1*> \brief \b SGEBD2 reduces a general matrix to bidiagonal form using an unblocked algorithm. 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8*> \htmlonly 9*> Download SGEBD2 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/sgebd2.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/sgebd2.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/sgebd2.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE SGEBD2( 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* REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), 28* $ TAUQ( * ), WORK( * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> SGEBD2 reduces a real general m by n matrix A to upper or lower 38*> bidiagonal form B by an orthogonal transformation: Q**T * 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 REAL 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 orthogonal matrix Q as a product of elementary 67*> reflectors, and the elements above the first superdiagonal, 68*> with the array TAUP, represent the orthogonal 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 orthogonal matrix Q as a product of 74*> elementary reflectors, and the elements above the diagonal, 75*> with the array TAUP, represent the orthogonal 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 REAL 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 REAL 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 REAL array, dimension (min(M,N)) 104*> The scalar factors of the elementary reflectors which 105*> represent the orthogonal matrix Q. See Further Details. 106*> \endverbatim 107*> 108*> \param[out] TAUP 109*> \verbatim 110*> TAUP is REAL array, dimension (min(M,N)) 111*> The scalar factors of the elementary reflectors which 112*> represent the orthogonal matrix P. See Further Details. 113*> \endverbatim 114*> 115*> \param[out] WORK 116*> \verbatim 117*> WORK is REAL 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*> \ingroup realGEcomputational 136* 137*> \par Further Details: 138* ===================== 139*> 140*> \verbatim 141*> 142*> The matrices Q and P are represented as products of elementary 143*> reflectors: 144*> 145*> If m >= n, 146*> 147*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 148*> 149*> Each H(i) and G(i) has the form: 150*> 151*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T 152*> 153*> where tauq and taup are real scalars, and v and u are real vectors; 154*> v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(i+1:m,i); 155*> u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(i,i+2:n); 156*> tauq is stored in TAUQ(i) and taup in TAUP(i). 157*> 158*> If m < n, 159*> 160*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 161*> 162*> Each H(i) and G(i) has the form: 163*> 164*> H(i) = I - tauq * v * v**T and G(i) = I - taup * u * u**T 165*> 166*> where tauq and taup are real scalars, and v and u are real vectors; 167*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 168*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 169*> tauq is stored in TAUQ(i) and taup in TAUP(i). 170*> 171*> The contents of A on exit are illustrated by the following examples: 172*> 173*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 174*> 175*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 176*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 177*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 178*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 179*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 180*> ( v1 v2 v3 v4 v5 ) 181*> 182*> where d and e denote diagonal and off-diagonal elements of B, vi 183*> denotes an element of the vector defining H(i), and ui an element of 184*> the vector defining G(i). 185*> \endverbatim 186*> 187* ===================================================================== 188 SUBROUTINE SGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 189* 190* -- LAPACK computational routine -- 191* -- LAPACK is a software package provided by Univ. of Tennessee, -- 192* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 193* 194* .. Scalar Arguments .. 195 INTEGER INFO, LDA, M, N 196* .. 197* .. Array Arguments .. 198 REAL A( LDA, * ), D( * ), E( * ), TAUP( * ), 199 $ TAUQ( * ), WORK( * ) 200* .. 201* 202* ===================================================================== 203* 204* .. Parameters .. 205 REAL ZERO, ONE 206 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 207* .. 208* .. Local Scalars .. 209 INTEGER I 210* .. 211* .. External Subroutines .. 212 EXTERNAL SLARF, SLARFG, XERBLA 213* .. 214* .. Intrinsic Functions .. 215 INTRINSIC MAX, MIN 216* .. 217* .. Executable Statements .. 218* 219* Test the input parameters 220* 221 INFO = 0 222 IF( M.LT.0 ) THEN 223 INFO = -1 224 ELSE IF( N.LT.0 ) THEN 225 INFO = -2 226 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 227 INFO = -4 228 END IF 229 IF( INFO.LT.0 ) THEN 230 CALL XERBLA( 'SGEBD2', -INFO ) 231 RETURN 232 END IF 233* 234 IF( M.GE.N ) THEN 235* 236* Reduce to upper bidiagonal form 237* 238 DO 10 I = 1, N 239* 240* Generate elementary reflector H(i) to annihilate A(i+1:m,i) 241* 242 CALL SLARFG( M-I+1, A( I, I ), A( MIN( I+1, M ), I ), 1, 243 $ TAUQ( I ) ) 244 D( I ) = A( I, I ) 245 A( I, I ) = ONE 246* 247* Apply H(i) to A(i:m,i+1:n) from the left 248* 249 IF( I.LT.N ) 250 $ CALL SLARF( 'Left', M-I+1, N-I, A( I, I ), 1, TAUQ( I ), 251 $ A( I, I+1 ), LDA, WORK ) 252 A( I, I ) = D( I ) 253* 254 IF( I.LT.N ) THEN 255* 256* Generate elementary reflector G(i) to annihilate 257* A(i,i+2:n) 258* 259 CALL SLARFG( N-I, A( I, I+1 ), A( I, MIN( I+2, N ) ), 260 $ LDA, TAUP( I ) ) 261 E( I ) = A( I, I+1 ) 262 A( I, I+1 ) = ONE 263* 264* Apply G(i) to A(i+1:m,i+1:n) from the right 265* 266 CALL SLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 267 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 268 A( I, I+1 ) = E( I ) 269 ELSE 270 TAUP( I ) = ZERO 271 END IF 272 10 CONTINUE 273 ELSE 274* 275* Reduce to lower bidiagonal form 276* 277 DO 20 I = 1, M 278* 279* Generate elementary reflector G(i) to annihilate A(i,i+1:n) 280* 281 CALL SLARFG( N-I+1, A( I, I ), A( I, MIN( I+1, N ) ), LDA, 282 $ TAUP( I ) ) 283 D( I ) = A( I, I ) 284 A( I, I ) = ONE 285* 286* Apply G(i) to A(i+1:m,i:n) from the right 287* 288 IF( I.LT.M ) 289 $ CALL SLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 290 $ TAUP( I ), A( I+1, I ), LDA, WORK ) 291 A( I, I ) = D( I ) 292* 293 IF( I.LT.M ) THEN 294* 295* Generate elementary reflector H(i) to annihilate 296* A(i+2:m,i) 297* 298 CALL SLARFG( M-I, A( I+1, I ), A( MIN( I+2, M ), I ), 1, 299 $ TAUQ( I ) ) 300 E( I ) = A( I+1, I ) 301 A( I+1, I ) = ONE 302* 303* Apply H(i) to A(i+1:m,i+1:n) from the left 304* 305 CALL SLARF( 'Left', M-I, N-I, A( I+1, I ), 1, TAUQ( I ), 306 $ A( I+1, I+1 ), LDA, WORK ) 307 A( I+1, I ) = E( I ) 308 ELSE 309 TAUQ( I ) = ZERO 310 END IF 311 20 CONTINUE 312 END IF 313 RETURN 314* 315* End of SGEBD2 316* 317 END 318