1*> \brief \b CGEBD2 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 CGEBD2 + dependencies 10*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgebd2.f"> 11*> [TGZ]</a> 12*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgebd2.f"> 13*> [ZIP]</a> 14*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgebd2.f"> 15*> [TXT]</a> 16*> \endhtmlonly 17* 18* Definition: 19* =========== 20* 21* SUBROUTINE CGEBD2( 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 D( * ), E( * ) 28* COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) 29* .. 30* 31* 32*> \par Purpose: 33* ============= 34*> 35*> \verbatim 36*> 37*> CGEBD2 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 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 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 COMPLEX 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 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 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 complexGEcomputational 136* @precisions normal c -> s d z 137* 138*> \par Further Details: 139* ===================== 140*> 141*> \verbatim 142*> 143*> The matrices Q and P are represented as products of elementary 144*> reflectors: 145*> 146*> If m >= n, 147*> 148*> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 149*> 150*> Each H(i) and G(i) has the form: 151*> 152*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 153*> 154*> where tauq and taup are complex scalars, and v and u are complex 155*> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in 156*> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in 157*> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 158*> 159*> If m < n, 160*> 161*> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 162*> 163*> Each H(i) and G(i) has the form: 164*> 165*> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 166*> 167*> where tauq and taup are complex scalars, v and u are complex vectors; 168*> v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(i+2:m,i); 169*> u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in A(i,i+1:n); 170*> tauq is stored in TAUQ(i) and taup in TAUP(i). 171*> 172*> The contents of A on exit are illustrated by the following examples: 173*> 174*> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 175*> 176*> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 177*> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 178*> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 179*> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 180*> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 181*> ( v1 v2 v3 v4 v5 ) 182*> 183*> where d and e denote diagonal and off-diagonal elements of B, vi 184*> denotes an element of the vector defining H(i), and ui an element of 185*> the vector defining G(i). 186*> \endverbatim 187*> 188* ===================================================================== 189 SUBROUTINE CGEBD2( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, INFO ) 190* 191* -- LAPACK computational routine -- 192* -- LAPACK is a software package provided by Univ. of Tennessee, -- 193* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 194* 195* .. Scalar Arguments .. 196 INTEGER INFO, LDA, M, N 197* .. 198* .. Array Arguments .. 199 REAL D( * ), E( * ) 200 COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) 201* .. 202* 203* ===================================================================== 204* 205* .. Parameters .. 206 COMPLEX ZERO, ONE 207 PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), 208 $ ONE = ( 1.0E+0, 0.0E+0 ) ) 209* .. 210* .. Local Scalars .. 211 INTEGER I 212 COMPLEX ALPHA 213* .. 214* .. External Subroutines .. 215 EXTERNAL CLACGV, CLARF, CLARFG, XERBLA 216* .. 217* .. Intrinsic Functions .. 218 INTRINSIC CONJG, MAX, MIN 219* .. 220* .. Executable Statements .. 221* 222* Test the input parameters 223* 224 INFO = 0 225 IF( M.LT.0 ) THEN 226 INFO = -1 227 ELSE IF( N.LT.0 ) THEN 228 INFO = -2 229 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 230 INFO = -4 231 END IF 232 IF( INFO.LT.0 ) THEN 233 CALL XERBLA( 'CGEBD2', -INFO ) 234 RETURN 235 END IF 236* 237 IF( M.GE.N ) THEN 238* 239* Reduce to upper bidiagonal form 240* 241 DO 10 I = 1, N 242* 243* Generate elementary reflector H(i) to annihilate A(i+1:m,i) 244* 245 ALPHA = A( I, I ) 246 CALL CLARFG( M-I+1, ALPHA, A( MIN( I+1, M ), I ), 1, 247 $ TAUQ( I ) ) 248 D( I ) = REAL( ALPHA ) 249 A( I, I ) = ONE 250* 251* Apply H(i)**H to A(i:m,i+1:n) from the left 252* 253 IF( I.LT.N ) 254 $ CALL CLARF( 'Left', M-I+1, N-I, A( I, I ), 1, 255 $ CONJG( TAUQ( I ) ), A( I, I+1 ), LDA, WORK ) 256 A( I, I ) = D( I ) 257* 258 IF( I.LT.N ) THEN 259* 260* Generate elementary reflector G(i) to annihilate 261* A(i,i+2:n) 262* 263 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 264 ALPHA = A( I, I+1 ) 265 CALL CLARFG( N-I, ALPHA, A( I, MIN( I+2, N ) ), 266 $ LDA, TAUP( I ) ) 267 E( I ) = REAL( ALPHA ) 268 A( I, I+1 ) = ONE 269* 270* Apply G(i) to A(i+1:m,i+1:n) from the right 271* 272 CALL CLARF( 'Right', M-I, N-I, A( I, I+1 ), LDA, 273 $ TAUP( I ), A( I+1, I+1 ), LDA, WORK ) 274 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 275 A( I, I+1 ) = E( I ) 276 ELSE 277 TAUP( I ) = ZERO 278 END IF 279 10 CONTINUE 280 ELSE 281* 282* Reduce to lower bidiagonal form 283* 284 DO 20 I = 1, M 285* 286* Generate elementary reflector G(i) to annihilate A(i,i+1:n) 287* 288 CALL CLACGV( N-I+1, A( I, I ), LDA ) 289 ALPHA = A( I, I ) 290 CALL CLARFG( N-I+1, ALPHA, A( I, MIN( I+1, N ) ), LDA, 291 $ TAUP( I ) ) 292 D( I ) = REAL( ALPHA ) 293 A( I, I ) = ONE 294* 295* Apply G(i) to A(i+1:m,i:n) from the right 296* 297 IF( I.LT.M ) 298 $ CALL CLARF( 'Right', M-I, N-I+1, A( I, I ), LDA, 299 $ TAUP( I ), A( I+1, I ), LDA, WORK ) 300 CALL CLACGV( N-I+1, A( I, I ), LDA ) 301 A( I, I ) = D( I ) 302* 303 IF( I.LT.M ) THEN 304* 305* Generate elementary reflector H(i) to annihilate 306* A(i+2:m,i) 307* 308 ALPHA = A( I+1, I ) 309 CALL CLARFG( M-I, ALPHA, A( MIN( I+2, M ), I ), 1, 310 $ TAUQ( I ) ) 311 E( I ) = REAL( ALPHA ) 312 A( I+1, I ) = ONE 313* 314* Apply H(i)**H to A(i+1:m,i+1:n) from the left 315* 316 CALL CLARF( 'Left', M-I, N-I, A( I+1, I ), 1, 317 $ CONJG( TAUQ( I ) ), A( I+1, I+1 ), LDA, 318 $ WORK ) 319 A( I+1, I ) = E( I ) 320 ELSE 321 TAUQ( I ) = ZERO 322 END IF 323 20 CONTINUE 324 END IF 325 RETURN 326* 327* End of CGEBD2 328* 329 END 330