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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CGEBRD 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CGEBRD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cgebrd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cgebrd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cgebrd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CGEBRD( 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 * REAL D( * ), E( * ) 00029 * COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), 00030 * $ WORK( * ) 00031 * .. 00032 * 00033 * 00034 *> \par Purpose: 00035 * ============= 00036 *> 00037 *> \verbatim 00038 *> 00039 *> CGEBRD reduces a general complex M-by-N matrix A to upper or lower 00040 *> bidiagonal form B by a unitary transformation: Q**H * A * P = B. 00041 *> 00042 *> If m >= n, B is upper bidiagonal; if m < n, B is lower bidiagonal. 00043 *> \endverbatim 00044 * 00045 * Arguments: 00046 * ========== 00047 * 00048 *> \param[in] M 00049 *> \verbatim 00050 *> M is INTEGER 00051 *> The number of rows in the matrix A. M >= 0. 00052 *> \endverbatim 00053 *> 00054 *> \param[in] N 00055 *> \verbatim 00056 *> N is INTEGER 00057 *> The number of columns in the matrix A. N >= 0. 00058 *> \endverbatim 00059 *> 00060 *> \param[in,out] A 00061 *> \verbatim 00062 *> A is COMPLEX array, dimension (LDA,N) 00063 *> On entry, the M-by-N general matrix to be reduced. 00064 *> On exit, 00065 *> if m >= n, the diagonal and the first superdiagonal are 00066 *> overwritten with the upper bidiagonal matrix B; the 00067 *> elements below the diagonal, with the array TAUQ, represent 00068 *> the unitary matrix Q as a product of elementary 00069 *> reflectors, and the elements above the first superdiagonal, 00070 *> with the array TAUP, represent the unitary matrix P as 00071 *> a product of elementary reflectors; 00072 *> if m < n, the diagonal and the first subdiagonal are 00073 *> overwritten with the lower bidiagonal matrix B; the 00074 *> elements below the first subdiagonal, with the array TAUQ, 00075 *> represent the unitary matrix Q as a product of 00076 *> elementary reflectors, and the elements above the diagonal, 00077 *> with the array TAUP, represent the unitary matrix P as 00078 *> a product of elementary reflectors. 00079 *> See Further Details. 00080 *> \endverbatim 00081 *> 00082 *> \param[in] LDA 00083 *> \verbatim 00084 *> LDA is INTEGER 00085 *> The leading dimension of the array A. LDA >= max(1,M). 00086 *> \endverbatim 00087 *> 00088 *> \param[out] D 00089 *> \verbatim 00090 *> D is REAL array, dimension (min(M,N)) 00091 *> The diagonal elements of the bidiagonal matrix B: 00092 *> D(i) = A(i,i). 00093 *> \endverbatim 00094 *> 00095 *> \param[out] E 00096 *> \verbatim 00097 *> E is REAL array, dimension (min(M,N)-1) 00098 *> The off-diagonal elements of the bidiagonal matrix B: 00099 *> if m >= n, E(i) = A(i,i+1) for i = 1,2,...,n-1; 00100 *> if m < n, E(i) = A(i+1,i) for i = 1,2,...,m-1. 00101 *> \endverbatim 00102 *> 00103 *> \param[out] TAUQ 00104 *> \verbatim 00105 *> TAUQ is COMPLEX array dimension (min(M,N)) 00106 *> The scalar factors of the elementary reflectors which 00107 *> represent the unitary matrix Q. See Further Details. 00108 *> \endverbatim 00109 *> 00110 *> \param[out] TAUP 00111 *> \verbatim 00112 *> TAUP is COMPLEX array, dimension (min(M,N)) 00113 *> The scalar factors of the elementary reflectors which 00114 *> represent the unitary matrix P. See Further Details. 00115 *> \endverbatim 00116 *> 00117 *> \param[out] WORK 00118 *> \verbatim 00119 *> WORK is COMPLEX array, dimension (MAX(1,LWORK)) 00120 *> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00121 *> \endverbatim 00122 *> 00123 *> \param[in] LWORK 00124 *> \verbatim 00125 *> LWORK is INTEGER 00126 *> The length of the array WORK. LWORK >= max(1,M,N). 00127 *> For optimum performance LWORK >= (M+N)*NB, where NB 00128 *> is the optimal blocksize. 00129 *> 00130 *> If LWORK = -1, then a workspace query is assumed; the routine 00131 *> only calculates the optimal size of the WORK array, returns 00132 *> this value as the first entry of the WORK array, and no error 00133 *> message related to LWORK is issued by XERBLA. 00134 *> \endverbatim 00135 *> 00136 *> \param[out] INFO 00137 *> \verbatim 00138 *> INFO is INTEGER 00139 *> = 0: successful exit. 00140 *> < 0: if INFO = -i, the i-th argument had an illegal value. 00141 *> \endverbatim 00142 * 00143 * Authors: 00144 * ======== 00145 * 00146 *> \author Univ. of Tennessee 00147 *> \author Univ. of California Berkeley 00148 *> \author Univ. of Colorado Denver 00149 *> \author NAG Ltd. 00150 * 00151 *> \date November 2011 00152 * 00153 *> \ingroup complexGEcomputational 00154 * 00155 *> \par Further Details: 00156 * ===================== 00157 *> 00158 *> \verbatim 00159 *> 00160 *> The matrices Q and P are represented as products of elementary 00161 *> reflectors: 00162 *> 00163 *> If m >= n, 00164 *> 00165 *> Q = H(1) H(2) . . . H(n) and P = G(1) G(2) . . . G(n-1) 00166 *> 00167 *> Each H(i) and G(i) has the form: 00168 *> 00169 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 00170 *> 00171 *> where tauq and taup are complex scalars, and v and u are complex 00172 *> vectors; v(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in 00173 *> A(i+1:m,i); u(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in 00174 *> A(i,i+2:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00175 *> 00176 *> If m < n, 00177 *> 00178 *> Q = H(1) H(2) . . . H(m-1) and P = G(1) G(2) . . . G(m) 00179 *> 00180 *> Each H(i) and G(i) has the form: 00181 *> 00182 *> H(i) = I - tauq * v * v**H and G(i) = I - taup * u * u**H 00183 *> 00184 *> where tauq and taup are complex scalars, and v and u are complex 00185 *> vectors; v(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in 00186 *> A(i+2:m,i); u(1:i-1) = 0, u(i) = 1, and u(i+1:n) is stored on exit in 00187 *> A(i,i+1:n); tauq is stored in TAUQ(i) and taup in TAUP(i). 00188 *> 00189 *> The contents of A on exit are illustrated by the following examples: 00190 *> 00191 *> m = 6 and n = 5 (m > n): m = 5 and n = 6 (m < n): 00192 *> 00193 *> ( d e u1 u1 u1 ) ( d u1 u1 u1 u1 u1 ) 00194 *> ( v1 d e u2 u2 ) ( e d u2 u2 u2 u2 ) 00195 *> ( v1 v2 d e u3 ) ( v1 e d u3 u3 u3 ) 00196 *> ( v1 v2 v3 d e ) ( v1 v2 e d u4 u4 ) 00197 *> ( v1 v2 v3 v4 d ) ( v1 v2 v3 e d u5 ) 00198 *> ( v1 v2 v3 v4 v5 ) 00199 *> 00200 *> where d and e denote diagonal and off-diagonal elements of B, vi 00201 *> denotes an element of the vector defining H(i), and ui an element of 00202 *> the vector defining G(i). 00203 *> \endverbatim 00204 *> 00205 * ===================================================================== 00206 SUBROUTINE CGEBRD( M, N, A, LDA, D, E, TAUQ, TAUP, WORK, LWORK, 00207 $ INFO ) 00208 * 00209 * -- LAPACK computational routine (version 3.4.0) -- 00210 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00211 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00212 * November 2011 00213 * 00214 * .. Scalar Arguments .. 00215 INTEGER INFO, LDA, LWORK, M, N 00216 * .. 00217 * .. Array Arguments .. 00218 REAL D( * ), E( * ) 00219 COMPLEX A( LDA, * ), TAUP( * ), TAUQ( * ), 00220 $ WORK( * ) 00221 * .. 00222 * 00223 * ===================================================================== 00224 * 00225 * .. Parameters .. 00226 COMPLEX ONE 00227 PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ) ) 00228 * .. 00229 * .. Local Scalars .. 00230 LOGICAL LQUERY 00231 INTEGER I, IINFO, J, LDWRKX, LDWRKY, LWKOPT, MINMN, NB, 00232 $ NBMIN, NX 00233 REAL WS 00234 * .. 00235 * .. External Subroutines .. 00236 EXTERNAL CGEBD2, CGEMM, CLABRD, XERBLA 00237 * .. 00238 * .. Intrinsic Functions .. 00239 INTRINSIC MAX, MIN, REAL 00240 * .. 00241 * .. External Functions .. 00242 INTEGER ILAENV 00243 EXTERNAL ILAENV 00244 * .. 00245 * .. Executable Statements .. 00246 * 00247 * Test the input parameters 00248 * 00249 INFO = 0 00250 NB = MAX( 1, ILAENV( 1, 'CGEBRD', ' ', M, N, -1, -1 ) ) 00251 LWKOPT = ( M+N )*NB 00252 WORK( 1 ) = REAL( LWKOPT ) 00253 LQUERY = ( LWORK.EQ.-1 ) 00254 IF( M.LT.0 ) THEN 00255 INFO = -1 00256 ELSE IF( N.LT.0 ) THEN 00257 INFO = -2 00258 ELSE IF( LDA.LT.MAX( 1, M ) ) THEN 00259 INFO = -4 00260 ELSE IF( LWORK.LT.MAX( 1, M, N ) .AND. .NOT.LQUERY ) THEN 00261 INFO = -10 00262 END IF 00263 IF( INFO.LT.0 ) THEN 00264 CALL XERBLA( 'CGEBRD', -INFO ) 00265 RETURN 00266 ELSE IF( LQUERY ) THEN 00267 RETURN 00268 END IF 00269 * 00270 * Quick return if possible 00271 * 00272 MINMN = MIN( M, N ) 00273 IF( MINMN.EQ.0 ) THEN 00274 WORK( 1 ) = 1 00275 RETURN 00276 END IF 00277 * 00278 WS = MAX( M, N ) 00279 LDWRKX = M 00280 LDWRKY = N 00281 * 00282 IF( NB.GT.1 .AND. NB.LT.MINMN ) THEN 00283 * 00284 * Set the crossover point NX. 00285 * 00286 NX = MAX( NB, ILAENV( 3, 'CGEBRD', ' ', M, N, -1, -1 ) ) 00287 * 00288 * Determine when to switch from blocked to unblocked code. 00289 * 00290 IF( NX.LT.MINMN ) THEN 00291 WS = ( M+N )*NB 00292 IF( LWORK.LT.WS ) THEN 00293 * 00294 * Not enough work space for the optimal NB, consider using 00295 * a smaller block size. 00296 * 00297 NBMIN = ILAENV( 2, 'CGEBRD', ' ', M, N, -1, -1 ) 00298 IF( LWORK.GE.( M+N )*NBMIN ) THEN 00299 NB = LWORK / ( M+N ) 00300 ELSE 00301 NB = 1 00302 NX = MINMN 00303 END IF 00304 END IF 00305 END IF 00306 ELSE 00307 NX = MINMN 00308 END IF 00309 * 00310 DO 30 I = 1, MINMN - NX, NB 00311 * 00312 * Reduce rows and columns i:i+ib-1 to bidiagonal form and return 00313 * the matrices X and Y which are needed to update the unreduced 00314 * part of the matrix 00315 * 00316 CALL CLABRD( M-I+1, N-I+1, NB, A( I, I ), LDA, D( I ), E( I ), 00317 $ TAUQ( I ), TAUP( I ), WORK, LDWRKX, 00318 $ WORK( LDWRKX*NB+1 ), LDWRKY ) 00319 * 00320 * Update the trailing submatrix A(i+ib:m,i+ib:n), using 00321 * an update of the form A := A - V*Y**H - X*U**H 00322 * 00323 CALL CGEMM( 'No transpose', 'Conjugate transpose', M-I-NB+1, 00324 $ N-I-NB+1, NB, -ONE, A( I+NB, I ), LDA, 00325 $ WORK( LDWRKX*NB+NB+1 ), LDWRKY, ONE, 00326 $ A( I+NB, I+NB ), LDA ) 00327 CALL CGEMM( 'No transpose', 'No transpose', M-I-NB+1, N-I-NB+1, 00328 $ NB, -ONE, WORK( NB+1 ), LDWRKX, A( I, I+NB ), LDA, 00329 $ ONE, A( I+NB, I+NB ), LDA ) 00330 * 00331 * Copy diagonal and off-diagonal elements of B back into A 00332 * 00333 IF( M.GE.N ) THEN 00334 DO 10 J = I, I + NB - 1 00335 A( J, J ) = D( J ) 00336 A( J, J+1 ) = E( J ) 00337 10 CONTINUE 00338 ELSE 00339 DO 20 J = I, I + NB - 1 00340 A( J, J ) = D( J ) 00341 A( J+1, J ) = E( J ) 00342 20 CONTINUE 00343 END IF 00344 30 CONTINUE 00345 * 00346 * Use unblocked code to reduce the remainder of the matrix 00347 * 00348 CALL CGEBD2( M-I+1, N-I+1, A( I, I ), LDA, D( I ), E( I ), 00349 $ TAUQ( I ), TAUP( I ), WORK, IINFO ) 00350 WORK( 1 ) = WS 00351 RETURN 00352 * 00353 * End of CGEBRD 00354 * 00355 END