1: *> \brief \b ZUNBDB2
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
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9: *> Download ZUNBDB2 + dependencies
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11: *> [TGZ]</a>
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13: *> [ZIP]</a>
14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zunbdb2.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZUNBDB2( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
22: * TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
23: *
24: * .. Scalar Arguments ..
25: * INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
26: * ..
27: * .. Array Arguments ..
28: * DOUBLE PRECISION PHI(*), THETA(*)
29: * COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
30: * $ X11(LDX11,*), X21(LDX21,*)
31: * ..
32: *
33: *
34: *> \par Purpose:
35: *> =============
36: *>
37: *>\verbatim
38: *>
39: *> ZUNBDB2 simultaneously bidiagonalizes the blocks of a tall and skinny
40: *> matrix X with orthonomal columns:
41: *>
42: *> [ B11 ]
43: *> [ X11 ] [ P1 | ] [ 0 ]
44: *> [-----] = [---------] [-----] Q1**T .
45: *> [ X21 ] [ | P2 ] [ B21 ]
46: *> [ 0 ]
47: *>
48: *> X11 is P-by-Q, and X21 is (M-P)-by-Q. P must be no larger than M-P,
49: *> Q, or M-Q. Routines ZUNBDB1, ZUNBDB3, and ZUNBDB4 handle cases in
50: *> which P is not the minimum dimension.
51: *>
52: *> The unitary matrices P1, P2, and Q1 are P-by-P, (M-P)-by-(M-P),
53: *> and (M-Q)-by-(M-Q), respectively. They are represented implicitly by
54: *> Householder vectors.
55: *>
56: *> B11 and B12 are P-by-P bidiagonal matrices represented implicitly by
57: *> angles THETA, PHI.
58: *>
59: *>\endverbatim
60: *
61: * Arguments:
62: * ==========
63: *
64: *> \param[in] M
65: *> \verbatim
66: *> M is INTEGER
67: *> The number of rows X11 plus the number of rows in X21.
68: *> \endverbatim
69: *>
70: *> \param[in] P
71: *> \verbatim
72: *> P is INTEGER
73: *> The number of rows in X11. 0 <= P <= min(M-P,Q,M-Q).
74: *> \endverbatim
75: *>
76: *> \param[in] Q
77: *> \verbatim
78: *> Q is INTEGER
79: *> The number of columns in X11 and X21. 0 <= Q <= M.
80: *> \endverbatim
81: *>
82: *> \param[in,out] X11
83: *> \verbatim
84: *> X11 is COMPLEX*16 array, dimension (LDX11,Q)
85: *> On entry, the top block of the matrix X to be reduced. On
86: *> exit, the columns of tril(X11) specify reflectors for P1 and
87: *> the rows of triu(X11,1) specify reflectors for Q1.
88: *> \endverbatim
89: *>
90: *> \param[in] LDX11
91: *> \verbatim
92: *> LDX11 is INTEGER
93: *> The leading dimension of X11. LDX11 >= P.
94: *> \endverbatim
95: *>
96: *> \param[in,out] X21
97: *> \verbatim
98: *> X21 is COMPLEX*16 array, dimension (LDX21,Q)
99: *> On entry, the bottom block of the matrix X to be reduced. On
100: *> exit, the columns of tril(X21) specify reflectors for P2.
101: *> \endverbatim
102: *>
103: *> \param[in] LDX21
104: *> \verbatim
105: *> LDX21 is INTEGER
106: *> The leading dimension of X21. LDX21 >= M-P.
107: *> \endverbatim
108: *>
109: *> \param[out] THETA
110: *> \verbatim
111: *> THETA is DOUBLE PRECISION array, dimension (Q)
112: *> The entries of the bidiagonal blocks B11, B21 are defined by
113: *> THETA and PHI. See Further Details.
114: *> \endverbatim
115: *>
116: *> \param[out] PHI
117: *> \verbatim
118: *> PHI is DOUBLE PRECISION array, dimension (Q-1)
119: *> The entries of the bidiagonal blocks B11, B21 are defined by
120: *> THETA and PHI. See Further Details.
121: *> \endverbatim
122: *>
123: *> \param[out] TAUP1
124: *> \verbatim
125: *> TAUP1 is COMPLEX*16 array, dimension (P)
126: *> The scalar factors of the elementary reflectors that define
127: *> P1.
128: *> \endverbatim
129: *>
130: *> \param[out] TAUP2
131: *> \verbatim
132: *> TAUP2 is COMPLEX*16 array, dimension (M-P)
133: *> The scalar factors of the elementary reflectors that define
134: *> P2.
135: *> \endverbatim
136: *>
137: *> \param[out] TAUQ1
138: *> \verbatim
139: *> TAUQ1 is COMPLEX*16 array, dimension (Q)
140: *> The scalar factors of the elementary reflectors that define
141: *> Q1.
142: *> \endverbatim
143: *>
144: *> \param[out] WORK
145: *> \verbatim
146: *> WORK is COMPLEX*16 array, dimension (LWORK)
147: *> \endverbatim
148: *>
149: *> \param[in] LWORK
150: *> \verbatim
151: *> LWORK is INTEGER
152: *> The dimension of the array WORK. LWORK >= M-Q.
153: *>
154: *> If LWORK = -1, then a workspace query is assumed; the routine
155: *> only calculates the optimal size of the WORK array, returns
156: *> this value as the first entry of the WORK array, and no error
157: *> message related to LWORK is issued by XERBLA.
158: *> \endverbatim
159: *>
160: *> \param[out] INFO
161: *> \verbatim
162: *> INFO is INTEGER
163: *> = 0: successful exit.
164: *> < 0: if INFO = -i, the i-th argument had an illegal value.
165: *> \endverbatim
166: *
167: * Authors:
168: * ========
169: *
170: *> \author Univ. of Tennessee
171: *> \author Univ. of California Berkeley
172: *> \author Univ. of Colorado Denver
173: *> \author NAG Ltd.
174: *
175: *> \date July 2012
176: *
177: *> \ingroup complex16OTHERcomputational
178: *
179: *> \par Further Details:
180: * =====================
181: *>
182: *> \verbatim
183: *>
184: *> The upper-bidiagonal blocks B11, B21 are represented implicitly by
185: *> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
186: *> in each bidiagonal band is a product of a sine or cosine of a THETA
187: *> with a sine or cosine of a PHI. See [1] or ZUNCSD for details.
188: *>
189: *> P1, P2, and Q1 are represented as products of elementary reflectors.
190: *> See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
191: *> and ZUNGLQ.
192: *> \endverbatim
193: *
194: *> \par References:
195: * ================
196: *>
197: *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
198: *> Algorithms, 50(1):33-65, 2009.
199: *>
200: * =====================================================================
201: SUBROUTINE ZUNBDB2( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
202: $ TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
203: *
204: * -- LAPACK computational routine (version 3.7.0) --
205: * -- LAPACK is a software package provided by Univ. of Tennessee, --
206: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
207: * July 2012
208: *
209: * .. Scalar Arguments ..
210: INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
211: * ..
212: * .. Array Arguments ..
213: DOUBLE PRECISION PHI(*), THETA(*)
214: COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
215: $ X11(LDX11,*), X21(LDX21,*)
216: * ..
217: *
218: * ====================================================================
219: *
220: * .. Parameters ..
221: COMPLEX*16 NEGONE, ONE
222: PARAMETER ( NEGONE = (-1.0D0,0.0D0),
223: $ ONE = (1.0D0,0.0D0) )
224: * ..
225: * .. Local Scalars ..
226: DOUBLE PRECISION C, S
227: INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
228: $ LWORKMIN, LWORKOPT
229: LOGICAL LQUERY
230: * ..
231: * .. External Subroutines ..
232: EXTERNAL ZLARF, ZLARFGP, ZUNBDB5, ZDROT, ZSCAL, XERBLA
233: * ..
234: * .. External Functions ..
235: DOUBLE PRECISION DZNRM2
236: EXTERNAL DZNRM2
237: * ..
238: * .. Intrinsic Function ..
239: INTRINSIC ATAN2, COS, MAX, SIN, SQRT
240: * ..
241: * .. Executable Statements ..
242: *
243: * Test input arguments
244: *
245: INFO = 0
246: LQUERY = LWORK .EQ. -1
247: *
248: IF( M .LT. 0 ) THEN
249: INFO = -1
250: ELSE IF( P .LT. 0 .OR. P .GT. M-P ) THEN
251: INFO = -2
252: ELSE IF( Q .LT. 0 .OR. Q .LT. P .OR. M-Q .LT. P ) THEN
253: INFO = -3
254: ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
255: INFO = -5
256: ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
257: INFO = -7
258: END IF
259: *
260: * Compute workspace
261: *
262: IF( INFO .EQ. 0 ) THEN
263: ILARF = 2
264: LLARF = MAX( P-1, M-P, Q-1 )
265: IORBDB5 = 2
266: LORBDB5 = Q-1
267: LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
268: LWORKMIN = LWORKOPT
269: WORK(1) = LWORKOPT
270: IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
271: INFO = -14
272: END IF
273: END IF
274: IF( INFO .NE. 0 ) THEN
275: CALL XERBLA( 'ZUNBDB2', -INFO )
276: RETURN
277: ELSE IF( LQUERY ) THEN
278: RETURN
279: END IF
280: *
281: * Reduce rows 1, ..., P of X11 and X21
282: *
283: DO I = 1, P
284: *
285: IF( I .GT. 1 ) THEN
286: CALL ZDROT( Q-I+1, X11(I,I), LDX11, X21(I-1,I), LDX21, C,
287: $ S )
288: END IF
289: CALL ZLACGV( Q-I+1, X11(I,I), LDX11 )
290: CALL ZLARFGP( Q-I+1, X11(I,I), X11(I,I+1), LDX11, TAUQ1(I) )
291: C = DBLE( X11(I,I) )
292: X11(I,I) = ONE
293: CALL ZLARF( 'R', P-I, Q-I+1, X11(I,I), LDX11, TAUQ1(I),
294: $ X11(I+1,I), LDX11, WORK(ILARF) )
295: CALL ZLARF( 'R', M-P-I+1, Q-I+1, X11(I,I), LDX11, TAUQ1(I),
296: $ X21(I,I), LDX21, WORK(ILARF) )
297: CALL ZLACGV( Q-I+1, X11(I,I), LDX11 )
298: S = SQRT( DZNRM2( P-I, X11(I+1,I), 1 )**2
299: $ + DZNRM2( M-P-I+1, X21(I,I), 1 )**2 )
300: THETA(I) = ATAN2( S, C )
301: *
302: CALL ZUNBDB5( P-I, M-P-I+1, Q-I, X11(I+1,I), 1, X21(I,I), 1,
303: $ X11(I+1,I+1), LDX11, X21(I,I+1), LDX21,
304: $ WORK(IORBDB5), LORBDB5, CHILDINFO )
305: CALL ZSCAL( P-I, NEGONE, X11(I+1,I), 1 )
306: CALL ZLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
307: IF( I .LT. P ) THEN
308: CALL ZLARFGP( P-I, X11(I+1,I), X11(I+2,I), 1, TAUP1(I) )
309: PHI(I) = ATAN2( DBLE( X11(I+1,I) ), DBLE( X21(I,I) ) )
310: C = COS( PHI(I) )
311: S = SIN( PHI(I) )
312: X11(I+1,I) = ONE
313: CALL ZLARF( 'L', P-I, Q-I, X11(I+1,I), 1, DCONJG(TAUP1(I)),
314: $ X11(I+1,I+1), LDX11, WORK(ILARF) )
315: END IF
316: X21(I,I) = ONE
317: CALL ZLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, DCONJG(TAUP2(I)),
318: $ X21(I,I+1), LDX21, WORK(ILARF) )
319: *
320: END DO
321: *
322: * Reduce the bottom-right portion of X21 to the identity matrix
323: *
324: DO I = P + 1, Q
325: CALL ZLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
326: X21(I,I) = ONE
327: CALL ZLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, DCONJG(TAUP2(I)),
328: $ X21(I,I+1), LDX21, WORK(ILARF) )
329: END DO
330: *
331: RETURN
332: *
333: * End of ZUNBDB2
334: *
335: END
336:
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