github.com/gonum/lapack@v0.0.0-20181123203213-e4cdc5a0bff9/internal/testdata/netlib/dlahr2.f (about) 1 *> \brief \b DLAHR2 reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A. 2 * 3 * =========== DOCUMENTATION =========== 4 * 5 * Online html documentation available at 6 * http://www.netlib.org/lapack/explore-html/ 7 * 8 *> \htmlonly 9 *> Download DLAHR2 + dependencies 10 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlahr2.f"> 11 *> [TGZ]</a> 12 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlahr2.f"> 13 *> [ZIP]</a> 14 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlahr2.f"> 15 *> [TXT]</a> 16 *> \endhtmlonly 17 * 18 * Definition: 19 * =========== 20 * 21 * SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) 22 * 23 * .. Scalar Arguments .. 24 * INTEGER K, LDA, LDT, LDY, N, NB 25 * .. 26 * .. Array Arguments .. 27 * DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), 28 * $ Y( LDY, NB ) 29 * .. 30 * 31 * 32 *> \par Purpose: 33 * ============= 34 *> 35 *> \verbatim 36 *> 37 *> DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) 38 *> matrix A so that elements below the k-th subdiagonal are zero. The 39 *> reduction is performed by an orthogonal similarity transformation 40 *> Q**T * A * Q. The routine returns the matrices V and T which determine 41 *> Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T. 42 *> 43 *> This is an auxiliary routine called by DGEHRD. 44 *> \endverbatim 45 * 46 * Arguments: 47 * ========== 48 * 49 *> \param[in] N 50 *> \verbatim 51 *> N is INTEGER 52 *> The order of the matrix A. 53 *> \endverbatim 54 *> 55 *> \param[in] K 56 *> \verbatim 57 *> K is INTEGER 58 *> The offset for the reduction. Elements below the k-th 59 *> subdiagonal in the first NB columns are reduced to zero. 60 *> K < N. 61 *> \endverbatim 62 *> 63 *> \param[in] NB 64 *> \verbatim 65 *> NB is INTEGER 66 *> The number of columns to be reduced. 67 *> \endverbatim 68 *> 69 *> \param[in,out] A 70 *> \verbatim 71 *> A is DOUBLE PRECISION array, dimension (LDA,N-K+1) 72 *> On entry, the n-by-(n-k+1) general matrix A. 73 *> On exit, the elements on and above the k-th subdiagonal in 74 *> the first NB columns are overwritten with the corresponding 75 *> elements of the reduced matrix; the elements below the k-th 76 *> subdiagonal, with the array TAU, represent the matrix Q as a 77 *> product of elementary reflectors. The other columns of A are 78 *> unchanged. See Further Details. 79 *> \endverbatim 80 *> 81 *> \param[in] LDA 82 *> \verbatim 83 *> LDA is INTEGER 84 *> The leading dimension of the array A. LDA >= max(1,N). 85 *> \endverbatim 86 *> 87 *> \param[out] TAU 88 *> \verbatim 89 *> TAU is DOUBLE PRECISION array, dimension (NB) 90 *> The scalar factors of the elementary reflectors. See Further 91 *> Details. 92 *> \endverbatim 93 *> 94 *> \param[out] T 95 *> \verbatim 96 *> T is DOUBLE PRECISION array, dimension (LDT,NB) 97 *> The upper triangular matrix T. 98 *> \endverbatim 99 *> 100 *> \param[in] LDT 101 *> \verbatim 102 *> LDT is INTEGER 103 *> The leading dimension of the array T. LDT >= NB. 104 *> \endverbatim 105 *> 106 *> \param[out] Y 107 *> \verbatim 108 *> Y is DOUBLE PRECISION array, dimension (LDY,NB) 109 *> The n-by-nb matrix Y. 110 *> \endverbatim 111 *> 112 *> \param[in] LDY 113 *> \verbatim 114 *> LDY is INTEGER 115 *> The leading dimension of the array Y. LDY >= N. 116 *> \endverbatim 117 * 118 * Authors: 119 * ======== 120 * 121 *> \author Univ. of Tennessee 122 *> \author Univ. of California Berkeley 123 *> \author Univ. of Colorado Denver 124 *> \author NAG Ltd. 125 * 126 *> \date September 2012 127 * 128 *> \ingroup doubleOTHERauxiliary 129 * 130 *> \par Further Details: 131 * ===================== 132 *> 133 *> \verbatim 134 *> 135 *> The matrix Q is represented as a product of nb elementary reflectors 136 *> 137 *> Q = H(1) H(2) . . . H(nb). 138 *> 139 *> Each H(i) has the form 140 *> 141 *> H(i) = I - tau * v * v**T 142 *> 143 *> where tau is a real scalar, and v is a real vector with 144 *> v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in 145 *> A(i+k+1:n,i), and tau in TAU(i). 146 *> 147 *> The elements of the vectors v together form the (n-k+1)-by-nb matrix 148 *> V which is needed, with T and Y, to apply the transformation to the 149 *> unreduced part of the matrix, using an update of the form: 150 *> A := (I - V*T*V**T) * (A - Y*V**T). 151 *> 152 *> The contents of A on exit are illustrated by the following example 153 *> with n = 7, k = 3 and nb = 2: 154 *> 155 *> ( a a a a a ) 156 *> ( a a a a a ) 157 *> ( a a a a a ) 158 *> ( h h a a a ) 159 *> ( v1 h a a a ) 160 *> ( v1 v2 a a a ) 161 *> ( v1 v2 a a a ) 162 *> 163 *> where a denotes an element of the original matrix A, h denotes a 164 *> modified element of the upper Hessenberg matrix H, and vi denotes an 165 *> element of the vector defining H(i). 166 *> 167 *> This subroutine is a slight modification of LAPACK-3.0's DLAHRD 168 *> incorporating improvements proposed by Quintana-Orti and Van de 169 *> Gejin. Note that the entries of A(1:K,2:NB) differ from those 170 *> returned by the original LAPACK-3.0's DLAHRD routine. (This 171 *> subroutine is not backward compatible with LAPACK-3.0's DLAHRD.) 172 *> \endverbatim 173 * 174 *> \par References: 175 * ================ 176 *> 177 *> Gregorio Quintana-Orti and Robert van de Geijn, "Improving the 178 *> performance of reduction to Hessenberg form," ACM Transactions on 179 *> Mathematical Software, 32(2):180-194, June 2006. 180 *> 181 * ===================================================================== 182 SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) 183 * 184 * -- LAPACK auxiliary routine (version 3.4.2) -- 185 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 186 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 187 * September 2012 188 * 189 * .. Scalar Arguments .. 190 INTEGER K, LDA, LDT, LDY, N, NB 191 * .. 192 * .. Array Arguments .. 193 DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), 194 $ Y( LDY, NB ) 195 * .. 196 * 197 * ===================================================================== 198 * 199 * .. Parameters .. 200 DOUBLE PRECISION ZERO, ONE 201 PARAMETER ( ZERO = 0.0D+0, 202 $ ONE = 1.0D+0 ) 203 * .. 204 * .. Local Scalars .. 205 INTEGER I 206 DOUBLE PRECISION EI 207 * .. 208 * .. External Subroutines .. 209 EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY, 210 $ DLARFG, DSCAL, DTRMM, DTRMV 211 * .. 212 * .. Intrinsic Functions .. 213 INTRINSIC MIN 214 * .. 215 * .. Executable Statements .. 216 * 217 * Quick return if possible 218 * 219 IF( N.LE.1 ) 220 $ RETURN 221 * 222 DO 10 I = 1, NB 223 IF( I.GT.1 ) THEN 224 * 225 * Update A(K+1:N,I) 226 * 227 * Update I-th column of A - Y * V**T 228 * 229 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY, 230 $ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) 231 * 232 * Apply I - V * T**T * V**T to this column (call it b) from the 233 * left, using the last column of T as workspace 234 * 235 * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) 236 * ( V2 ) ( b2 ) 237 * 238 * where V1 is unit lower triangular 239 * 240 * w := V1**T * b1 241 * 242 CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) 243 CALL DTRMV( 'Lower', 'Transpose', 'UNIT', 244 $ I-1, A( K+1, 1 ), 245 $ LDA, T( 1, NB ), 1 ) 246 * 247 * w := w + V2**T * b2 248 * 249 CALL DGEMV( 'Transpose', N-K-I+1, I-1, 250 $ ONE, A( K+I, 1 ), 251 $ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) 252 * 253 * w := T**T * w 254 * 255 CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT', 256 $ I-1, T, LDT, 257 $ T( 1, NB ), 1 ) 258 * 259 * b2 := b2 - V2*w 260 * 261 CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE, 262 $ A( K+I, 1 ), 263 $ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) 264 * 265 * b1 := b1 - V1*w 266 * 267 CALL DTRMV( 'Lower', 'NO TRANSPOSE', 268 $ 'UNIT', I-1, 269 $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) 270 CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) 271 * 272 A( K+I-1, I-1 ) = EI 273 END IF 274 * 275 * Generate the elementary reflector H(I) to annihilate 276 * A(K+I+1:N,I) 277 * 278 CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, 279 $ TAU( I ) ) 280 EI = A( K+I, I ) 281 A( K+I, I ) = ONE 282 * 283 * Compute Y(K+1:N,I) 284 * 285 CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1, 286 $ ONE, A( K+1, I+1 ), 287 $ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 ) 288 CALL DGEMV( 'Transpose', N-K-I+1, I-1, 289 $ ONE, A( K+I, 1 ), LDA, 290 $ A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) 291 CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, 292 $ Y( K+1, 1 ), LDY, 293 $ T( 1, I ), 1, ONE, Y( K+1, I ), 1 ) 294 CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 ) 295 * 296 * Compute T(1:I,I) 297 * 298 CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) 299 CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT', 300 $ I-1, T, LDT, 301 $ T( 1, I ), 1 ) 302 T( I, I ) = TAU( I ) 303 * 304 10 CONTINUE 305 A( K+NB, NB ) = EI 306 * 307 * Compute Y(1:K,1:NB) 308 * 309 CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY ) 310 CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE', 311 $ 'UNIT', K, NB, 312 $ ONE, A( K+1, 1 ), LDA, Y, LDY ) 313 IF( N.GT.K+NB ) 314 $ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K, 315 $ NB, N-K-NB, ONE, 316 $ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y, 317 $ LDY ) 318 CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE', 319 $ 'NON-UNIT', K, NB, 320 $ ONE, T, LDT, Y, LDY ) 321 * 322 RETURN 323 * 324 * End of DLAHR2 325 * 326 END