Annotation of rpl/lapack/lapack/dgebrd.f, revision 1.9
1.9 ! bertrand 1: *> \brief \b DGEBRD
! 2: *
! 3: * =========== DOCUMENTATION ===========
! 4: *
! 5: * Online html documentation available at
! 6: * http://www.netlib.org/lapack/explore-html/
! 7: *
! 8: *> \htmlonly
! 9: *> Download DGEBRD + dependencies
! 10: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dgebrd.f">
! 11: *> [TGZ]</a>
! 12: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dgebrd.f">
! 13: *> [ZIP]</a>
! 14: *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dgebrd.f">
! 15: *> [TXT]</a>
! 16: *> \endhtmlonly
! 17: *
! 18: * Definition:
! 19: * ===========
! 20: *
! 21: * SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
! 22: * INFO )
! 23: *
! 24: * .. Scalar Arguments ..
! 25: * INTEGER INFO, LDA, LWORK, M, N
! 26: * ..
! 27: * .. Array Arguments ..
! 28: * DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
! 29: * $ TAUQ( * ), WORK( * )
! 30: * ..
! 31: *
! 32: *
! 33: *> \par Purpose:
! 34: * =============
! 35: *>
! 36: *> \verbatim
! 37: *>
! 38: *> DGEBRD reduces a general real M-by-N matrix A to upper or lower
! 39: *> bidiagonal form B by an orthogonal transformation: Q**T * A * P = B.
! 40: *>
! 41: *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal.
! 42: *> \endverbatim
! 43: *
! 44: * Arguments:
! 45: * ==========
! 46: *
! 47: *> \param[in] M
! 48: *> \verbatim
! 49: *> M is INTEGER
! 50: *> The number of rows in the matrix A. M >= 0.
! 51: *> \endverbatim
! 52: *>
! 53: *> \param[in] N
! 54: *> \verbatim
! 55: *> N is INTEGER
! 56: *> The number of columns in the matrix A. N >= 0.
! 57: *> \endverbatim
! 58: *>
! 59: *> \param[in,out] A
! 60: *> \verbatim
! 61: *> A is DOUBLE PRECISION array, dimension (LDA,N)
! 62: *> On entry, the M-by-N general matrix to be reduced.
! 63: *> On exit,
! 64: *> if m >= n, the diagonal and the first superdiagonal are
! 65: *> overwritten with the upper bidiagonal matrix B; the
! 66: *> elements below the diagonal, with the array TAUQ, represent
! 67: *> the orthogonal matrix Q as a product of elementary
! 68: *> reflectors, and the elements above the first superdiagonal,
! 69: *> with the array TAUP, represent the orthogonal matrix P as
! 70: *> a product of elementary reflectors;
! 71: *> if m < n, the diagonal and the first subdiagonal are
! 72: *> overwritten with the lower bidiagonal matrix B; the
! 73: *> elements below the first subdiagonal, with the array TAUQ,
! 74: *> represent the orthogonal matrix Q as a product of
! 75: *> elementary reflectors, and the elements above the diagonal,
! 76: *> with the array TAUP, represent the orthogonal matrix P as
! 77: *> a product of elementary reflectors.
! 78: *> See Further Details.
! 79: *> \endverbatim
! 80: *>
! 81: *> \param[in] LDA
! 82: *> \verbatim
! 83: *> LDA is INTEGER
! 84: *> The leading dimension of the array A. LDA >= max(1,M).
! 85: *> \endverbatim
! 86: *>
! 87: *> \param[out] D
! 88: *> \verbatim
! 89: *> D is DOUBLE PRECISION array, dimension (min(M,N))
! 90: *> The diagonal elements of the bidiagonal matrix B:
! 91: *> D(i) = A(i,i).
! 92: *> \endverbatim
! 93: *>
! 94: *> \param[out] E
! 95: *> \verbatim
! 96: *> E is DOUBLE PRECISION array, dimension (min(M,N)-1)
! 97: *> The off-diagonal elements of the bidiagonal matrix B:
! 98: *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1;
! 99: *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1.
! 100: *> \endverbatim
! 101: *>
! 102: *> \param[out] TAUQ
! 103: *> \verbatim
! 104: *> TAUQ is DOUBLE PRECISION array dimension (min(M,N))
! 105: *> The scalar factors of the elementary reflectors which
! 106: *> represent the orthogonal matrix Q. See Further Details.
! 107: *> \endverbatim
! 108: *>
! 109: *> \param[out] TAUP
! 110: *> \verbatim
! 111: *> TAUP is DOUBLE PRECISION array, dimension (min(M,N))
! 112: *> The scalar factors of the elementary reflectors which
! 113: *> represent the orthogonal matrix P. See Further Details.
! 114: *> \endverbatim
! 115: *>
! 116: *> \param[out] WORK
! 117: *> \verbatim
! 118: *> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
! 119: *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
! 120: *> \endverbatim
! 121: *>
! 122: *> \param[in] LWORK
! 123: *> \verbatim
! 124: *> LWORK is INTEGER
! 125: *> The length of the array WORK. LWORK >= max(1,M,N).
! 126: *> For optimum performance LWORK >= (M+N)*NB, where NB
! 127: *> is the optimal blocksize.
! 128: *>
! 129: *> If LWORK = -1, then a workspace query is assumed; the routine
! 130: *> only calculates the optimal size of the WORK array, returns
! 131: *> this value as the first entry of the WORK array, and no error
! 132: *> message related to LWORK is issued by XERBLA.
! 133: *> \endverbatim
! 134: *>
! 135: *> \param[out] INFO
! 136: *> \verbatim
! 137: *> INFO is INTEGER
! 138: *> = 0: successful exit
! 139: *> < 0: if INFO = -i, the i-th argument had an illegal value.
! 140: *> \endverbatim
! 141: *
! 142: * Authors:
! 143: * ========
! 144: *
! 145: *> \author Univ. of Tennessee
! 146: *> \author Univ. of California Berkeley
! 147: *> \author Univ. of Colorado Denver
! 148: *> \author NAG Ltd.
! 149: *
! 150: *> \date November 2011
! 151: *
! 152: *> \ingroup doubleGEcomputational
! 153: *
! 154: *> \par Further Details:
! 155: * =====================
! 156: *>
! 157: *> \verbatim
! 158: *>
! 159: *> The matrices Q and P are represented as products of elementary
! 160: *> reflectors:
! 161: *>
! 162: *> If m >= n,
! 163: *>
! 164: *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1)
! 165: *>
! 166: *> Each H(i) and G(i) has the form:
! 167: *>
! 168: *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
! 169: *>
! 170: *> where tauq and taup are real scalars, and v and u are real vectors;
! 171: *> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i);
! 172: *> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n);
! 173: *> tauq is stored in TAUQ(i) and taup in TAUP(i).
! 174: *>
! 175: *> If m < n,
! 176: *>
! 177: *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m)
! 178: *>
! 179: *> Each H(i) and G(i) has the form:
! 180: *>
! 181: *> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T
! 182: *>
! 183: *> where tauq and taup are real scalars, and v and u are real vectors;
! 184: *> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i);
! 185: *> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n);
! 186: *> tauq is stored in TAUQ(i) and taup in TAUP(i).
! 187: *>
! 188: *> The contents of A on exit are illustrated by the following examples:
! 189: *>
! 190: *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n):
! 191: *>
! 192: *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 )
! 193: *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 )
! 194: *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 )
! 195: *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 )
! 196: *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 )
! 197: *> ( v1 v2 v3 v4 v5 )
! 198: *>
! 199: *> where d and e denote diagonal and off-diagonal elements of B, vi
! 200: *> denotes an element of the vector defining H(i), and ui an element of
! 201: *> the vector defining G(i).
! 202: *> \endverbatim
! 203: *>
! 204: * =====================================================================
1.1 bertrand 205: SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK,
206: $ INFO )
207: *
1.9 ! bertrand 208: * -- LAPACK computational routine (version 3.4.0) --
1.1 bertrand 209: * -- LAPACK is a software package provided by Univ. of Tennessee, --
210: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
1.9 ! bertrand 211: * November 2011
1.1 bertrand 212: *
213: * .. Scalar Arguments ..
214: INTEGER INFO, LDA, LWORK, M, N
215: * ..
216: * .. Array Arguments ..
217: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAUP( * ),
218: $ TAUQ( * ), WORK( * )
219: * ..
220: *
221: * =====================================================================
222: *
223: * .. Parameters ..
224: DOUBLE PRECISION ONE
225: PARAMETER ( ONE = 1.0D+0 )
226: * ..
227: * .. Local Scalars ..
228: LOGICAL LQUERY
229: INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB,
230: $ NBMIN, NX
231: DOUBLE PRECISION WS
232: * ..
233: * .. External Subroutines ..
234: EXTERNAL DGEBD2, DGEMM, DLABRD, XERBLA
235: * ..
236: * .. Intrinsic Functions ..
237: INTRINSIC DBLE, MAX, MIN
238: * ..
239: * .. External Functions ..
240: INTEGER ILAENV
241: EXTERNAL ILAENV
242: * ..
243: * .. Executable Statements ..
244: *
245: * Test the input parameters
246: *
247: INFO = 0
248: NB = MAX( 1, ILAENV( 1, 'DGEBRD', ' ', M, N, -1, -1 ) )
249: LWKOPT = ( M+N )*NB
250: WORK( 1 ) = DBLE( LWKOPT )
251: LQUERY = ( LWORK.EQ.-1 )
252: IF( M.LT.0 ) THEN
253: INFO = -1
254: ELSE IF( N.LT.0 ) THEN
255: INFO = -2
256: ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
257: INFO = -4
258: ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN
259: INFO = -10
260: END IF
261: IF( INFO.LT.0 ) THEN
262: CALL XERBLA( 'DGEBRD', -INFO )
263: RETURN
264: ELSE IF( LQUERY ) THEN
265: RETURN
266: END IF
267: *
268: * Quick return if possible
269: *
270: MINMN = MIN( M, N )
271: IF( MINMN.EQ.0 ) THEN
272: WORK( 1 ) = 1
273: RETURN
274: END IF
275: *
276: WS = MAX( M, N )
277: LDWRKX = M
278: LDWRKY = N
279: *
280: IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN
281: *
282: * Set the crossover point NX.
283: *
284: NX = MAX( NB, ILAENV( 3, 'DGEBRD', ' ', M, N, -1, -1 ) )
285: *
286: * Determine when to switch from blocked to unblocked code.
287: *
288: IF( NX.LT.MINMN ) THEN
289: WS = ( M+N )*NB
290: IF( LWORK.LT.WS ) THEN
291: *
292: * Not enough work space for the optimal NB, consider using
293: * a smaller block size.
294: *
295: NBMIN = ILAENV( 2, 'DGEBRD', ' ', M, N, -1, -1 )
296: IF( LWORK.GE.( M+N )*NBMIN ) THEN
297: NB = LWORK / ( M+N )
298: ELSE
299: NB = 1
300: NX = MINMN
301: END IF
302: END IF
303: END IF
304: ELSE
305: NX = MINMN
306: END IF
307: *
308: DO 30 I = 1, MINMN - NX, NB
309: *
310: * Reduce rows and columns i:i+nb-1 to bidiagonal form and return
311: * the matrices X and Y which are needed to update the unreduced
312: * part of the matrix
313: *
314: CALL DLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ),
315: $ TAUQ( I ), TAUP( I ), WORK, LDWRKX,
316: $ WORK( LDWRKX*NB+1 ), LDWRKY )
317: *
318: * Update the trailing submatrix A(i+nb:m,i+nb:n), using an update
1.8 bertrand 319: * of the form A := A - V*Y**T - X*U**T
1.1 bertrand 320: *
321: CALL DGEMM( 'No transpose', 'Transpose', M-I-NB+1, N-I-NB+1,
322: $ NB, -ONE, A( I+NB, I ), LDA,
323: $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE,
324: $ A( I+NB, I+NB ), LDA )
325: CALL DGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1,
326: $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA,
327: $ ONE, A( I+NB, I+NB ), LDA )
328: *
329: * Copy diagonal and off-diagonal elements of B back into A
330: *
331: IF( M.GE.N ) THEN
332: DO 10 J = I, I + NB - 1
333: A( J, J ) = D( J )
334: A( J, J+1 ) = E( J )
335: 10 CONTINUE
336: ELSE
337: DO 20 J = I, I + NB - 1
338: A( J, J ) = D( J )
339: A( J+1, J ) = E( J )
340: 20 CONTINUE
341: END IF
342: 30 CONTINUE
343: *
344: * Use unblocked code to reduce the remainder of the matrix
345: *
346: CALL DGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ),
347: $ TAUQ( I ), TAUP( I ), WORK, IINFO )
348: WORK( 1 ) = WS
349: RETURN
350: *
351: * End of DGEBRD
352: *
353: END
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