|
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 |
SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
* * -- LAPACK 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 INFO, LDA, LWORK, N * .. * .. Array Arguments .. DOUBLE PRECISION D( * ), E( * ) COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) * .. * * Purpose * ======= * * ZHETRD reduces a complex Hermitian matrix A to real symmetric * tridiagonal form T by a unitary similarity transformation: * Q**H * A * Q = T. * * Arguments * ========= * * UPLO (input) CHARACTER*1 * = 'U': Upper triangle of A is stored; * = 'L': Lower triangle of A is stored. * * N (input) INTEGER * The order of the matrix A. N >= 0. * * A (input/output) COMPLEX*16 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 diagonal and first superdiagonal * of A are overwritten by the corresponding elements of the * tridiagonal matrix T, and the elements above the first * superdiagonal, with the array TAU, represent the unitary * matrix Q as a product of elementary reflectors; if UPLO * = 'L', the diagonal and first subdiagonal of A are over- * written by the corresponding elements of the tridiagonal * matrix T, and the elements below the first subdiagonal, 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). * * D (output) DOUBLE PRECISION array, dimension (N) * The diagonal elements of the tridiagonal matrix T: * D(i) = A(i,i). * * E (output) DOUBLE PRECISION array, dimension (N-1) * The off-diagonal elements of the tridiagonal matrix T: * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. * * TAU (output) COMPLEX*16 array, dimension (N-1) * The scalar factors of the elementary reflectors (see Further * Details). * * WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK)) * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. * * LWORK (input) INTEGER * The dimension of the array WORK. LWORK >= 1. * For optimum performance LWORK >= N*NB, where NB is the * optimal blocksize. * * If LWORK = -1, then a workspace query is assumed; the routine * only calculates the optimal size of the WORK array, returns * this value as the first entry of the WORK array, and no error * message related to LWORK is issued by XERBLA. * * INFO (output) INTEGER * = 0: successful exit * < 0: if INFO = -i, the i-th argument had an illegal value * * Further Details * =============== * * If UPLO = 'U', the matrix Q is represented as a product of elementary * reflectors * * Q = H(n-1) . . . H(2) H(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+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in * A(1:i-1,i+1), and tau in TAU(i). * * If UPLO = 'L', the matrix Q is represented as a product of elementary * reflectors * * Q = H(1) H(2) . . . H(n-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(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), * and tau in TAU(i). * * The contents of A on exit are illustrated by the following examples * with n = 5: * * if UPLO = 'U': if UPLO = 'L': * * ( d e v2 v3 v4 ) ( d ) * ( d e v3 v4 ) ( e d ) * ( d e v4 ) ( v1 e d ) * ( d e ) ( v1 v2 e d ) * ( d ) ( v1 v2 v3 e d ) * * where d and e denote diagonal and off-diagonal elements of T, and vi * denotes an element of the vector defining H(i). * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ONE PARAMETER ( ONE = 1.0D+0 ) COMPLEX*16 CONE PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. LOGICAL LQUERY, UPPER INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, $ NBMIN, NX * .. * .. External Subroutines .. EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD * .. * .. Intrinsic Functions .. INTRINSIC MAX * .. * .. External Functions .. LOGICAL LSAME INTEGER ILAENV EXTERNAL LSAME, ILAENV * .. * .. Executable Statements .. * * Test the input parameters * INFO = 0 UPPER = LSAME( UPLO, 'U' ) LQUERY = ( LWORK.EQ.-1 ) IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN INFO = -1 ELSE IF( N.LT.0 ) THEN INFO = -2 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN INFO = -4 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN INFO = -9 END IF * IF( INFO.EQ.0 ) THEN * * Determine the block size. * NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) LWKOPT = N*NB WORK( 1 ) = LWKOPT END IF * IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZHETRD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Quick return if possible * IF( N.EQ.0 ) THEN WORK( 1 ) = 1 RETURN END IF * NX = N IWS = 1 IF( NB.GT.1 .AND. NB.LT.N ) THEN * * Determine when to cross over from blocked to unblocked code * (last block is always handled by unblocked code). * NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) ) IF( NX.LT.N ) THEN * * Determine if workspace is large enough for blocked code. * LDWORK = N IWS = LDWORK*NB IF( LWORK.LT.IWS ) THEN * * Not enough workspace to use optimal NB: determine the * minimum value of NB, and reduce NB or force use of * unblocked code by setting NX = N. * NB = MAX( LWORK / LDWORK, 1 ) NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 ) IF( NB.LT.NBMIN ) $ NX = N END IF ELSE NX = N END IF ELSE NB = 1 END IF * IF( UPPER ) THEN * * Reduce the upper triangle of A. * Columns 1:kk are handled by the unblocked method. * KK = N - ( ( N-NX+NB-1 ) / NB )*NB DO 20 I = N - NB + 1, KK + 1, -NB * * Reduce columns i:i+nb-1 to tridiagonal form and form the * matrix W which is needed to update the unreduced part of * the matrix * CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, $ LDWORK ) * * Update the unreduced submatrix A(1:i-1,1:i-1), using an * update of the form: A := A - V*W**H - W*V**H * CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE, $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) * * Copy superdiagonal elements back into A, and diagonal * elements into D * DO 10 J = I, I + NB - 1 A( J-1, J ) = E( J-1 ) D( J ) = A( J, J ) 10 CONTINUE 20 CONTINUE * * Use unblocked code to reduce the last or only block * CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) ELSE * * Reduce the lower triangle of A * DO 40 I = 1, N - NX, NB * * Reduce columns i:i+nb-1 to tridiagonal form and form the * matrix W which is needed to update the unreduced part of * the matrix * CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), $ TAU( I ), WORK, LDWORK ) * * Update the unreduced submatrix A(i+nb:n,i+nb:n), using * an update of the form: A := A - V*W**H - W*V**H * CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE, $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, $ A( I+NB, I+NB ), LDA ) * * Copy subdiagonal elements back into A, and diagonal * elements into D * DO 30 J = I, I + NB - 1 A( J+1, J ) = E( J ) D( J ) = A( J, J ) 30 CONTINUE 40 CONTINUE * * Use unblocked code to reduce the last or only block * CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), $ TAU( I ), IINFO ) END IF * WORK( 1 ) = LWKOPT RETURN * * End of ZHETRD * END |