1: *> \brief \b ZUNBDB1
2: *
3: * =========== DOCUMENTATION ===========
4: *
5: * Online html documentation available at
6: * http://www.netlib.org/lapack/explore-html/
7: *
8: *> \htmlonly
9: *> Download ZUNBDB1 + 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/zunbdb1.f">
15: *> [TXT]</a>
16: *> \endhtmlonly
17: *
18: * Definition:
19: * ===========
20: *
21: * SUBROUTINE ZUNBDB1( 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: *> ZUNBDB1 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. Q must be no larger than P,
49: *> M-P, or M-Q. Routines ZUNBDB2, ZUNBDB3, and ZUNBDB4 handle cases in
50: *> which Q 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 Q-by-Q 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 <= M.
74: *> \endverbatim
75: *>
76: *> \param[in] Q
77: *> \verbatim
78: *> Q is INTEGER
79: *> The number of columns in X11 and X21. 0 <= Q <=
80: *> MIN(P,M-P,M-Q).
81: *> \endverbatim
82: *>
83: *> \param[in,out] X11
84: *> \verbatim
85: *> X11 is COMPLEX*16 array, dimension (LDX11,Q)
86: *> On entry, the top block of the matrix X to be reduced. On
87: *> exit, the columns of tril(X11) specify reflectors for P1 and
88: *> the rows of triu(X11,1) specify reflectors for Q1.
89: *> \endverbatim
90: *>
91: *> \param[in] LDX11
92: *> \verbatim
93: *> LDX11 is INTEGER
94: *> The leading dimension of X11. LDX11 >= P.
95: *> \endverbatim
96: *>
97: *> \param[in,out] X21
98: *> \verbatim
99: *> X21 is COMPLEX*16 array, dimension (LDX21,Q)
100: *> On entry, the bottom block of the matrix X to be reduced. On
101: *> exit, the columns of tril(X21) specify reflectors for P2.
102: *> \endverbatim
103: *>
104: *> \param[in] LDX21
105: *> \verbatim
106: *> LDX21 is INTEGER
107: *> The leading dimension of X21. LDX21 >= M-P.
108: *> \endverbatim
109: *>
110: *> \param[out] THETA
111: *> \verbatim
112: *> THETA is DOUBLE PRECISION array, dimension (Q)
113: *> The entries of the bidiagonal blocks B11, B21 are defined by
114: *> THETA and PHI. See Further Details.
115: *> \endverbatim
116: *>
117: *> \param[out] PHI
118: *> \verbatim
119: *> PHI is DOUBLE PRECISION array, dimension (Q-1)
120: *> The entries of the bidiagonal blocks B11, B21 are defined by
121: *> THETA and PHI. See Further Details.
122: *> \endverbatim
123: *>
124: *> \param[out] TAUP1
125: *> \verbatim
126: *> TAUP1 is COMPLEX*16 array, dimension (P)
127: *> The scalar factors of the elementary reflectors that define
128: *> P1.
129: *> \endverbatim
130: *>
131: *> \param[out] TAUP2
132: *> \verbatim
133: *> TAUP2 is COMPLEX*16 array, dimension (M-P)
134: *> The scalar factors of the elementary reflectors that define
135: *> P2.
136: *> \endverbatim
137: *>
138: *> \param[out] TAUQ1
139: *> \verbatim
140: *> TAUQ1 is COMPLEX*16 array, dimension (Q)
141: *> The scalar factors of the elementary reflectors that define
142: *> Q1.
143: *> \endverbatim
144: *>
145: *> \param[out] WORK
146: *> \verbatim
147: *> WORK is COMPLEX*16 array, dimension (LWORK)
148: *> \endverbatim
149: *>
150: *> \param[in] LWORK
151: *> \verbatim
152: *> LWORK is INTEGER
153: *> The dimension of the array WORK. LWORK >= M-Q.
154: *>
155: *> If LWORK = -1, then a workspace query is assumed; the routine
156: *> only calculates the optimal size of the WORK array, returns
157: *> this value as the first entry of the WORK array, and no error
158: *> message related to LWORK is issued by XERBLA.
159: *> \endverbatim
160: *>
161: *> \param[out] INFO
162: *> \verbatim
163: *> INFO is INTEGER
164: *> = 0: successful exit.
165: *> < 0: if INFO = -i, the i-th argument had an illegal value.
166: *> \endverbatim
167: *>
168: *
169: * Authors:
170: * ========
171: *
172: *> \author Univ. of Tennessee
173: *> \author Univ. of California Berkeley
174: *> \author Univ. of Colorado Denver
175: *> \author NAG Ltd.
176: *
177: *> \date July 2012
178: *
179: *> \ingroup complex16OTHERcomputational
180: *
181: *> \par Further Details:
182: * =====================
183: *>
184: *> \verbatim
185: *>
186: *> The upper-bidiagonal blocks B11, B21 are represented implicitly by
187: *> angles THETA(1), ..., THETA(Q) and PHI(1), ..., PHI(Q-1). Every entry
188: *> in each bidiagonal band is a product of a sine or cosine of a THETA
189: *> with a sine or cosine of a PHI. See [1] or ZUNCSD for details.
190: *>
191: *> P1, P2, and Q1 are represented as products of elementary reflectors.
192: *> See ZUNCSD2BY1 for details on generating P1, P2, and Q1 using ZUNGQR
193: *> and ZUNGLQ.
194: *> \endverbatim
195: *
196: *> \par References:
197: * ================
198: *>
199: *> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
200: *> Algorithms, 50(1):33-65, 2009.
201: *>
202: * =====================================================================
203: SUBROUTINE ZUNBDB1( M, P, Q, X11, LDX11, X21, LDX21, THETA, PHI,
204: $ TAUP1, TAUP2, TAUQ1, WORK, LWORK, INFO )
205: *
206: * -- LAPACK computational routine (version 3.7.0) --
207: * -- LAPACK is a software package provided by Univ. of Tennessee, --
208: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
209: * July 2012
210: *
211: * .. Scalar Arguments ..
212: INTEGER INFO, LWORK, M, P, Q, LDX11, LDX21
213: * ..
214: * .. Array Arguments ..
215: DOUBLE PRECISION PHI(*), THETA(*)
216: COMPLEX*16 TAUP1(*), TAUP2(*), TAUQ1(*), WORK(*),
217: $ X11(LDX11,*), X21(LDX21,*)
218: * ..
219: *
220: * ====================================================================
221: *
222: * .. Parameters ..
223: COMPLEX*16 ONE
224: PARAMETER ( ONE = (1.0D0,0.0D0) )
225: * ..
226: * .. Local Scalars ..
227: DOUBLE PRECISION C, S
228: INTEGER CHILDINFO, I, ILARF, IORBDB5, LLARF, LORBDB5,
229: $ LWORKMIN, LWORKOPT
230: LOGICAL LQUERY
231: * ..
232: * .. External Subroutines ..
233: EXTERNAL ZLARF, ZLARFGP, ZUNBDB5, ZDROT, XERBLA
234: EXTERNAL ZLACGV
235: * ..
236: * .. External Functions ..
237: DOUBLE PRECISION DZNRM2
238: EXTERNAL DZNRM2
239: * ..
240: * .. Intrinsic Function ..
241: INTRINSIC ATAN2, COS, MAX, SIN, SQRT
242: * ..
243: * .. Executable Statements ..
244: *
245: * Test input arguments
246: *
247: INFO = 0
248: LQUERY = LWORK .EQ. -1
249: *
250: IF( M .LT. 0 ) THEN
251: INFO = -1
252: ELSE IF( P .LT. Q .OR. M-P .LT. Q ) THEN
253: INFO = -2
254: ELSE IF( Q .LT. 0 .OR. M-Q .LT. Q ) THEN
255: INFO = -3
256: ELSE IF( LDX11 .LT. MAX( 1, P ) ) THEN
257: INFO = -5
258: ELSE IF( LDX21 .LT. MAX( 1, M-P ) ) THEN
259: INFO = -7
260: END IF
261: *
262: * Compute workspace
263: *
264: IF( INFO .EQ. 0 ) THEN
265: ILARF = 2
266: LLARF = MAX( P-1, M-P-1, Q-1 )
267: IORBDB5 = 2
268: LORBDB5 = Q-2
269: LWORKOPT = MAX( ILARF+LLARF-1, IORBDB5+LORBDB5-1 )
270: LWORKMIN = LWORKOPT
271: WORK(1) = LWORKOPT
272: IF( LWORK .LT. LWORKMIN .AND. .NOT.LQUERY ) THEN
273: INFO = -14
274: END IF
275: END IF
276: IF( INFO .NE. 0 ) THEN
277: CALL XERBLA( 'ZUNBDB1', -INFO )
278: RETURN
279: ELSE IF( LQUERY ) THEN
280: RETURN
281: END IF
282: *
283: * Reduce columns 1, ..., Q of X11 and X21
284: *
285: DO I = 1, Q
286: *
287: CALL ZLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
288: CALL ZLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
289: THETA(I) = ATAN2( DBLE( X21(I,I) ), DBLE( X11(I,I) ) )
290: C = COS( THETA(I) )
291: S = SIN( THETA(I) )
292: X11(I,I) = ONE
293: X21(I,I) = ONE
294: CALL ZLARF( 'L', P-I+1, Q-I, X11(I,I), 1, DCONJG(TAUP1(I)),
295: $ X11(I,I+1), LDX11, WORK(ILARF) )
296: CALL ZLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, DCONJG(TAUP2(I)),
297: $ X21(I,I+1), LDX21, WORK(ILARF) )
298: *
299: IF( I .LT. Q ) THEN
300: CALL ZDROT( Q-I, X11(I,I+1), LDX11, X21(I,I+1), LDX21, C,
301: $ S )
302: CALL ZLACGV( Q-I, X21(I,I+1), LDX21 )
303: CALL ZLARFGP( Q-I, X21(I,I+1), X21(I,I+2), LDX21, TAUQ1(I) )
304: S = DBLE( X21(I,I+1) )
305: X21(I,I+1) = ONE
306: CALL ZLARF( 'R', P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
307: $ X11(I+1,I+1), LDX11, WORK(ILARF) )
308: CALL ZLARF( 'R', M-P-I, Q-I, X21(I,I+1), LDX21, TAUQ1(I),
309: $ X21(I+1,I+1), LDX21, WORK(ILARF) )
310: CALL ZLACGV( Q-I, X21(I,I+1), LDX21 )
311: C = SQRT( DZNRM2( P-I, X11(I+1,I+1), 1 )**2
312: $ + DZNRM2( M-P-I, X21(I+1,I+1), 1 )**2 )
313: PHI(I) = ATAN2( S, C )
314: CALL ZUNBDB5( P-I, M-P-I, Q-I-1, X11(I+1,I+1), 1,
315: $ X21(I+1,I+1), 1, X11(I+1,I+2), LDX11,
316: $ X21(I+1,I+2), LDX21, WORK(IORBDB5), LORBDB5,
317: $ CHILDINFO )
318: END IF
319: *
320: END DO
321: *
322: RETURN
323: *
324: * End of ZUNBDB1
325: *
326: END
327:
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