001:       SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
002: *
003: *  -- LAPACK routine (version 3.2) --
004: *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
005: *     November 2006
006: *
007: *     .. Scalar Arguments ..
008:       INTEGER            INFO, LDA, M, N
009: *     ..
010: *     .. Array Arguments ..
011:       DOUBLE PRECISION   D( * ), E( * )
012:       COMPLEX*16         A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
013: *     ..
014: *
015: *  Purpose
016: *  =======
017: *
018: *  ZGEBD2 reduces a complex general m by n matrix A to upper or lower
019: *  real bidiagonal form B by a unitary transformation: Q' * A * P = B.
020: *
021: *  If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
022: *
023: *  Arguments
024: *  =========
025: *
026: *  M       (input) INTEGER
027: *          The number of rows in the matrix A.  M >= 0.
028: *
029: *  N       (input) INTEGER
030: *          The number of columns in the matrix A.  N >= 0.
031: *
032: *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
033: *          On entry, the m by n general matrix to be reduced.
034: *          On exit,
035: *          if m >= n, the diagonal and the first superdiagonal are
036: *            overwritten with the upper bidiagonal matrix B; the
037: *            elements below the diagonal, with the array TAUQ, represent
038: *            the unitary matrix Q as a product of elementary
039: *            reflectors, and the elements above the first superdiagonal,
040: *            with the array TAUP, represent the unitary matrix P as
041: *            a product of elementary reflectors;
042: *          if m < n, the diagonal and the first subdiagonal are
043: *            overwritten with the lower bidiagonal matrix B; the
044: *            elements below the first subdiagonal, with the array TAUQ,
045: *            represent the unitary matrix Q as a product of
046: *            elementary reflectors, and the elements above the diagonal,
047: *            with the array TAUP, represent the unitary matrix P as
048: *            a product of elementary reflectors.
049: *          See Further Details.
050: *
051: *  LDA     (input) INTEGER
052: *          The leading dimension of the array A.  LDA >= max(1,M).
053: *
054: *  D       (output) DOUBLE PRECISION array, dimension (min(M,N))
055: *          The diagonal elements of the bidiagonal matrix B:
056: *          D(i) = A(i,i).
057: *
058: *  E       (output) DOUBLE PRECISION array, dimension (min(M,N)-1)
059: *          The off-diagonal elements of the bidiagonal matrix B:
060: *          if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
061: *          if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
062: *
063: *  TAUQ    (output) COMPLEX*16 array dimension (min(M,N))
064: *          The scalar factors of the elementary reflectors which
065: *          represent the unitary matrix Q. See Further Details.
066: *
067: *  TAUP    (output) COMPLEX*16 array, dimension (min(M,N))
068: *          The scalar factors of the elementary reflectors which
069: *          represent the unitary matrix P. See Further Details.
070: *
071: *  WORK    (workspace) COMPLEX*16 array, dimension (max(M,N))
072: *
073: *  INFO    (output) INTEGER
074: *          = 0: successful exit
075: *          < 0: if INFO = -i, the i-th argument had an illegal value.
076: *
077: *  Further Details
078: *  ===============
079: *
080: *  The matrices Q and P are represented as products of elementary
081: *  reflectors:
082: *
083: *  If m >= n,
084: *
085: *     Q = H(1) H(2) . . . H(n)  and  P = G(1) G(2) . . . G(n-1)
086: *
087: *  Each H(i) and G(i) has the form:
088: *
089: *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
090: *
091: *  where tauq and taup are complex scalars, and v and u are complex
092: *  vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
093: *  A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
094: *  A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
095: *
096: *  If m < n,
097: *
098: *     Q = H(1) H(2) . . . H(m-1)  and  P = G(1) G(2) . . . G(m)
099: *
100: *  Each H(i) and G(i) has the form:
101: *
102: *     H(i) = I - tauq * v * v'  and G(i) = I - taup * u * u'
103: *
104: *  where tauq and taup are complex scalars, v and u are complex vectors;
105: *  v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
106: *  u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
107: *  tauq is stored in TAUQ(i) and taup in TAUP(i).
108: *
109: *  The contents of A on exit are illustrated by the following examples:
110: *
111: *  m = 6 and n = 5 (m > n):          m = 5 and n = 6 (m < n):
112: *
113: *    (  d   e   u1  u1  u1 )           (  d   u1  u1  u1  u1  u1 )
114: *    (  v1  d   e   u2  u2 )           (  e   d   u2  u2  u2  u2 )
115: *    (  v1  v2  d   e   u3 )           (  v1  e   d   u3  u3  u3 )
116: *    (  v1  v2  v3  d   e  )           (  v1  v2  e   d   u4  u4 )
117: *    (  v1  v2  v3  v4  d  )           (  v1  v2  v3  e   d   u5 )
118: *    (  v1  v2  v3  v4  v5 )
119: *
120: *  where d and e denote diagonal and off-diagonal elements of B, vi
121: *  denotes an element of the vector defining H(i), and ui an element of
122: *  the vector defining G(i).
123: *
124: *  =====================================================================
125: *
126: *     .. Parameters ..
127:       COMPLEX*16         ZERO, ONE
128:       PARAMETER          ( ZERO = ( 0.0D+0, 0.0D+0 ),
129:      $                   ONE = ( 1.0D+0, 0.0D+0 ) )
130: *     ..
131: *     .. Local Scalars ..
132:       INTEGER            I
133:       COMPLEX*16         ALPHA
134: *     ..
135: *     .. External Subroutines ..
136:       EXTERNAL           XERBLA, ZLACGV, ZLARF, ZLARFG
137: *     ..
138: *     .. Intrinsic Functions ..
139:       INTRINSIC          DCONJG, MAX, MIN
140: *     ..
141: *     .. Executable Statements ..
142: *
143: *     Test the input parameters
144: *
145:       INFO = 0
146:       IF( M.LT.0 ) THEN
147:          INFO = -1
148:       ELSE IF( N.LT.0 ) THEN
149:          INFO = -2
150:       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
151:          INFO = -4
152:       END IF
153:       IF( INFO.LT.0 ) THEN
154:          CALL XERBLA( 'ZGEBD2', -INFO )
155:          RETURN
156:       END IF
157: *
158:       IF( M.GE.N ) THEN
159: *
160: *        Reduce to upper bidiagonal form
161: *
162:          DO 10 I = 1, N
163: *
164: *           Generate elementary reflector H(i) to annihilate A(i+1:m,i)
165: *
166:             ALPHA = A( I, I )
167:             CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
168:      $                   TAUQ( I ) )
169:             D( I ) = ALPHA
170:             A( I, I ) = ONE
171: *
172: *           Apply H(i)' to A(i:m,i+1:n) from the left
173: *
174:             IF( I.LT.N )
175:      $         CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
176:      $                     DCONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK )
177:             A( I, I ) = D( I )
178: *
179:             IF( I.LT.N ) THEN
180: *
181: *              Generate elementary reflector G(i) to annihilate
182: *              A(i,i+2:n)
183: *
184:                CALL ZLACGV( N-I, A( I, I+1 ), LDA )
185:                ALPHA = A( I, I+1 )
186:                CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
187:      $                      TAUP( I ) )
188:                E( I ) = ALPHA
189:                A( I, I+1 ) = ONE
190: *
191: *              Apply G(i) to A(i+1:m,i+1:n) from the right
192: *
193:                CALL ZLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
194:      $                     TAUP( I ), A( I+1, I+1 ), LDA, WORK )
195:                CALL ZLACGV( N-I, A( I, I+1 ), LDA )
196:                A( I, I+1 ) = E( I )
197:             ELSE
198:                TAUP( I ) = ZERO
199:             END IF
200:    10    CONTINUE
201:       ELSE
202: *
203: *        Reduce to lower bidiagonal form
204: *
205:          DO 20 I = 1, M
206: *
207: *           Generate elementary reflector G(i) to annihilate A(i,i+1:n)
208: *
209:             CALL ZLACGV( N-I+1, A( I, I ), LDA )
210:             ALPHA = A( I, I )
211:             CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
212:      $                   TAUP( I ) )
213:             D( I ) = ALPHA
214:             A( I, I ) = ONE
215: *
216: *           Apply G(i) to A(i+1:m,i:n) from the right
217: *
218:             IF( I.LT.M )
219:      $         CALL ZLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
220:      $                     TAUP( I ), A( I+1, I ), LDA, WORK )
221:             CALL ZLACGV( N-I+1, A( I, I ), LDA )
222:             A( I, I ) = D( I )
223: *
224:             IF( I.LT.M ) THEN
225: *
226: *              Generate elementary reflector H(i) to annihilate
227: *              A(i+2:m,i)
228: *
229:                ALPHA = A( I+1, I )
230:                CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
231:      $                      TAUQ( I ) )
232:                E( I ) = ALPHA
233:                A( I+1, I ) = ONE
234: *
235: *              Apply H(i)' to A(i+1:m,i+1:n) from the left
236: *
237:                CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1,
238:      $                     DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,
239:      $                     WORK )
240:                A( I+1, I ) = E( I )
241:             ELSE
242:                TAUQ( I ) = ZERO
243:             END IF
244:    20    CONTINUE
245:       END IF
246:       RETURN
247: *
248: *     End of ZGEBD2
249: *
250:       END
251: