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* ===================================================================== 205 SUBROUTINE DGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 206 $ INFO ) 207* 208* -- LAPACK computational routine (version 3.4.0) -- 209* -- LAPACK is a software package provided by Univ. of Tennessee, -- 210* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 211* November 2011 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 319* of the form A := A - V*Y**T - X*U**T 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 354c $Id$ 355