![]() |
LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
|
00001 *> \brief \b ZGEBRD 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download ZGEBRD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgebrd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgebrd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgebrd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 00022 * INFO ) 00023 * 00024 * .. Scalar Arguments .. 00025 * INTEGER INFO, LDA, LWORK, M, N 00026 * .. 00027 * .. Array Arguments .. 00028 * DOUBLE PRECISION D( * ), E( * ) 00029 * COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) 00030 * .. 00031 * 00032 * 00033 *> \par Purpose: 00034 * ============= 00035 *> 00036 *> \verbatim 00037 *> 00038 *> ZGEBRD reduces a general complex M-by-N matrix A to upper or lower 00039 *> bidiagonal form B by a unitary transformation: Q**H * A * P = B. 00040 *> 00041 *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 00042 *> \endverbatim 00043 * 00044 * Arguments: 00045 * ========== 00046 * 00047 *> \param[in] M 00048 *> \verbatim 00049 *> M is INTEGER 00050 *> The number of rows in the matrix A. M >= 0. 00051 *> \endverbatim 00052 *> 00053 *> \param[in] N 00054 *> \verbatim 00055 *> N is INTEGER 00056 *> The number of columns in the matrix A. N >= 0. 00057 *> \endverbatim 00058 *> 00059 *> \param[in,out] A 00060 *> \verbatim 00061 *> A is COMPLEX*16 array, dimension (LDA,N) 00062 *> On entry, the M-by-N general matrix to be reduced. 00063 *> On exit, 00064 *> if m >= n, the diagonal and the first superdiagonal are 00065 *> overwritten with the upper bidiagonal matrix B; the 00066 *> elements below the diagonal, with the array TAUQ, represent 00067 *> the unitary matrix Q as a product of elementary 00068 *> reflectors, and the elements above the first superdiagonal, 00069 *> with the array TAUP, represent the unitary matrix P as 00070 *> a product of elementary reflectors; 00071 *> if m < n, the diagonal and the first subdiagonal are 00072 *> overwritten with the lower bidiagonal matrix B; the 00073 *> elements below the first subdiagonal, with the array TAUQ, 00074 *> represent the unitary matrix Q as a product of 00075 *> elementary reflectors, and the elements above the diagonal, 00076 *> with the array TAUP, represent the unitary matrix P as 00077 *> a product of elementary reflectors. 00078 *> See Further Details. 00079 *> \endverbatim 00080 *> 00081 *> \param[in] LDA 00082 *> \verbatim 00083 *> LDA is INTEGER 00084 *> The leading dimension of the array A. LDA >= max(1,M). 00085 *> \endverbatim 00086 *> 00087 *> \param[out] D 00088 *> \verbatim 00089 *> D is DOUBLE PRECISION array, dimension (min(M,N)) 00090 *> The diagonal elements of the bidiagonal matrix B: 00091 *> D(i) = A(i,i). 00092 *> \endverbatim 00093 *> 00094 *> \param[out] E 00095 *> \verbatim 00096 *> E is DOUBLE PRECISION array, dimension (min(M,N)-1) 00097 *> The off-diagonal elements of the bidiagonal matrix B: 00098 *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 00099 *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 00100 *> \endverbatim 00101 *> 00102 *> \param[out] TAUQ 00103 *> \verbatim 00104 *> TAUQ is COMPLEX*16 array dimension (min(M,N)) 00105 *> The scalar factors of the elementary reflectors which 00106 *> represent the unitary matrix Q. See Further Details. 00107 *> \endverbatim 00108 *> 00109 *> \param[out] TAUP 00110 *> \verbatim 00111 *> TAUP is COMPLEX*16 array, dimension (min(M,N)) 00112 *> The scalar factors of the elementary reflectors which 00113 *> represent the unitary matrix P. See Further Details. 00114 *> \endverbatim 00115 *> 00116 *> \param[out] WORK 00117 *> \verbatim 00118 *> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) 00119 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00120 *> \endverbatim 00121 *> 00122 *> \param[in] LWORK 00123 *> \verbatim 00124 *> LWORK is INTEGER 00125 *> The length of the array WORK. LWORK >= max(1,M,N). 00126 *> For optimum performance LWORK >= (M+N)*NB, where NB 00127 *> is the optimal blocksize. 00128 *> 00129 *> If LWORK = -1, then a workspace query is assumed; the routine 00130 *> only calculates the optimal size of the WORK array, returns 00131 *> this value as the first entry of the WORK array, and no error 00132 *> message related to LWORK is issued by XERBLA. 00133 *> \endverbatim 00134 *> 00135 *> \param[out] INFO 00136 *> \verbatim 00137 *> INFO is INTEGER 00138 *> = 0: successful exit. 00139 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00140 *> \endverbatim 00141 * 00142 * Authors: 00143 * ======== 00144 * 00145 *> \author Univ. of Tennessee 00146 *> \author Univ. of California Berkeley 00147 *> \author Univ. of Colorado Denver 00148 *> \author NAG Ltd. 00149 * 00150 *> \date November 2011 00151 * 00152 *> \ingroup complex16GEcomputational 00153 * 00154 *> \par Further Details: 00155 * ===================== 00156 *> 00157 *> \verbatim 00158 *> 00159 *> The matrices Q and P are represented as products of elementary 00160 *> reflectors: 00161 *> 00162 *> If m >= n, 00163 *> 00164 *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 00165 *> 00166 *> Each H(i) and G(i) has the form: 00167 *> 00168 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 00169 *> 00170 *> where tauq and taup are complex scalars, and v and u are complex 00171 *> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in 00172 *> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in 00173 *> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00174 *> 00175 *> If m < n, 00176 *> 00177 *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 00178 *> 00179 *> Each H(i) and G(i) has the form: 00180 *> 00181 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 00182 *> 00183 *> where tauq and taup are complex scalars, and v and u are complex 00184 *> vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in 00185 *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in 00186 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00187 *> 00188 *> The contents of A on exit are illustrated by the following examples: 00189 *> 00190 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00191 *> 00192 *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 00193 *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 00194 *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 00195 *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 00196 *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 00197 *> ( v1 v2 v3 v4 v5 ) 00198 *> 00199 *> where d and e denote diagonal and off-diagonal elements of B, vi 00200 *> denotes an element of the vector defining H(i), and ui an element of 00201 *> the vector defining G(i). 00202 *> \endverbatim 00203 *> 00204 * ===================================================================== 00205 SUBROUTINE ZGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 00206 $ INFO ) 00207 * 00208 * -- LAPACK computational routine (version 3.4.0) -- 00209 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00210 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00211 * November 2011 00212 * 00213 * .. Scalar Arguments .. 00214 INTEGER INFO, LDA, LWORK, M, N 00215 * .. 00216 * .. Array Arguments .. 00217 DOUBLE PRECISION D( * ), E( * ) 00218 COMPLEX*16 A( LDA, * ), TAUP( * ), TAUQ( * ), WORK( * ) 00219 * .. 00220 * 00221 * ===================================================================== 00222 * 00223 * .. Parameters .. 00224 COMPLEX*16 ONE 00225 PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ) ) 00226 * .. 00227 * .. Local Scalars .. 00228 LOGICAL LQUERY 00229 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, 00230 $ NBMIN, NX 00231 DOUBLE PRECISION WS 00232 * .. 00233 * .. External Subroutines .. 00234 EXTERNAL XERBLA, ZGEBD2, ZGEMM, ZLABRD 00235 * .. 00236 * .. Intrinsic Functions .. 00237 INTRINSIC DBLE, MAX, MIN 00238 * .. 00239 * .. External Functions .. 00240 INTEGER ILAENV 00241 EXTERNAL ILAENV 00242 * .. 00243 * .. Executable Statements .. 00244 * 00245 * Test the input parameters 00246 * 00247 INFO = 0 00248 NB = MAX( 1, ILAENV( 1, 'ZGEBRD', ' ', M, N, -1, -1 ) ) 00249 LWKOPT = ( M+N )*NB 00250 WORK( 1 ) = DBLE( LWKOPT ) 00251 LQUERY = ( LWORK.EQ.-1 ) 00252 IF( M.LT.0 ) THEN 00253 INFO = -1 00254 ELSE IF( N.LT.0 ) THEN 00255 INFO = -2 00256 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00257 INFO = -4 00258 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN 00259 INFO = -10 00260 END IF 00261 IF( INFO.LT.0 ) THEN 00262 CALL XERBLA( 'ZGEBRD', -INFO ) 00263 RETURN 00264 ELSE IF( LQUERY ) THEN 00265 RETURN 00266 END IF 00267 * 00268 * Quick return if possible 00269 * 00270 MINMN = MIN( M, N ) 00271 IF( MINMN.EQ.0 ) THEN 00272 WORK( 1 ) = 1 00273 RETURN 00274 END IF 00275 * 00276 WS = MAX( M, N ) 00277 LDWRKX = M 00278 LDWRKY = N 00279 * 00280 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN 00281 * 00282 * Set the crossover point NX. 00283 * 00284 NX = MAX( NB, ILAENV( 3, 'ZGEBRD', ' ', M, N, -1, -1 ) ) 00285 * 00286 * Determine when to switch from blocked to unblocked code. 00287 * 00288 IF( NX.LT.MINMN ) THEN 00289 WS = ( M+N )*NB 00290 IF( LWORK.LT.WS ) THEN 00291 * 00292 * Not enough work space for the optimal NB, consider using 00293 * a smaller block size. 00294 * 00295 NBMIN = ILAENV( 2, 'ZGEBRD', ' ', M, N, -1, -1 ) 00296 IF( LWORK.GE.( M+N )*NBMIN ) THEN 00297 NB = LWORK / ( M+N ) 00298 ELSE 00299 NB = 1 00300 NX = MINMN 00301 END IF 00302 END IF 00303 END IF 00304 ELSE 00305 NX = MINMN 00306 END IF 00307 * 00308 DO 30 I = 1, MINMN - NX, NB 00309 * 00310 * Reduce rows and columns i:i+ib-1 to bidiagonal form and return 00311 * the matrices X and Y which are needed to update the unreduced 00312 * part of the matrix 00313 * 00314 CALL ZLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), 00315 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX, 00316 $ WORK( LDWRKX*NB+1 ), LDWRKY ) 00317 * 00318 * Update the trailing submatrix A(i+ib:m,i+ib:n), using 00319 * an update of the form A := A - V*Y**H - X*U**H 00320 * 00321 CALL ZGEMM( 'No transpose', 'Conjugate transpose', M-I-NB+1, 00322 $ N-I-NB+1, NB, -ONE, A( I+NB, I ), LDA, 00323 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, 00324 $ A( I+NB, I+NB ), LDA ) 00325 CALL ZGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, 00326 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, 00327 $ ONE, A( I+NB, I+NB ), LDA ) 00328 * 00329 * Copy diagonal and off-diagonal elements of B back into A 00330 * 00331 IF( M.GE.N ) THEN 00332 DO 10 J = I, I + NB - 1 00333 A( J, J ) = D( J ) 00334 A( J, J+1 ) = E( J ) 00335 10 CONTINUE 00336 ELSE 00337 DO 20 J = I, I + NB - 1 00338 A( J, J ) = D( J ) 00339 A( J+1, J ) = E( J ) 00340 20 CONTINUE 00341 END IF 00342 30 CONTINUE 00343 * 00344 * Use unblocked code to reduce the remainder of the matrix 00345 * 00346 CALL ZGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), 00347 $ TAUQ( I ), TAUP( I ), WORK, IINFO ) 00348 WORK( 1 ) = WS 00349 RETURN 00350 * 00351 * End of ZGEBRD 00352 * 00353 END