|
1
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 |
SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
* * -- LAPACK auxiliary routine (version 3.3.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2011 -- * * .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDW, N, NB * .. * .. Array Arguments .. REAL E( * ) COMPLEX A( LDA, * ), TAU( * ), W( LDW, * ) * .. * * Purpose * ======= * * CLATRD reduces NB rows and columns of a complex Hermitian matrix A to * Hermitian tridiagonal form by a unitary similarity * transformation Q**H * A * Q, and returns the matrices V and W which are * needed to apply the transformation to the unreduced part of A. * * If UPLO = 'U', CLATRD reduces the last NB rows and columns of a * matrix, of which the upper triangle is supplied; * if UPLO = 'L', CLATRD reduces the first NB rows and columns of a * matrix, of which the lower triangle is supplied. * * This is an auxiliary routine called by CHETRD. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * Specifies whether the upper or lower triangular part of the * Hermitian matrix A is stored: * = 'U': Upper triangular * = 'L': Lower triangular * * N (input) INTEGER * The order of the matrix A. * * NB (input) INTEGER * The number of rows and columns to be reduced. * * A (input/output) COMPLEX array, dimension (LDA,N) * On entry, the Hermitian matrix A. If UPLO = 'U', the leading * n-by-n upper triangular part of A contains the upper * triangular part of the matrix A, and the strictly lower * triangular part of A is not referenced. If UPLO = 'L', the * leading n-by-n lower triangular part of A contains the lower * triangular part of the matrix A, and the strictly upper * triangular part of A is not referenced. * On exit: * if UPLO = 'U', the last NB columns have been reduced to * tridiagonal form, with the diagonal elements overwriting * the diagonal elements of A; the elements above the diagonal * with the array TAU, represent the unitary matrix Q as a * product of elementary reflectors; * if UPLO = 'L', the first NB columns have been reduced to * tridiagonal form, with the diagonal elements overwriting * the diagonal elements of A; the elements below the diagonal * with the array TAU, represent the unitary matrix Q as a * product of elementary reflectors. * See Further Details. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * E (output) REAL array, dimension (N-1) * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal * elements of the last NB columns of the reduced matrix; * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of * the first NB columns of the reduced matrix. * * TAU (output) COMPLEX array, dimension (N-1) * The scalar factors of the elementary reflectors, stored in * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. * See Further Details. * * W (output) COMPLEX array, dimension (LDW,NB) * The n-by-nb matrix W required to update the unreduced part * of A. * * LDW (input) INTEGER * The leading dimension of the array W. LDW >= max(1,N). * * Further Details * =============== * * If UPLO = 'U', the matrix Q is represented as a product of elementary * reflectors * * Q = H(n) H(n-1) . . . H(n-nb+1). * * Each H(i) has the form * * H(i) = I - tau * v * v**H * * where tau is a complex scalar, and v is a complex vector with * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), * and tau in TAU(i-1). * * If UPLO = 'L', the matrix Q is represented as a product of elementary * reflectors * * Q = H(1) H(2) . . . H(nb). * * Each H(i) has the form * * H(i) = I - tau * v * v**H * * where tau is a complex scalar, and v is a complex vector with * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), * and tau in TAU(i). * * The elements of the vectors v together form the n-by-nb matrix V * which is needed, with W, to apply the transformation to the unreduced * part of the matrix, using a Hermitian rank-2k update of the form: * A := A - V*W**H - W*V**H. * * The contents of A on exit are illustrated by the following examples * with n = 5 and nb = 2: * * if UPLO = 'U': if UPLO = 'L': * * ( a a a v4 v5 ) ( d ) * ( a a v4 v5 ) ( 1 d ) * ( a 1 v5 ) ( v1 1 a ) * ( d 1 ) ( v1 v2 a a ) * ( d ) ( v1 v2 a a a ) * * where d denotes a diagonal element of the reduced matrix, a denotes * an element of the original matrix that is unchanged, and vi denotes * an element of the vector defining H(i). * * ===================================================================== * * .. Parameters .. COMPLEX ZERO, ONE, HALF PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), $ ONE = ( 1.0E+0, 0.0E+0 ), $ HALF = ( 0.5E+0, 0.0E+0 ) ) * .. * .. Local Scalars .. INTEGER I, IW COMPLEX ALPHA * .. * .. External Subroutines .. EXTERNAL CAXPY, CGEMV, CHEMV, CLACGV, CLARFG, CSCAL * .. * .. External Functions .. LOGICAL LSAME COMPLEX CDOTC EXTERNAL LSAME, CDOTC * .. * .. Intrinsic Functions .. INTRINSIC MIN, REAL * .. * .. Executable Statements .. * * Quick return if possible * IF( N.LE.0 ) $ RETURN * IF( LSAME( UPLO, 'U' ) ) THEN * * Reduce last NB columns of upper triangle * DO 10 I = N, N - NB + 1, -1 IW = I - N + NB IF( I.LT.N ) THEN * * Update A(1:i,i) * A( I, I ) = REAL( A( I, I ) ) CALL CLACGV( N-I, W( I, IW+1 ), LDW ) CALL CGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) CALL CLACGV( N-I, W( I, IW+1 ), LDW ) CALL CLACGV( N-I, A( I, I+1 ), LDA ) CALL CGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) CALL CLACGV( N-I, A( I, I+1 ), LDA ) A( I, I ) = REAL( A( I, I ) ) END IF IF( I.GT.1 ) THEN * * Generate elementary reflector H(i) to annihilate * A(1:i-2,i) * ALPHA = A( I-1, I ) CALL CLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) ) E( I-1 ) = ALPHA A( I-1, I ) = ONE * * Compute W(1:i-1,i) * CALL CHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, $ ZERO, W( 1, IW ), 1 ) IF( I.LT.N ) THEN CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO, $ W( I+1, IW ), 1 ) CALL CGEMV( 'No transpose', I-1, N-I, -ONE, $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, $ W( 1, IW ), 1 ) CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO, $ W( I+1, IW ), 1 ) CALL CGEMV( 'No transpose', I-1, N-I, -ONE, $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, $ W( 1, IW ), 1 ) END IF CALL CSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) ALPHA = -HALF*TAU( I-1 )*CDOTC( I-1, W( 1, IW ), 1, $ A( 1, I ), 1 ) CALL CAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) END IF * 10 CONTINUE ELSE * * Reduce first NB columns of lower triangle * DO 20 I = 1, NB * * Update A(i:n,i) * A( I, I ) = REAL( A( I, I ) ) CALL CLACGV( I-1, W( I, 1 ), LDW ) CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) CALL CLACGV( I-1, W( I, 1 ), LDW ) CALL CLACGV( I-1, A( I, 1 ), LDA ) CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) CALL CLACGV( I-1, A( I, 1 ), LDA ) A( I, I ) = REAL( A( I, I ) ) IF( I.LT.N ) THEN * * Generate elementary reflector H(i) to annihilate * A(i+2:n,i) * ALPHA = A( I+1, I ) CALL CLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, $ TAU( I ) ) E( I ) = ALPHA A( I+1, I ) = ONE * * Compute W(i+1:n,i) * CALL CHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO, $ W( 1, I ), 1 ) CALL CGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, $ W( 1, I ), 1 ) CALL CGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL CSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) ALPHA = -HALF*TAU( I )*CDOTC( N-I, W( I+1, I ), 1, $ A( I+1, I ), 1 ) CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) END IF * 20 CONTINUE END IF * RETURN * * End of CLATRD * END |