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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CLATRD 00002 * 00003 * =========== DOCUMENTATION =========== 00004 * 00005 * Online html documentation available at 00006 * http://www.netlib.org/lapack/explore-html/ 00007 * 00008 *> \htmlonly 00009 *> Download CLATRD + dependencies 00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clatrd.f"> 00011 *> [TGZ]</a> 00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clatrd.f"> 00013 *> [ZIP]</a> 00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clatrd.f"> 00015 *> [TXT]</a> 00016 *> \endhtmlonly 00017 * 00018 * Definition: 00019 * =========== 00020 * 00021 * SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) 00022 * 00023 * .. Scalar Arguments .. 00024 * CHARACTER UPLO 00025 * INTEGER LDA, LDW, N, NB 00026 * .. 00027 * .. Array Arguments .. 00028 * REAL E( * ) 00029 * COMPLEX A( LDA, * ), TAU( * ), W( LDW, * ) 00030 * .. 00031 * 00032 * 00033 *> \par Purpose: 00034 * ============= 00035 *> 00036 *> \verbatim 00037 *> 00038 *> CLATRD reduces NB rows and columns of a complex Hermitian matrix A to 00039 *> Hermitian tridiagonal form by a unitary similarity 00040 *> transformation Q**H * A * Q, and returns the matrices V and W which are 00041 *> needed to apply the transformation to the unreduced part of A. 00042 *> 00043 *> If UPLO = 'U', CLATRD reduces the last NB rows and columns of a 00044 *> matrix, of which the upper triangle is supplied; 00045 *> if UPLO = 'L', CLATRD reduces the first NB rows and columns of a 00046 *> matrix, of which the lower triangle is supplied. 00047 *> 00048 *> This is an auxiliary routine called by CHETRD. 00049 *> \endverbatim 00050 * 00051 * Arguments: 00052 * ========== 00053 * 00054 *> \param[in] UPLO 00055 *> \verbatim 00056 *> UPLO is CHARACTER*1 00057 *> Specifies whether the upper or lower triangular part of the 00058 *> Hermitian matrix A is stored: 00059 *> = 'U': Upper triangular 00060 *> = 'L': Lower triangular 00061 *> \endverbatim 00062 *> 00063 *> \param[in] N 00064 *> \verbatim 00065 *> N is INTEGER 00066 *> The order of the matrix A. 00067 *> \endverbatim 00068 *> 00069 *> \param[in] NB 00070 *> \verbatim 00071 *> NB is INTEGER 00072 *> The number of rows and columns to be reduced. 00073 *> \endverbatim 00074 *> 00075 *> \param[in,out] A 00076 *> \verbatim 00077 *> A is COMPLEX array, dimension (LDA,N) 00078 *> On entry, the Hermitian matrix A. If UPLO = 'U', the leading 00079 *> n-by-n upper triangular part of A contains the upper 00080 *> triangular part of the matrix A, and the strictly lower 00081 *> triangular part of A is not referenced. If UPLO = 'L', the 00082 *> leading n-by-n lower triangular part of A contains the lower 00083 *> triangular part of the matrix A, and the strictly upper 00084 *> triangular part of A is not referenced. 00085 *> On exit: 00086 *> if UPLO = 'U', the last NB columns have been reduced to 00087 *> tridiagonal form, with the diagonal elements overwriting 00088 *> the diagonal elements of A; the elements above the diagonal 00089 *> with the array TAU, represent the unitary matrix Q as a 00090 *> product of elementary reflectors; 00091 *> if UPLO = 'L', the first NB columns have been reduced to 00092 *> tridiagonal form, with the diagonal elements overwriting 00093 *> the diagonal elements of A; the elements below the diagonal 00094 *> with the array TAU, represent the unitary matrix Q as a 00095 *> product of elementary reflectors. 00096 *> See Further Details. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] LDA 00100 *> \verbatim 00101 *> LDA is INTEGER 00102 *> The leading dimension of the array A. LDA >= max(1,N). 00103 *> \endverbatim 00104 *> 00105 *> \param[out] E 00106 *> \verbatim 00107 *> E is REAL array, dimension (N-1) 00108 *> If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal 00109 *> elements of the last NB columns of the reduced matrix; 00110 *> if UPLO = 'L', E(1:nb) contains the subdiagonal elements of 00111 *> the first NB columns of the reduced matrix. 00112 *> \endverbatim 00113 *> 00114 *> \param[out] TAU 00115 *> \verbatim 00116 *> TAU is COMPLEX array, dimension (N-1) 00117 *> The scalar factors of the elementary reflectors, stored in 00118 *> TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. 00119 *> See Further Details. 00120 *> \endverbatim 00121 *> 00122 *> \param[out] W 00123 *> \verbatim 00124 *> W is COMPLEX array, dimension (LDW,NB) 00125 *> The n-by-nb matrix W required to update the unreduced part 00126 *> of A. 00127 *> \endverbatim 00128 *> 00129 *> \param[in] LDW 00130 *> \verbatim 00131 *> LDW is INTEGER 00132 *> The leading dimension of the array W. LDW >= max(1,N). 00133 *> \endverbatim 00134 * 00135 * Authors: 00136 * ======== 00137 * 00138 *> \author Univ. of Tennessee 00139 *> \author Univ. of California Berkeley 00140 *> \author Univ. of Colorado Denver 00141 *> \author NAG Ltd. 00142 * 00143 *> \date November 2011 00144 * 00145 *> \ingroup complexOTHERauxiliary 00146 * 00147 *> \par Further Details: 00148 * ===================== 00149 *> 00150 *> \verbatim 00151 *> 00152 *> If UPLO = 'U', the matrix Q is represented as a product of elementary 00153 *> reflectors 00154 *> 00155 *> Q = H(n) H(n-1) . . . H(n-nb+1). 00156 *> 00157 *> Each H(i) has the form 00158 *> 00159 *> H(i) = I - tau * v * v**H 00160 *> 00161 *> where tau is a complex scalar, and v is a complex vector with 00162 *> v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), 00163 *> and tau in TAU(i-1). 00164 *> 00165 *> If UPLO = 'L', the matrix Q is represented as a product of elementary 00166 *> reflectors 00167 *> 00168 *> Q = H(1) H(2) . . . H(nb). 00169 *> 00170 *> Each H(i) has the form 00171 *> 00172 *> H(i) = I - tau * v * v**H 00173 *> 00174 *> where tau is a complex scalar, and v is a complex vector with 00175 *> v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), 00176 *> and tau in TAU(i). 00177 *> 00178 *> The elements of the vectors v together form the n-by-nb matrix V 00179 *> which is needed, with W, to apply the transformation to the unreduced 00180 *> part of the matrix, using a Hermitian rank-2k update of the form: 00181 *> A := A - V*W**H - W*V**H. 00182 *> 00183 *> The contents of A on exit are illustrated by the following examples 00184 *> with n = 5 and nb = 2: 00185 *> 00186 *> if UPLO = 'U': if UPLO = 'L': 00187 *> 00188 *> ( a a a v4 v5 ) ( d ) 00189 *> ( a a v4 v5 ) ( 1 d ) 00190 *> ( a 1 v5 ) ( v1 1 a ) 00191 *> ( d 1 ) ( v1 v2 a a ) 00192 *> ( d ) ( v1 v2 a a a ) 00193 *> 00194 *> where d denotes a diagonal element of the reduced matrix, a denotes 00195 *> an element of the original matrix that is unchanged, and vi denotes 00196 *> an element of the vector defining H(i). 00197 *> \endverbatim 00198 *> 00199 * ===================================================================== 00200 SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) 00201 * 00202 * -- LAPACK auxiliary routine (version 3.4.0) -- 00203 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00204 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00205 * November 2011 00206 * 00207 * .. Scalar Arguments .. 00208 CHARACTER UPLO 00209 INTEGER LDA, LDW, N, NB 00210 * .. 00211 * .. Array Arguments .. 00212 REAL E( * ) 00213 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * ) 00214 * .. 00215 * 00216 * ===================================================================== 00217 * 00218 * .. Parameters .. 00219 COMPLEX ZERO, ONE, HALF 00220 PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), 00221 $ ONE = ( 1.0E+0, 0.0E+0 ), 00222 $ HALF = ( 0.5E+0, 0.0E+0 ) ) 00223 * .. 00224 * .. Local Scalars .. 00225 INTEGER I, IW 00226 COMPLEX ALPHA 00227 * .. 00228 * .. External Subroutines .. 00229 EXTERNAL CAXPY, CGEMV, CHEMV, CLACGV, CLARFG, CSCAL 00230 * .. 00231 * .. External Functions .. 00232 LOGICAL LSAME 00233 COMPLEX CDOTC 00234 EXTERNAL LSAME, CDOTC 00235 * .. 00236 * .. Intrinsic Functions .. 00237 INTRINSIC MIN, REAL 00238 * .. 00239 * .. Executable Statements .. 00240 * 00241 * Quick return if possible 00242 * 00243 IF( N.LE.0 ) 00244 $ RETURN 00245 * 00246 IF( LSAME( UPLO, 'U' ) ) THEN 00247 * 00248 * Reduce last NB columns of upper triangle 00249 * 00250 DO 10 I = N, N - NB + 1, -1 00251 IW = I - N + NB 00252 IF( I.LT.N ) THEN 00253 * 00254 * Update A(1:i,i) 00255 * 00256 A( I, I ) = REAL( A( I, I ) ) 00257 CALL CLACGV( N-I, W( I, IW+1 ), LDW ) 00258 CALL CGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), 00259 $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) 00260 CALL CLACGV( N-I, W( I, IW+1 ), LDW ) 00261 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 00262 CALL CGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), 00263 $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) 00264 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 00265 A( I, I ) = REAL( A( I, I ) ) 00266 END IF 00267 IF( I.GT.1 ) THEN 00268 * 00269 * Generate elementary reflector H(i) to annihilate 00270 * A(1:i-2,i) 00271 * 00272 ALPHA = A( I-1, I ) 00273 CALL CLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) ) 00274 E( I-1 ) = ALPHA 00275 A( I-1, I ) = ONE 00276 * 00277 * Compute W(1:i-1,i) 00278 * 00279 CALL CHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, 00280 $ ZERO, W( 1, IW ), 1 ) 00281 IF( I.LT.N ) THEN 00282 CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, 00283 $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO, 00284 $ W( I+1, IW ), 1 ) 00285 CALL CGEMV( 'No transpose', I-1, N-I, -ONE, 00286 $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, 00287 $ W( 1, IW ), 1 ) 00288 CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, 00289 $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO, 00290 $ W( I+1, IW ), 1 ) 00291 CALL CGEMV( 'No transpose', I-1, N-I, -ONE, 00292 $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, 00293 $ W( 1, IW ), 1 ) 00294 END IF 00295 CALL CSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) 00296 ALPHA = -HALF*TAU( I-1 )*CDOTC( I-1, W( 1, IW ), 1, 00297 $ A( 1, I ), 1 ) 00298 CALL CAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) 00299 END IF 00300 * 00301 10 CONTINUE 00302 ELSE 00303 * 00304 * Reduce first NB columns of lower triangle 00305 * 00306 DO 20 I = 1, NB 00307 * 00308 * Update A(i:n,i) 00309 * 00310 A( I, I ) = REAL( A( I, I ) ) 00311 CALL CLACGV( I-1, W( I, 1 ), LDW ) 00312 CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), 00313 $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) 00314 CALL CLACGV( I-1, W( I, 1 ), LDW ) 00315 CALL CLACGV( I-1, A( I, 1 ), LDA ) 00316 CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), 00317 $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) 00318 CALL CLACGV( I-1, A( I, 1 ), LDA ) 00319 A( I, I ) = REAL( A( I, I ) ) 00320 IF( I.LT.N ) THEN 00321 * 00322 * Generate elementary reflector H(i) to annihilate 00323 * A(i+2:n,i) 00324 * 00325 ALPHA = A( I+1, I ) 00326 CALL CLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, 00327 $ TAU( I ) ) 00328 E( I ) = ALPHA 00329 A( I+1, I ) = ONE 00330 * 00331 * Compute W(i+1:n,i) 00332 * 00333 CALL CHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, 00334 $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) 00335 CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, 00336 $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO, 00337 $ W( 1, I ), 1 ) 00338 CALL CGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), 00339 $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 00340 CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, 00341 $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, 00342 $ W( 1, I ), 1 ) 00343 CALL CGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), 00344 $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 00345 CALL CSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) 00346 ALPHA = -HALF*TAU( I )*CDOTC( N-I, W( I+1, I ), 1, 00347 $ A( I+1, I ), 1 ) 00348 CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) 00349 END IF 00350 * 00351 20 CONTINUE 00352 END IF 00353 * 00354 RETURN 00355 * 00356 * End of CLATRD 00357 * 00358 END