSP ) THEN
1252: CALL ZLACPY( 'All', N, N, V, LDV, U, LDU )
1253: END IF
1254: *
1255: ELSE IF ( LSVEC .AND. ( .NOT. RSVEC ) ) THEN
1256: *
1257: * .. Singular Values and Left Singular Vectors ..
1258: *
1259: * .. second preconditioning step to avoid need to accumulate
1260: * Jacobi rotations in the Jacobi iterations.
1261: DO 1965 p = 1, NR
1262: CALL ZCOPY( N-p+1, A(p,p), LDA, U(p,p), 1 )
1263: CALL ZLACGV( N-p+1, U(p,p), 1 )
1264: 1965 CONTINUE
1265: CALL ZLASET( 'Upper', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
1266: *
1267: CALL ZGEQRF( N, NR, U, LDU, CWORK(N+1), CWORK(2*N+1),
1268: $ LWORK-2*N, IERR )
1269: *
1270: DO 1967 p = 1, NR - 1
1271: CALL ZCOPY( NR-p, U(p,p+1), LDU, U(p+1,p), 1 )
1272: CALL ZLACGV( N-p+1, U(p,p), 1 )
1273: 1967 CONTINUE
1274: CALL ZLASET( 'Upper', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
1275: *
1276: CALL ZGESVJ( 'Lower', 'U', 'N', NR,NR, U, LDU, SVA, NR, A,
1277: $ LDA, CWORK(N+1), LWORK-N, RWORK, LRWORK, INFO )
1278: SCALEM = RWORK(1)
1279: NUMRANK = NINT(RWORK(2))
1280: *
1281: IF ( NR .LT. M ) THEN
1282: CALL ZLASET( 'A', M-NR, NR,CZERO, CZERO, U(NR+1,1), LDU )
1283: IF ( NR .LT. N1 ) THEN
1284: CALL ZLASET( 'A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU )
1285: CALL ZLASET( 'A',M-NR,N1-NR,CZERO,CONE,U(NR+1,NR+1),LDU )
1286: END IF
1287: END IF
1288: *
1289: CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
1290: $ LDU, CWORK(N+1), LWORK-N, IERR )
1291: *
1292: IF ( ROWPIV )
1293: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1294: *
1295: DO 1974 p = 1, N1
1296: XSC = ONE / DZNRM2( M, U(1,p), 1 )
1297: CALL ZDSCAL( M, XSC, U(1,p), 1 )
1298: 1974 CONTINUE
1299: *
1300: IF ( TRANSP ) THEN
1301: CALL ZLACPY( 'All', N, N, U, LDU, V, LDV )
1302: END IF
1303: *
1304: ELSE
1305: *
1306: * .. Full SVD ..
1307: *
1308: IF ( .NOT. JRACC ) THEN
1309: *
1310: IF ( .NOT. ALMORT ) THEN
1311: *
1312: * Second Preconditioning Step (QRF [with pivoting])
1313: * Note that the composition of TRANSPOSE, QRF and TRANSPOSE is
1314: * equivalent to an LQF CALL. Since in many libraries the QRF
1315: * seems to be better optimized than the LQF, we do explicit
1316: * transpose and use the QRF. This is subject to changes in an
1317: * optimized implementation of ZGEJSV.
1318: *
1319: DO 1968 p = 1, NR
1320: CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1321: CALL ZLACGV( N-p+1, V(p,p), 1 )
1322: 1968 CONTINUE
1323: *
1324: * .. the following two loops perturb small entries to avoid
1325: * denormals in the second QR factorization, where they are
1326: * as good as zeros. This is done to avoid painfully slow
1327: * computation with denormals. The relative size of the perturbation
1328: * is a parameter that can be changed by the implementer.
1329: * This perturbation device will be obsolete on machines with
1330: * properly implemented arithmetic.
1331: * To switch it off, set L2PERT=.FALSE. To remove it from the
1332: * code, remove the action under L2PERT=.TRUE., leave the ELSE part.
1333: * The following two loops should be blocked and fused with the
1334: * transposed copy above.
1335: *
1336: IF ( L2PERT ) THEN
1337: XSC = DSQRT(SMALL)
1338: DO 2969 q = 1, NR
1339: CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
1340: DO 2968 p = 1, N
1341: IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
1342: $ .OR. ( p .LT. q ) )
1343: * $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
1344: $ V(p,q) = CTEMP
1345: IF ( p .LT. q ) V(p,q) = - V(p,q)
1346: 2968 CONTINUE
1347: 2969 CONTINUE
1348: ELSE
1349: CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
1350: END IF
1351: *
1352: * Estimate the row scaled condition number of R1
1353: * (If R1 is rectangular, N > NR, then the condition number
1354: * of the leading NR x NR submatrix is estimated.)
1355: *
1356: CALL ZLACPY( 'L', NR, NR, V, LDV, CWORK(2*N+1), NR )
1357: DO 3950 p = 1, NR
1358: TEMP1 = DZNRM2(NR-p+1,CWORK(2*N+(p-1)*NR+p),1)
1359: CALL ZDSCAL(NR-p+1,ONE/TEMP1,CWORK(2*N+(p-1)*NR+p),1)
1360: 3950 CONTINUE
1361: CALL ZPOCON('Lower',NR,CWORK(2*N+1),NR,ONE,TEMP1,
1362: $ CWORK(2*N+NR*NR+1),RWORK,IERR)
1363: CONDR1 = ONE / DSQRT(TEMP1)
1364: * .. here need a second oppinion on the condition number
1365: * .. then assume worst case scenario
1366: * R1 is OK for inverse <=> CONDR1 .LT. DFLOAT(N)
1367: * more conservative <=> CONDR1 .LT. SQRT(DFLOAT(N))
1368: *
1369: COND_OK = DSQRT(DSQRT(DFLOAT(NR)))
1370: *[TP] COND_OK is a tuning parameter.
1371: *
1372: IF ( CONDR1 .LT. COND_OK ) THEN
1373: * .. the second QRF without pivoting. Note: in an optimized
1374: * implementation, this QRF should be implemented as the QRF
1375: * of a lower triangular matrix.
1376: * R1^* = Q2 * R2
1377: CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
1378: $ LWORK-2*N, IERR )
1379: *
1380: IF ( L2PERT ) THEN
1381: XSC = DSQRT(SMALL)/EPSLN
1382: DO 3959 p = 2, NR
1383: DO 3958 q = 1, p - 1
1384: CTEMP=DCMPLX(XSC*DMIN1(ABS(V(p,p)),ABS(V(q,q))),
1385: $ ZERO)
1386: IF ( ABS(V(q,p)) .LE. TEMP1 )
1387: * $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
1388: $ V(q,p) = CTEMP
1389: 3958 CONTINUE
1390: 3959 CONTINUE
1391: END IF
1392: *
1393: IF ( NR .NE. N )
1394: $ CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
1395: * .. save ...
1396: *
1397: * .. this transposed copy should be better than naive
1398: DO 1969 p = 1, NR - 1
1399: CALL ZCOPY( NR-p, V(p,p+1), LDV, V(p+1,p), 1 )
1400: CALL ZLACGV(NR-p+1, V(p,p), 1 )
1401: 1969 CONTINUE
1402: V(NR,NR)=DCONJG(V(NR,NR))
1403: *
1404: CONDR2 = CONDR1
1405: *
1406: ELSE
1407: *
1408: * .. ill-conditioned case: second QRF with pivoting
1409: * Note that windowed pivoting would be equaly good
1410: * numerically, and more run-time efficient. So, in
1411: * an optimal implementation, the next call to ZGEQP3
1412: * should be replaced with eg. CALL ZGEQPX (ACM TOMS #782)
1413: * with properly (carefully) chosen parameters.
1414: *
1415: * R1^* * P2 = Q2 * R2
1416: DO 3003 p = 1, NR
1417: IWORK(N+p) = 0
1418: 3003 CONTINUE
1419: CALL ZGEQP3( N, NR, V, LDV, IWORK(N+1), CWORK(N+1),
1420: $ CWORK(2*N+1), LWORK-2*N, RWORK, IERR )
1421: ** CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
1422: ** $ LWORK-2*N, IERR )
1423: IF ( L2PERT ) THEN
1424: XSC = DSQRT(SMALL)
1425: DO 3969 p = 2, NR
1426: DO 3968 q = 1, p - 1
1427: CTEMP=DCMPLX(XSC*DMIN1(ABS(V(p,p)),ABS(V(q,q))),
1428: $ ZERO)
1429: IF ( ABS(V(q,p)) .LE. TEMP1 )
1430: * $ V(q,p) = TEMP1 * ( V(q,p) / ABS(V(q,p)) )
1431: $ V(q,p) = CTEMP
1432: 3968 CONTINUE
1433: 3969 CONTINUE
1434: END IF
1435: *
1436: CALL ZLACPY( 'A', N, NR, V, LDV, CWORK(2*N+1), N )
1437: *
1438: IF ( L2PERT ) THEN
1439: XSC = DSQRT(SMALL)
1440: DO 8970 p = 2, NR
1441: DO 8971 q = 1, p - 1
1442: CTEMP=DCMPLX(XSC*DMIN1(ABS(V(p,p)),ABS(V(q,q))),
1443: $ ZERO)
1444: * V(p,q) = - TEMP1*( V(q,p) / ABS(V(q,p)) )
1445: V(p,q) = - CTEMP
1446: 8971 CONTINUE
1447: 8970 CONTINUE
1448: ELSE
1449: CALL ZLASET( 'L',NR-1,NR-1,CZERO,CZERO,V(2,1),LDV )
1450: END IF
1451: * Now, compute R2 = L3 * Q3, the LQ factorization.
1452: CALL ZGELQF( NR, NR, V, LDV, CWORK(2*N+N*NR+1),
1453: $ CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR, IERR )
1454: * .. and estimate the condition number
1455: CALL ZLACPY( 'L',NR,NR,V,LDV,CWORK(2*N+N*NR+NR+1),NR )
1456: DO 4950 p = 1, NR
1457: TEMP1 = DZNRM2( p, CWORK(2*N+N*NR+NR+p), NR )
1458: CALL ZDSCAL( p, ONE/TEMP1, CWORK(2*N+N*NR+NR+p), NR )
1459: 4950 CONTINUE
1460: CALL ZPOCON( 'L',NR,CWORK(2*N+N*NR+NR+1),NR,ONE,TEMP1,
1461: $ CWORK(2*N+N*NR+NR+NR*NR+1),RWORK,IERR )
1462: CONDR2 = ONE / DSQRT(TEMP1)
1463: *
1464: *
1465: IF ( CONDR2 .GE. COND_OK ) THEN
1466: * .. save the Householder vectors used for Q3
1467: * (this overwrittes the copy of R2, as it will not be
1468: * needed in this branch, but it does not overwritte the
1469: * Huseholder vectors of Q2.).
1470: CALL ZLACPY( 'U', NR, NR, V, LDV, CWORK(2*N+1), N )
1471: * .. and the rest of the information on Q3 is in
1472: * WORK(2*N+N*NR+1:2*N+N*NR+N)
1473: END IF
1474: *
1475: END IF
1476: *
1477: IF ( L2PERT ) THEN
1478: XSC = DSQRT(SMALL)
1479: DO 4968 q = 2, NR
1480: CTEMP = XSC * V(q,q)
1481: DO 4969 p = 1, q - 1
1482: * V(p,q) = - TEMP1*( V(p,q) / ABS(V(p,q)) )
1483: V(p,q) = - CTEMP
1484: 4969 CONTINUE
1485: 4968 CONTINUE
1486: ELSE
1487: CALL ZLASET( 'U', NR-1,NR-1, CZERO,CZERO, V(1,2), LDV )
1488: END IF
1489: *
1490: * Second preconditioning finished; continue with Jacobi SVD
1491: * The input matrix is lower trinagular.
1492: *
1493: * Recover the right singular vectors as solution of a well
1494: * conditioned triangular matrix equation.
1495: *
1496: IF ( CONDR1 .LT. COND_OK ) THEN
1497: *
1498: CALL ZGESVJ( 'L','U','N',NR,NR,V,LDV,SVA,NR,U, LDU,
1499: $ CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,RWORK,
1500: $ LRWORK, INFO )
1501: SCALEM = RWORK(1)
1502: NUMRANK = NINT(RWORK(2))
1503: DO 3970 p = 1, NR
1504: CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
1505: CALL ZDSCAL( NR, SVA(p), V(1,p), 1 )
1506: 3970 CONTINUE
1507:
1508: * .. pick the right matrix equation and solve it
1509: *
1510: IF ( NR .EQ. N ) THEN
1511: * :)) .. best case, R1 is inverted. The solution of this matrix
1512: * equation is Q2*V2 = the product of the Jacobi rotations
1513: * used in ZGESVJ, premultiplied with the orthogonal matrix
1514: * from the second QR factorization.
1515: CALL ZTRSM('L','U','N','N', NR,NR,CONE, A,LDA, V,LDV)
1516: ELSE
1517: * .. R1 is well conditioned, but non-square. Adjoint of R2
1518: * is inverted to get the product of the Jacobi rotations
1519: * used in ZGESVJ. The Q-factor from the second QR
1520: * factorization is then built in explicitly.
1521: CALL ZTRSM('L','U','C','N',NR,NR,CONE,CWORK(2*N+1),
1522: $ N,V,LDV)
1523: IF ( NR .LT. N ) THEN
1524: CALL ZLASET('A',N-NR,NR,ZERO,CZERO,V(NR+1,1),LDV)
1525: CALL ZLASET('A',NR,N-NR,ZERO,CZERO,V(1,NR+1),LDV)
1526: CALL ZLASET('A',N-NR,N-NR,ZERO,CONE,V(NR+1,NR+1),LDV)
1527: END IF
1528: CALL ZUNMQR('L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
1529: $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR)
1530: END IF
1531: *
1532: ELSE IF ( CONDR2 .LT. COND_OK ) THEN
1533: *
1534: * The matrix R2 is inverted. The solution of the matrix equation
1535: * is Q3^* * V3 = the product of the Jacobi rotations (appplied to
1536: * the lower triangular L3 from the LQ factorization of
1537: * R2=L3*Q3), pre-multiplied with the transposed Q3.
1538: CALL ZGESVJ( 'L', 'U', 'N', NR, NR, V, LDV, SVA, NR, U,
1539: $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
1540: $ RWORK, LRWORK, INFO )
1541: SCALEM = RWORK(1)
1542: NUMRANK = NINT(RWORK(2))
1543: DO 3870 p = 1, NR
1544: CALL ZCOPY( NR, V(1,p), 1, U(1,p), 1 )
1545: CALL ZDSCAL( NR, SVA(p), U(1,p), 1 )
1546: 3870 CONTINUE
1547: CALL ZTRSM('L','U','N','N',NR,NR,CONE,CWORK(2*N+1),N,
1548: $ U,LDU)
1549: * .. apply the permutation from the second QR factorization
1550: DO 873 q = 1, NR
1551: DO 872 p = 1, NR
1552: CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
1553: 872 CONTINUE
1554: DO 874 p = 1, NR
1555: U(p,q) = CWORK(2*N+N*NR+NR+p)
1556: 874 CONTINUE
1557: 873 CONTINUE
1558: IF ( NR .LT. N ) THEN
1559: CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
1560: CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
1561: CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
1562: END IF
1563: CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
1564: $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1565: ELSE
1566: * Last line of defense.
1567: * #:( This is a rather pathological case: no scaled condition
1568: * improvement after two pivoted QR factorizations. Other
1569: * possibility is that the rank revealing QR factorization
1570: * or the condition estimator has failed, or the COND_OK
1571: * is set very close to ONE (which is unnecessary). Normally,
1572: * this branch should never be executed, but in rare cases of
1573: * failure of the RRQR or condition estimator, the last line of
1574: * defense ensures that ZGEJSV completes the task.
1575: * Compute the full SVD of L3 using ZGESVJ with explicit
1576: * accumulation of Jacobi rotations.
1577: CALL ZGESVJ( 'L', 'U', 'V', NR, NR, V, LDV, SVA, NR, U,
1578: $ LDU, CWORK(2*N+N*NR+NR+1), LWORK-2*N-N*NR-NR,
1579: $ RWORK, LRWORK, INFO )
1580: SCALEM = RWORK(1)
1581: NUMRANK = NINT(RWORK(2))
1582: IF ( NR .LT. N ) THEN
1583: CALL ZLASET( 'A',N-NR,NR,CZERO,CZERO,V(NR+1,1),LDV )
1584: CALL ZLASET( 'A',NR,N-NR,CZERO,CZERO,V(1,NR+1),LDV )
1585: CALL ZLASET('A',N-NR,N-NR,CZERO,CONE,V(NR+1,NR+1),LDV)
1586: END IF
1587: CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
1588: $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1589: *
1590: CALL ZUNMLQ( 'L', 'C', NR, NR, NR, CWORK(2*N+1), N,
1591: $ CWORK(2*N+N*NR+1), U, LDU, CWORK(2*N+N*NR+NR+1),
1592: $ LWORK-2*N-N*NR-NR, IERR )
1593: DO 773 q = 1, NR
1594: DO 772 p = 1, NR
1595: CWORK(2*N+N*NR+NR+IWORK(N+p)) = U(p,q)
1596: 772 CONTINUE
1597: DO 774 p = 1, NR
1598: U(p,q) = CWORK(2*N+N*NR+NR+p)
1599: 774 CONTINUE
1600: 773 CONTINUE
1601: *
1602: END IF
1603: *
1604: * Permute the rows of V using the (column) permutation from the
1605: * first QRF. Also, scale the columns to make them unit in
1606: * Euclidean norm. This applies to all cases.
1607: *
1608: TEMP1 = DSQRT(DFLOAT(N)) * EPSLN
1609: DO 1972 q = 1, N
1610: DO 972 p = 1, N
1611: CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
1612: 972 CONTINUE
1613: DO 973 p = 1, N
1614: V(p,q) = CWORK(2*N+N*NR+NR+p)
1615: 973 CONTINUE
1616: XSC = ONE / DZNRM2( N, V(1,q), 1 )
1617: IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1618: $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
1619: 1972 CONTINUE
1620: * At this moment, V contains the right singular vectors of A.
1621: * Next, assemble the left singular vector matrix U (M x N).
1622: IF ( NR .LT. M ) THEN
1623: CALL ZLASET('A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU)
1624: IF ( NR .LT. N1 ) THEN
1625: CALL ZLASET('A',NR,N1-NR,CZERO,CZERO,U(1,NR+1),LDU)
1626: CALL ZLASET('A',M-NR,N1-NR,CZERO,CONE,
1627: $ U(NR+1,NR+1),LDU)
1628: END IF
1629: END IF
1630: *
1631: * The Q matrix from the first QRF is built into the left singular
1632: * matrix U. This applies to all cases.
1633: *
1634: CALL ZUNMQR( 'Left', 'No_Tr', M, N1, N, A, LDA, CWORK, U,
1635: $ LDU, CWORK(N+1), LWORK-N, IERR )
1636:
1637: * The columns of U are normalized. The cost is O(M*N) flops.
1638: TEMP1 = DSQRT(DFLOAT(M)) * EPSLN
1639: DO 1973 p = 1, NR
1640: XSC = ONE / DZNRM2( M, U(1,p), 1 )
1641: IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1642: $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
1643: 1973 CONTINUE
1644: *
1645: * If the initial QRF is computed with row pivoting, the left
1646: * singular vectors must be adjusted.
1647: *
1648: IF ( ROWPIV )
1649: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1650: *
1651: ELSE
1652: *
1653: * .. the initial matrix A has almost orthogonal columns and
1654: * the second QRF is not needed
1655: *
1656: CALL ZLACPY( 'Upper', N, N, A, LDA, CWORK(N+1), N )
1657: IF ( L2PERT ) THEN
1658: XSC = DSQRT(SMALL)
1659: DO 5970 p = 2, N
1660: CTEMP = XSC * CWORK( N + (p-1)*N + p )
1661: DO 5971 q = 1, p - 1
1662: * CWORK(N+(q-1)*N+p)=-TEMP1 * ( CWORK(N+(p-1)*N+q) /
1663: * $ ABS(CWORK(N+(p-1)*N+q)) )
1664: CWORK(N+(q-1)*N+p)=-CTEMP
1665: 5971 CONTINUE
1666: 5970 CONTINUE
1667: ELSE
1668: CALL ZLASET( 'Lower',N-1,N-1,CZERO,CZERO,CWORK(N+2),N )
1669: END IF
1670: *
1671: CALL ZGESVJ( 'Upper', 'U', 'N', N, N, CWORK(N+1), N, SVA,
1672: $ N, U, LDU, CWORK(N+N*N+1), LWORK-N-N*N, RWORK, LRWORK,
1673: $ INFO )
1674: *
1675: SCALEM = RWORK(1)
1676: NUMRANK = NINT(RWORK(2))
1677: DO 6970 p = 1, N
1678: CALL ZCOPY( N, CWORK(N+(p-1)*N+1), 1, U(1,p), 1 )
1679: CALL ZDSCAL( N, SVA(p), CWORK(N+(p-1)*N+1), 1 )
1680: 6970 CONTINUE
1681: *
1682: CALL ZTRSM( 'Left', 'Upper', 'NoTrans', 'No UD', N, N,
1683: $ CONE, A, LDA, CWORK(N+1), N )
1684: DO 6972 p = 1, N
1685: CALL ZCOPY( N, CWORK(N+p), N, V(IWORK(p),1), LDV )
1686: 6972 CONTINUE
1687: TEMP1 = DSQRT(DFLOAT(N))*EPSLN
1688: DO 6971 p = 1, N
1689: XSC = ONE / DZNRM2( N, V(1,p), 1 )
1690: IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1691: $ CALL ZDSCAL( N, XSC, V(1,p), 1 )
1692: 6971 CONTINUE
1693: *
1694: * Assemble the left singular vector matrix U (M x N).
1695: *
1696: IF ( N .LT. M ) THEN
1697: CALL ZLASET( 'A', M-N, N, CZERO, CZERO, U(N+1,1), LDU )
1698: IF ( N .LT. N1 ) THEN
1699: CALL ZLASET('A',N, N1-N, CZERO, CZERO, U(1,N+1),LDU)
1700: CALL ZLASET( 'A',M-N,N1-N, CZERO, CONE,U(N+1,N+1),LDU)
1701: END IF
1702: END IF
1703: CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
1704: $ LDU, CWORK(N+1), LWORK-N, IERR )
1705: TEMP1 = DSQRT(DFLOAT(M))*EPSLN
1706: DO 6973 p = 1, N1
1707: XSC = ONE / DZNRM2( M, U(1,p), 1 )
1708: IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1709: $ CALL ZDSCAL( M, XSC, U(1,p), 1 )
1710: 6973 CONTINUE
1711: *
1712: IF ( ROWPIV )
1713: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1714: *
1715: END IF
1716: *
1717: * end of the >> almost orthogonal case << in the full SVD
1718: *
1719: ELSE
1720: *
1721: * This branch deploys a preconditioned Jacobi SVD with explicitly
1722: * accumulated rotations. It is included as optional, mainly for
1723: * experimental purposes. It does perfom well, and can also be used.
1724: * In this implementation, this branch will be automatically activated
1725: * if the condition number sigma_max(A) / sigma_min(A) is predicted
1726: * to be greater than the overflow threshold. This is because the
1727: * a posteriori computation of the singular vectors assumes robust
1728: * implementation of BLAS and some LAPACK procedures, capable of working
1729: * in presence of extreme values. Since that is not always the case, ...
1730: *
1731: DO 7968 p = 1, NR
1732: CALL ZCOPY( N-p+1, A(p,p), LDA, V(p,p), 1 )
1733: CALL ZLACGV( N-p+1, V(p,p), 1 )
1734: 7968 CONTINUE
1735: *
1736: IF ( L2PERT ) THEN
1737: XSC = DSQRT(SMALL/EPSLN)
1738: DO 5969 q = 1, NR
1739: CTEMP = DCMPLX(XSC*ABS( V(q,q) ),ZERO)
1740: DO 5968 p = 1, N
1741: IF ( ( p .GT. q ) .AND. ( ABS(V(p,q)) .LE. TEMP1 )
1742: $ .OR. ( p .LT. q ) )
1743: * $ V(p,q) = TEMP1 * ( V(p,q) / ABS(V(p,q)) )
1744: $ V(p,q) = CTEMP
1745: IF ( p .LT. q ) V(p,q) = - V(p,q)
1746: 5968 CONTINUE
1747: 5969 CONTINUE
1748: ELSE
1749: CALL ZLASET( 'U', NR-1, NR-1, CZERO, CZERO, V(1,2), LDV )
1750: END IF
1751:
1752: CALL ZGEQRF( N, NR, V, LDV, CWORK(N+1), CWORK(2*N+1),
1753: $ LWORK-2*N, IERR )
1754: CALL ZLACPY( 'L', N, NR, V, LDV, CWORK(2*N+1), N )
1755: *
1756: DO 7969 p = 1, NR
1757: CALL ZCOPY( NR-p+1, V(p,p), LDV, U(p,p), 1 )
1758: CALL ZLACGV( NR-p+1, U(p,p), 1 )
1759: 7969 CONTINUE
1760:
1761: IF ( L2PERT ) THEN
1762: XSC = DSQRT(SMALL/EPSLN)
1763: DO 9970 q = 2, NR
1764: DO 9971 p = 1, q - 1
1765: CTEMP = DCMPLX(XSC * DMIN1(ABS(U(p,p)),ABS(U(q,q))),
1766: $ ZERO)
1767: * U(p,q) = - TEMP1 * ( U(q,p) / ABS(U(q,p)) )
1768: U(p,q) = - CTEMP
1769: 9971 CONTINUE
1770: 9970 CONTINUE
1771: ELSE
1772: CALL ZLASET('U', NR-1, NR-1, CZERO, CZERO, U(1,2), LDU )
1773: END IF
1774:
1775: CALL ZGESVJ( 'L', 'U', 'V', NR, NR, U, LDU, SVA,
1776: $ N, V, LDV, CWORK(2*N+N*NR+1), LWORK-2*N-N*NR,
1777: $ RWORK, LRWORK, INFO )
1778: SCALEM = RWORK(1)
1779: NUMRANK = NINT(RWORK(2))
1780:
1781: IF ( NR .LT. N ) THEN
1782: CALL ZLASET( 'A',N-NR,NR,ZERO,ZERO,V(NR+1,1),LDV )
1783: CALL ZLASET( 'A',NR,N-NR,ZERO,ZERO,V(1,NR+1),LDV )
1784: CALL ZLASET( 'A',N-NR,N-NR,ZERO,ONE,V(NR+1,NR+1),LDV )
1785: END IF
1786:
1787: CALL ZUNMQR( 'L','N',N,N,NR,CWORK(2*N+1),N,CWORK(N+1),
1788: $ V,LDV,CWORK(2*N+N*NR+NR+1),LWORK-2*N-N*NR-NR,IERR )
1789: *
1790: * Permute the rows of V using the (column) permutation from the
1791: * first QRF. Also, scale the columns to make them unit in
1792: * Euclidean norm. This applies to all cases.
1793: *
1794: TEMP1 = DSQRT(DFLOAT(N)) * EPSLN
1795: DO 7972 q = 1, N
1796: DO 8972 p = 1, N
1797: CWORK(2*N+N*NR+NR+IWORK(p)) = V(p,q)
1798: 8972 CONTINUE
1799: DO 8973 p = 1, N
1800: V(p,q) = CWORK(2*N+N*NR+NR+p)
1801: 8973 CONTINUE
1802: XSC = ONE / DZNRM2( N, V(1,q), 1 )
1803: IF ( (XSC .LT. (ONE-TEMP1)) .OR. (XSC .GT. (ONE+TEMP1)) )
1804: $ CALL ZDSCAL( N, XSC, V(1,q), 1 )
1805: 7972 CONTINUE
1806: *
1807: * At this moment, V contains the right singular vectors of A.
1808: * Next, assemble the left singular vector matrix U (M x N).
1809: *
1810: IF ( NR .LT. M ) THEN
1811: CALL ZLASET( 'A', M-NR, NR, CZERO, CZERO, U(NR+1,1), LDU )
1812: IF ( NR .LT. N1 ) THEN
1813: CALL ZLASET('A',NR, N1-NR, CZERO, CZERO, U(1,NR+1),LDU)
1814: CALL ZLASET('A',M-NR,N1-NR, CZERO, CONE,U(NR+1,NR+1),LDU)
1815: END IF
1816: END IF
1817: *
1818: CALL ZUNMQR( 'Left', 'No Tr', M, N1, N, A, LDA, CWORK, U,
1819: $ LDU, CWORK(N+1), LWORK-N, IERR )
1820: *
1821: IF ( ROWPIV )
1822: $ CALL ZLASWP( N1, U, LDU, 1, M-1, IWORK(2*N+1), -1 )
1823: *
1824: *
1825: END IF
1826: IF ( TRANSP ) THEN
1827: * .. swap U and V because the procedure worked on A^*
1828: DO 6974 p = 1, N
1829: CALL ZSWAP( N, U(1,p), 1, V(1,p), 1 )
1830: 6974 CONTINUE
1831: END IF
1832: *
1833: END IF
1834: * end of the full SVD
1835: *
1836: * Undo scaling, if necessary (and possible)
1837: *
1838: IF ( USCAL2 .LE. (BIG/SVA(1))*USCAL1 ) THEN
1839: CALL ZLASCL( 'G', 0, 0, USCAL1, USCAL2, NR, 1, SVA, N, IERR )
1840: USCAL1 = ONE
1841: USCAL2 = ONE
1842: END IF
1843: *
1844: IF ( NR .LT. N ) THEN
1845: DO 3004 p = NR+1, N
1846: SVA(p) = ZERO
1847: 3004 CONTINUE
1848: END IF
1849: *
1850: RWORK(1) = USCAL2 * SCALEM
1851: RWORK(2) = USCAL1
1852: IF ( ERREST ) RWORK(3) = SCONDA
1853: IF ( LSVEC .AND. RSVEC ) THEN
1854: RWORK(4) = CONDR1
1855: RWORK(5) = CONDR2
1856: END IF
1857: IF ( L2TRAN ) THEN
1858: RWORK(6) = ENTRA
1859: RWORK(7) = ENTRAT
1860: END IF
1861: *
1862: IWORK(1) = NR
1863: IWORK(2) = NUMRANK
1864: IWORK(3) = WARNING
1865: *
1866: RETURN
1867: * ..
1868: * .. END OF ZGEJSV
1869: * ..
1870: END
1871: *
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