Annotation of rpl/lapack/lapack/zgebd2.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b ZGEBD2
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download ZGEBD2 + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgebd2.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgebd2.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebd2.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
! 22: *
! 23: * .. Scalar Arguments ..
! 24: * INTEGER INFO, LDA, M, N
! 25: * ..
! 26: * .. Array Arguments ..
! 27: * DOUBLE PRECISION D( * ), E( * )
! 28: * COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
! 29: * ..
! 30: *
! 31: *
! 32: *> \par Purpose:
! 33: * =============
! 34: *>
! 35: *> \verbatim
! 36: *>
! 37: *> ZGEBD2 reduces a complex general m by n matrix A to upper or lower
! 38: *> real bidiagonal form B by a unitary transformation: Q**H * A * P = B.
! 39: *>
! 40: *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! 41: *> \endverbatim
! 42: *
! 43: * Arguments:
! 44: * ==========
! 45: *
! 46: *> \param[in] M
! 47: *> \verbatim
! 48: *> M is INTEGER
! 49: *> The number of rows in the matrix A. M >= 0.
! 50: *> \endverbatim
! 51: *>
! 52: *> \param[in] N
! 53: *> \verbatim
! 54: *> N is INTEGER
! 55: *> The number of columns in the matrix A. N >= 0.
! 56: *> \endverbatim
! 57: *>
! 58: *> \param[in,out] A
! 59: *> \verbatim
! 60: *> A is COMPLEX*16 array, dimension (LDA,N)
! 61: *> On entry, the m by n general matrix to be reduced.
! 62: *> On exit,
! 63: *> if m >= n, the diagonal and the first superdiagonal are
! 64: *> overwritten with the upper bidiagonal matrix B; the
! 65: *> elements below the diagonal, with the array TAUQ, represent
! 66: *> the unitary matrix Q as a product of elementary
! 67: *> reflectors, and the elements above the first superdiagonal,
! 68: *> with the array TAUP, represent the unitary matrix P as
! 69: *> a product of elementary reflectors;
! 70: *> if m < n, the diagonal and the first subdiagonal are
! 71: *> overwritten with the lower bidiagonal matrix B; the
! 72: *> elements below the first subdiagonal, with the array TAUQ,
! 73: *> represent the unitary matrix Q as a product of
! 74: *> elementary reflectors, and the elements above the diagonal,
! 75: *> with the array TAUP, represent the unitary matrix P as
! 76: *> a product of elementary reflectors.
! 77: *> See Further Details.
! 78: *> \endverbatim
! 79: *>
! 80: *> \param[in] LDA
! 81: *> \verbatim
! 82: *> LDA is INTEGER
! 83: *> The leading dimension of the array A. LDA >= max(1,M).
! 84: *> \endverbatim
! 85: *>
! 86: *> \param[out] D
! 87: *> \verbatim
! 88: *> D is DOUBLE PRECISION array, dimension (min(M,N))
! 89: *> The diagonal elements of the bidiagonal matrix B:
! 90: *> D(i) = A(i,i).
! 91: *> \endverbatim
! 92: *>
! 93: *> \param[out] E
! 94: *> \verbatim
! 95: *> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
! 96: *> The off-diagonal elements of the bidiagonal matrix B:
! 97: *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
! 98: *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
! 99: *> \endverbatim
! 100: *>
! 101: *> \param[out] TAUQ
! 102: *> \verbatim
! 103: *> TAUQ is COMPLEX*16 array dimension (min(M,N))
! 104: *> The scalar factors of the elementary reflectors which
! 105: *> represent the unitary matrix Q. See Further Details.
! 106: *> \endverbatim
! 107: *>
! 108: *> \param[out] TAUP
! 109: *> \verbatim
! 110: *> TAUP is COMPLEX*16 array, dimension (min(M,N))
! 111: *> The scalar factors of the elementary reflectors which
! 112: *> represent the unitary matrix P. See Further Details.
! 113: *> \endverbatim
! 114: *>
! 115: *> \param[out] WORK
! 116: *> \verbatim
! 117: *> WORK is COMPLEX*16 array, dimension (max(M,N))
! 118: *> \endverbatim
! 119: *>
! 120: *> \param[out] INFO
! 121: *> \verbatim
! 122: *> INFO is INTEGER
! 123: *> = 0: successful exit
! 124: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 125: *> \endverbatim
! 126: *
! 127: * Authors:
! 128: * ========
! 129: *
! 130: *> \author Univ. of Tennessee
! 131: *> \author Univ. of California Berkeley
! 132: *> \author Univ. of Colorado Denver
! 133: *> \author NAG Ltd.
! 134: *
! 135: *> \date November 2011
! 136: *
! 137: *> \ingroup complex16GEcomputational
! 138: *
! 139: *> \par Further Details:
! 140: * =====================
! 141: *>
! 142: *> \verbatim
! 143: *>
! 144: *> The matrices Q and P are represented as products of elementary
! 145: *> reflectors:
! 146: *>
! 147: *> If m >= n,
! 148: *>
! 149: *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
! 150: *>
! 151: *> Each H(i) and G(i) has the form:
! 152: *>
! 153: *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
! 154: *>
! 155: *> where tauq and taup are complex scalars, and v and u are complex
! 156: *> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in
! 157: *> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in
! 158: *> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i).
! 159: *>
! 160: *> If m < n,
! 161: *>
! 162: *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
! 163: *>
! 164: *> Each H(i) and G(i) has the form:
! 165: *>
! 166: *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H
! 167: *>
! 168: *> where tauq and taup are complex scalars, v and u are complex vectors;
! 169: *> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
! 170: *> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
! 171: *> tauq is stored in TAUQ(i) and taup in TAUP(i).
! 172: *>
! 173: *> The contents of A on exit are illustrated by the following examples:
! 174: *>
! 175: *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
! 176: *>
! 177: *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
! 178: *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
! 179: *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
! 180: *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
! 181: *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
! 182: *> ( v1 v2 v3 v4 v5 )
! 183: *>
! 184: *> where d and e denote diagonal and off-diagonal elements of B, vi
! 185: *> denotes an element of the vector defining H(i), and ui an element of
! 186: *> the vector defining G(i).
! 187: *> \endverbatim
! 188: *>
! 189: * =====================================================================
1.1 bertrand 190: SUBROUTINE ZGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO )
191: *
1.9 ! bertrand 192: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 193: * -- LAPACK is a software package provided by Univ. of Tennessee, --
194: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 195: * November 2011
1.1 bertrand 196: *
197: * .. Scalar Arguments ..
198: INTEGER INFO, LDA, M, N
199: * ..
200: * .. Array Arguments ..
201: DOUBLE PRECISION D( * ), E( * )
202: COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * )
203: * ..
204: *
205: * =====================================================================
206: *
207: * .. Parameters ..
208: COMPLEX*16 ZERO, ONE
209: PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ),
210: $ ONE = ( 1.0D+0, 0.0D+0 ) )
211: * ..
212: * .. Local Scalars ..
213: INTEGER I
214: COMPLEX*16 ALPHA
215: * ..
216: * .. External Subroutines ..
217: EXTERNAL XERBLA, ZLACGV, ZLARF, ZLARFG
218: * ..
219: * .. Intrinsic Functions ..
220: INTRINSIC DCONJG, MAX, MIN
221: * ..
222: * .. Executable Statements ..
223: *
224: * Test the input parameters
225: *
226: INFO = 0
227: IF( M.LT.0 ) THEN
228: INFO = -1
229: ELSE IF( N.LT.0 ) THEN
230: INFO = -2
231: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
232: INFO = -4
233: END IF
234: IF( INFO.LT.0 ) THEN
235: CALL XERBLA( 'ZGEBD2', -INFO )
236: RETURN
237: END IF
238: *
239: IF( M.GE.N ) THEN
240: *
241: * Reduce to upper bidiagonal form
242: *
243: DO 10 I = 1, N
244: *
245: * Generate elementary reflector H(i) to annihilate A(i+1:m,i)
246: *
247: ALPHA = A( I, I )
248: CALL ZLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1,
249: $ TAUQ( I ) )
250: D( I ) = ALPHA
251: A( I, I ) = ONE
252: *
1.8 bertrand 253: * Apply H(i)**H to A(i:m,i+1:n) from the left
1.1 bertrand 254: *
255: IF( I.LT.N )
256: $ CALL ZLARF( 'Left', M-I+1, N-I, A( I, I ), 1,
257: $ DCONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK )
258: A( I, I ) = D( I )
259: *
260: IF( I.LT.N ) THEN
261: *
262: * Generate elementary reflector G(i) to annihilate
263: * A(i,i+2:n)
264: *
265: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
266: ALPHA = A( I, I+1 )
267: CALL ZLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), LDA,
268: $ TAUP( I ) )
269: E( I ) = ALPHA
270: A( I, I+1 ) = ONE
271: *
272: * Apply G(i) to A(i+1:m,i+1:n) from the right
273: *
274: CALL ZLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA,
275: $ TAUP( I ), A( I+1, I+1 ), LDA, WORK )
276: CALL ZLACGV( N-I, A( I, I+1 ), LDA )
277: A( I, I+1 ) = E( I )
278: ELSE
279: TAUP( I ) = ZERO
280: END IF
281: 10 CONTINUE
282: ELSE
283: *
284: * Reduce to lower bidiagonal form
285: *
286: DO 20 I = 1, M
287: *
288: * Generate elementary reflector G(i) to annihilate A(i,i+1:n)
289: *
290: CALL ZLACGV( N-I+1, A( I, I ), LDA )
291: ALPHA = A( I, I )
292: CALL ZLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA,
293: $ TAUP( I ) )
294: D( I ) = ALPHA
295: A( I, I ) = ONE
296: *
297: * Apply G(i) to A(i+1:m,i:n) from the right
298: *
299: IF( I.LT.M )
300: $ CALL ZLARF( 'Right', M-I, N-I+1, A( I, I ), LDA,
301: $ TAUP( I ), A( I+1, I ), LDA, WORK )
302: CALL ZLACGV( N-I+1, A( I, I ), LDA )
303: A( I, I ) = D( I )
304: *
305: IF( I.LT.M ) THEN
306: *
307: * Generate elementary reflector H(i) to annihilate
308: * A(i+2:m,i)
309: *
310: ALPHA = A( I+1, I )
311: CALL ZLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1,
312: $ TAUQ( I ) )
313: E( I ) = ALPHA
314: A( I+1, I ) = ONE
315: *
1.8 bertrand 316: * Apply H(i)**H to A(i+1:m,i+1:n) from the left
1.1 bertrand 317: *
318: CALL ZLARF( 'Left', M-I, N-I, A( I+1, I ), 1,
319: $ DCONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA,
320: $ WORK )
321: A( I+1, I ) = E( I )
322: ELSE
323: TAUQ( I ) = ZERO
324: END IF
325: 20 CONTINUE
326: END IF
327: RETURN
328: *
329: * End of ZGEBD2
330: *
331: END
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