1: SUBROUTINE DLASDA( ICOMPQ, SMLSIZ, N, SQRE, D, E, U, LDU, VT, K,
2: $ DIFL, DIFR, Z, POLES, GIVPTR, GIVCOL, LDGCOL,
3: $ PERM, GIVNUM, C, S, WORK, IWORK, INFO )
4: *
5: * -- LAPACK auxiliary routine (version 3.2.2) --
6: * -- LAPACK is a software package provided by Univ. of Tennessee, --
7: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
8: * June 2010
9: *
10: * .. Scalar Arguments ..
11: INTEGER ICOMPQ, INFO, LDGCOL, LDU, N, SMLSIZ, SQRE
12: * ..
13: * .. Array Arguments ..
14: INTEGER GIVCOL( LDGCOL, * ), GIVPTR( * ), IWORK( * ),
15: $ K( * ), PERM( LDGCOL, * )
16: DOUBLE PRECISION C( * ), D( * ), DIFL( LDU, * ), DIFR( LDU, * ),
17: $ E( * ), GIVNUM( LDU, * ), POLES( LDU, * ),
18: $ S( * ), U( LDU, * ), VT( LDU, * ), WORK( * ),
19: $ Z( LDU, * )
20: * ..
21: *
22: * Purpose
23: * =======
24: *
25: * Using a divide and conquer approach, DLASDA computes the singular
26: * value decomposition (SVD) of a real upper bidiagonal N-by-M matrix
27: * B with diagonal D and offdiagonal E, where M = N + SQRE. The
28: * algorithm computes the singular values in the SVD B = U * S * VT.
29: * The orthogonal matrices U and VT are optionally computed in
30: * compact form.
31: *
32: * A related subroutine, DLASD0, computes the singular values and
33: * the singular vectors in explicit form.
34: *
35: * Arguments
36: * =========
37: *
38: * ICOMPQ (input) INTEGER
39: * Specifies whether singular vectors are to be computed
40: * in compact form, as follows
41: * = 0: Compute singular values only.
42: * = 1: Compute singular vectors of upper bidiagonal
43: * matrix in compact form.
44: *
45: * SMLSIZ (input) INTEGER
46: * The maximum size of the subproblems at the bottom of the
47: * computation tree.
48: *
49: * N (input) INTEGER
50: * The row dimension of the upper bidiagonal matrix. This is
51: * also the dimension of the main diagonal array D.
52: *
53: * SQRE (input) INTEGER
54: * Specifies the column dimension of the bidiagonal matrix.
55: * = 0: The bidiagonal matrix has column dimension M = N;
56: * = 1: The bidiagonal matrix has column dimension M = N + 1.
57: *
58: * D (input/output) DOUBLE PRECISION array, dimension ( N )
59: * On entry D contains the main diagonal of the bidiagonal
60: * matrix. On exit D, if INFO = 0, contains its singular values.
61: *
62: * E (input) DOUBLE PRECISION array, dimension ( M-1 )
63: * Contains the subdiagonal entries of the bidiagonal matrix.
64: * On exit, E has been destroyed.
65: *
66: * U (output) DOUBLE PRECISION array,
67: * dimension ( LDU, SMLSIZ ) if ICOMPQ = 1, and not referenced
68: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, U contains the left
69: * singular vector matrices of all subproblems at the bottom
70: * level.
71: *
72: * LDU (input) INTEGER, LDU = > N.
73: * The leading dimension of arrays U, VT, DIFL, DIFR, POLES,
74: * GIVNUM, and Z.
75: *
76: * VT (output) DOUBLE PRECISION array,
77: * dimension ( LDU, SMLSIZ+1 ) if ICOMPQ = 1, and not referenced
78: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, VT**T contains the right
79: * singular vector matrices of all subproblems at the bottom
80: * level.
81: *
82: * K (output) INTEGER array,
83: * dimension ( N ) if ICOMPQ = 1 and dimension 1 if ICOMPQ = 0.
84: * If ICOMPQ = 1, on exit, K(I) is the dimension of the I-th
85: * secular equation on the computation tree.
86: *
87: * DIFL (output) DOUBLE PRECISION array, dimension ( LDU, NLVL ),
88: * where NLVL = floor(log_2 (N/SMLSIZ))).
89: *
90: * DIFR (output) DOUBLE PRECISION array,
91: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1 and
92: * dimension ( N ) if ICOMPQ = 0.
93: * If ICOMPQ = 1, on exit, DIFL(1:N, I) and DIFR(1:N, 2 * I - 1)
94: * record distances between singular values on the I-th
95: * level and singular values on the (I -1)-th level, and
96: * DIFR(1:N, 2 * I ) contains the normalizing factors for
97: * the right singular vector matrix. See DLASD8 for details.
98: *
99: * Z (output) DOUBLE PRECISION array,
100: * dimension ( LDU, NLVL ) if ICOMPQ = 1 and
101: * dimension ( N ) if ICOMPQ = 0.
102: * The first K elements of Z(1, I) contain the components of
103: * the deflation-adjusted updating row vector for subproblems
104: * on the I-th level.
105: *
106: * POLES (output) DOUBLE PRECISION array,
107: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not referenced
108: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, POLES(1, 2*I - 1) and
109: * POLES(1, 2*I) contain the new and old singular values
110: * involved in the secular equations on the I-th level.
111: *
112: * GIVPTR (output) INTEGER array,
113: * dimension ( N ) if ICOMPQ = 1, and not referenced if
114: * ICOMPQ = 0. If ICOMPQ = 1, on exit, GIVPTR( I ) records
115: * the number of Givens rotations performed on the I-th
116: * problem on the computation tree.
117: *
118: * GIVCOL (output) INTEGER array,
119: * dimension ( LDGCOL, 2 * NLVL ) if ICOMPQ = 1, and not
120: * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
121: * GIVCOL(1, 2 *I - 1) and GIVCOL(1, 2 *I) record the locations
122: * of Givens rotations performed on the I-th level on the
123: * computation tree.
124: *
125: * LDGCOL (input) INTEGER, LDGCOL = > N.
126: * The leading dimension of arrays GIVCOL and PERM.
127: *
128: * PERM (output) INTEGER array,
129: * dimension ( LDGCOL, NLVL ) if ICOMPQ = 1, and not referenced
130: * if ICOMPQ = 0. If ICOMPQ = 1, on exit, PERM(1, I) records
131: * permutations done on the I-th level of the computation tree.
132: *
133: * GIVNUM (output) DOUBLE PRECISION array,
134: * dimension ( LDU, 2 * NLVL ) if ICOMPQ = 1, and not
135: * referenced if ICOMPQ = 0. If ICOMPQ = 1, on exit, for each I,
136: * GIVNUM(1, 2 *I - 1) and GIVNUM(1, 2 *I) record the C- and S-
137: * values of Givens rotations performed on the I-th level on
138: * the computation tree.
139: *
140: * C (output) DOUBLE PRECISION array,
141: * dimension ( N ) if ICOMPQ = 1, and dimension 1 if ICOMPQ = 0.
142: * If ICOMPQ = 1 and the I-th subproblem is not square, on exit,
143: * C( I ) contains the C-value of a Givens rotation related to
144: * the right null space of the I-th subproblem.
145: *
146: * S (output) DOUBLE PRECISION array, dimension ( N ) if
147: * ICOMPQ = 1, and dimension 1 if ICOMPQ = 0. If ICOMPQ = 1
148: * and the I-th subproblem is not square, on exit, S( I )
149: * contains the S-value of a Givens rotation related to
150: * the right null space of the I-th subproblem.
151: *
152: * WORK (workspace) DOUBLE PRECISION array, dimension
153: * (6 * N + (SMLSIZ + 1)*(SMLSIZ + 1)).
154: *
155: * IWORK (workspace) INTEGER array.
156: * Dimension must be at least (7 * N).
157: *
158: * INFO (output) INTEGER
159: * = 0: successful exit.
160: * < 0: if INFO = -i, the i-th argument had an illegal value.
161: * > 0: if INFO = 1, a singular value did not converge
162: *
163: * Further Details
164: * ===============
165: *
166: * Based on contributions by
167: * Ming Gu and Huan Ren, Computer Science Division, University of
168: * California at Berkeley, USA
169: *
170: * =====================================================================
171: *
172: * .. Parameters ..
173: DOUBLE PRECISION ZERO, ONE
174: PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
175: * ..
176: * .. Local Scalars ..
177: INTEGER I, I1, IC, IDXQ, IDXQI, IM1, INODE, ITEMP, IWK,
178: $ J, LF, LL, LVL, LVL2, M, NCC, ND, NDB1, NDIML,
179: $ NDIMR, NL, NLF, NLP1, NLVL, NR, NRF, NRP1, NRU,
180: $ NWORK1, NWORK2, SMLSZP, SQREI, VF, VFI, VL, VLI
181: DOUBLE PRECISION ALPHA, BETA
182: * ..
183: * .. External Subroutines ..
184: EXTERNAL DCOPY, DLASD6, DLASDQ, DLASDT, DLASET, XERBLA
185: * ..
186: * .. Executable Statements ..
187: *
188: * Test the input parameters.
189: *
190: INFO = 0
191: *
192: IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
193: INFO = -1
194: ELSE IF( SMLSIZ.LT.3 ) THEN
195: INFO = -2
196: ELSE IF( N.LT.0 ) THEN
197: INFO = -3
198: ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
199: INFO = -4
200: ELSE IF( LDU.LT.( N+SQRE ) ) THEN
201: INFO = -8
202: ELSE IF( LDGCOL.LT.N ) THEN
203: INFO = -17
204: END IF
205: IF( INFO.NE.0 ) THEN
206: CALL XERBLA( 'DLASDA', -INFO )
207: RETURN
208: END IF
209: *
210: M = N + SQRE
211: *
212: * If the input matrix is too small, call DLASDQ to find the SVD.
213: *
214: IF( N.LE.SMLSIZ ) THEN
215: IF( ICOMPQ.EQ.0 ) THEN
216: CALL DLASDQ( 'U', SQRE, N, 0, 0, 0, D, E, VT, LDU, U, LDU,
217: $ U, LDU, WORK, INFO )
218: ELSE
219: CALL DLASDQ( 'U', SQRE, N, M, N, 0, D, E, VT, LDU, U, LDU,
220: $ U, LDU, WORK, INFO )
221: END IF
222: RETURN
223: END IF
224: *
225: * Book-keeping and set up the computation tree.
226: *
227: INODE = 1
228: NDIML = INODE + N
229: NDIMR = NDIML + N
230: IDXQ = NDIMR + N
231: IWK = IDXQ + N
232: *
233: NCC = 0
234: NRU = 0
235: *
236: SMLSZP = SMLSIZ + 1
237: VF = 1
238: VL = VF + M
239: NWORK1 = VL + M
240: NWORK2 = NWORK1 + SMLSZP*SMLSZP
241: *
242: CALL DLASDT( N, NLVL, ND, IWORK( INODE ), IWORK( NDIML ),
243: $ IWORK( NDIMR ), SMLSIZ )
244: *
245: * for the nodes on bottom level of the tree, solve
246: * their subproblems by DLASDQ.
247: *
248: NDB1 = ( ND+1 ) / 2
249: DO 30 I = NDB1, ND
250: *
251: * IC : center row of each node
252: * NL : number of rows of left subproblem
253: * NR : number of rows of right subproblem
254: * NLF: starting row of the left subproblem
255: * NRF: starting row of the right subproblem
256: *
257: I1 = I - 1
258: IC = IWORK( INODE+I1 )
259: NL = IWORK( NDIML+I1 )
260: NLP1 = NL + 1
261: NR = IWORK( NDIMR+I1 )
262: NLF = IC - NL
263: NRF = IC + 1
264: IDXQI = IDXQ + NLF - 2
265: VFI = VF + NLF - 1
266: VLI = VL + NLF - 1
267: SQREI = 1
268: IF( ICOMPQ.EQ.0 ) THEN
269: CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, WORK( NWORK1 ),
270: $ SMLSZP )
271: CALL DLASDQ( 'U', SQREI, NL, NLP1, NRU, NCC, D( NLF ),
272: $ E( NLF ), WORK( NWORK1 ), SMLSZP,
273: $ WORK( NWORK2 ), NL, WORK( NWORK2 ), NL,
274: $ WORK( NWORK2 ), INFO )
275: ITEMP = NWORK1 + NL*SMLSZP
276: CALL DCOPY( NLP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
277: CALL DCOPY( NLP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
278: ELSE
279: CALL DLASET( 'A', NL, NL, ZERO, ONE, U( NLF, 1 ), LDU )
280: CALL DLASET( 'A', NLP1, NLP1, ZERO, ONE, VT( NLF, 1 ), LDU )
281: CALL DLASDQ( 'U', SQREI, NL, NLP1, NL, NCC, D( NLF ),
282: $ E( NLF ), VT( NLF, 1 ), LDU, U( NLF, 1 ), LDU,
283: $ U( NLF, 1 ), LDU, WORK( NWORK1 ), INFO )
284: CALL DCOPY( NLP1, VT( NLF, 1 ), 1, WORK( VFI ), 1 )
285: CALL DCOPY( NLP1, VT( NLF, NLP1 ), 1, WORK( VLI ), 1 )
286: END IF
287: IF( INFO.NE.0 ) THEN
288: RETURN
289: END IF
290: DO 10 J = 1, NL
291: IWORK( IDXQI+J ) = J
292: 10 CONTINUE
293: IF( ( I.EQ.ND ) .AND. ( SQRE.EQ.0 ) ) THEN
294: SQREI = 0
295: ELSE
296: SQREI = 1
297: END IF
298: IDXQI = IDXQI + NLP1
299: VFI = VFI + NLP1
300: VLI = VLI + NLP1
301: NRP1 = NR + SQREI
302: IF( ICOMPQ.EQ.0 ) THEN
303: CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, WORK( NWORK1 ),
304: $ SMLSZP )
305: CALL DLASDQ( 'U', SQREI, NR, NRP1, NRU, NCC, D( NRF ),
306: $ E( NRF ), WORK( NWORK1 ), SMLSZP,
307: $ WORK( NWORK2 ), NR, WORK( NWORK2 ), NR,
308: $ WORK( NWORK2 ), INFO )
309: ITEMP = NWORK1 + ( NRP1-1 )*SMLSZP
310: CALL DCOPY( NRP1, WORK( NWORK1 ), 1, WORK( VFI ), 1 )
311: CALL DCOPY( NRP1, WORK( ITEMP ), 1, WORK( VLI ), 1 )
312: ELSE
313: CALL DLASET( 'A', NR, NR, ZERO, ONE, U( NRF, 1 ), LDU )
314: CALL DLASET( 'A', NRP1, NRP1, ZERO, ONE, VT( NRF, 1 ), LDU )
315: CALL DLASDQ( 'U', SQREI, NR, NRP1, NR, NCC, D( NRF ),
316: $ E( NRF ), VT( NRF, 1 ), LDU, U( NRF, 1 ), LDU,
317: $ U( NRF, 1 ), LDU, WORK( NWORK1 ), INFO )
318: CALL DCOPY( NRP1, VT( NRF, 1 ), 1, WORK( VFI ), 1 )
319: CALL DCOPY( NRP1, VT( NRF, NRP1 ), 1, WORK( VLI ), 1 )
320: END IF
321: IF( INFO.NE.0 ) THEN
322: RETURN
323: END IF
324: DO 20 J = 1, NR
325: IWORK( IDXQI+J ) = J
326: 20 CONTINUE
327: 30 CONTINUE
328: *
329: * Now conquer each subproblem bottom-up.
330: *
331: J = 2**NLVL
332: DO 50 LVL = NLVL, 1, -1
333: LVL2 = LVL*2 - 1
334: *
335: * Find the first node LF and last node LL on
336: * the current level LVL.
337: *
338: IF( LVL.EQ.1 ) THEN
339: LF = 1
340: LL = 1
341: ELSE
342: LF = 2**( LVL-1 )
343: LL = 2*LF - 1
344: END IF
345: DO 40 I = LF, LL
346: IM1 = I - 1
347: IC = IWORK( INODE+IM1 )
348: NL = IWORK( NDIML+IM1 )
349: NR = IWORK( NDIMR+IM1 )
350: NLF = IC - NL
351: NRF = IC + 1
352: IF( I.EQ.LL ) THEN
353: SQREI = SQRE
354: ELSE
355: SQREI = 1
356: END IF
357: VFI = VF + NLF - 1
358: VLI = VL + NLF - 1
359: IDXQI = IDXQ + NLF - 1
360: ALPHA = D( IC )
361: BETA = E( IC )
362: IF( ICOMPQ.EQ.0 ) THEN
363: CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
364: $ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
365: $ IWORK( IDXQI ), PERM, GIVPTR( 1 ), GIVCOL,
366: $ LDGCOL, GIVNUM, LDU, POLES, DIFL, DIFR, Z,
367: $ K( 1 ), C( 1 ), S( 1 ), WORK( NWORK1 ),
368: $ IWORK( IWK ), INFO )
369: ELSE
370: J = J - 1
371: CALL DLASD6( ICOMPQ, NL, NR, SQREI, D( NLF ),
372: $ WORK( VFI ), WORK( VLI ), ALPHA, BETA,
373: $ IWORK( IDXQI ), PERM( NLF, LVL ),
374: $ GIVPTR( J ), GIVCOL( NLF, LVL2 ), LDGCOL,
375: $ GIVNUM( NLF, LVL2 ), LDU,
376: $ POLES( NLF, LVL2 ), DIFL( NLF, LVL ),
377: $ DIFR( NLF, LVL2 ), Z( NLF, LVL ), K( J ),
378: $ C( J ), S( J ), WORK( NWORK1 ),
379: $ IWORK( IWK ), INFO )
380: END IF
381: IF( INFO.NE.0 ) THEN
382: RETURN
383: END IF
384: 40 CONTINUE
385: 50 CONTINUE
386: *
387: RETURN
388: *
389: * End of DLASDA
390: *
391: END
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