00001 SUBROUTINE CLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) 00002 * 00003 * -- LAPACK auxiliary routine (version 3.2) -- 00004 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00005 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00006 * November 2006 00007 * 00008 * .. Scalar Arguments .. 00009 CHARACTER UPLO 00010 INTEGER LDA, LDW, N, NB 00011 * .. 00012 * .. Array Arguments .. 00013 REAL E( * ) 00014 COMPLEX A( LDA, * ), TAU( * ), W( LDW, * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * CLATRD reduces NB rows and columns of a complex Hermitian matrix A to 00021 * Hermitian tridiagonal form by a unitary similarity 00022 * transformation Q' * A * Q, and returns the matrices V and W which are 00023 * needed to apply the transformation to the unreduced part of A. 00024 * 00025 * If UPLO = 'U', CLATRD reduces the last NB rows and columns of a 00026 * matrix, of which the upper triangle is supplied; 00027 * if UPLO = 'L', CLATRD reduces the first NB rows and columns of a 00028 * matrix, of which the lower triangle is supplied. 00029 * 00030 * This is an auxiliary routine called by CHETRD. 00031 * 00032 * Arguments 00033 * ========= 00034 * 00035 * UPLO (input) CHARACTER*1 00036 * Specifies whether the upper or lower triangular part of the 00037 * Hermitian matrix A is stored: 00038 * = 'U': Upper triangular 00039 * = 'L': Lower triangular 00040 * 00041 * N (input) INTEGER 00042 * The order of the matrix A. 00043 * 00044 * NB (input) INTEGER 00045 * The number of rows and columns to be reduced. 00046 * 00047 * A (input/output) COMPLEX array, dimension (LDA,N) 00048 * On entry, the Hermitian matrix A. If UPLO = 'U', the leading 00049 * n-by-n upper triangular part of A contains the upper 00050 * triangular part of the matrix A, and the strictly lower 00051 * triangular part of A is not referenced. If UPLO = 'L', the 00052 * leading n-by-n lower triangular part of A contains the lower 00053 * triangular part of the matrix A, and the strictly upper 00054 * triangular part of A is not referenced. 00055 * On exit: 00056 * if UPLO = 'U', the last NB columns have been reduced to 00057 * tridiagonal form, with the diagonal elements overwriting 00058 * the diagonal elements of A; the elements above the diagonal 00059 * with the array TAU, represent the unitary matrix Q as a 00060 * product of elementary reflectors; 00061 * if UPLO = 'L', the first NB columns have been reduced to 00062 * tridiagonal form, with the diagonal elements overwriting 00063 * the diagonal elements of A; the elements below the diagonal 00064 * with the array TAU, represent the unitary matrix Q as a 00065 * product of elementary reflectors. 00066 * See Further Details. 00067 * 00068 * LDA (input) INTEGER 00069 * The leading dimension of the array A. LDA >= max(1,N). 00070 * 00071 * E (output) REAL array, dimension (N-1) 00072 * If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal 00073 * elements of the last NB columns of the reduced matrix; 00074 * if UPLO = 'L', E(1:nb) contains the subdiagonal elements of 00075 * the first NB columns of the reduced matrix. 00076 * 00077 * TAU (output) COMPLEX array, dimension (N-1) 00078 * The scalar factors of the elementary reflectors, stored in 00079 * TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. 00080 * See Further Details. 00081 * 00082 * W (output) COMPLEX array, dimension (LDW,NB) 00083 * The n-by-nb matrix W required to update the unreduced part 00084 * of A. 00085 * 00086 * LDW (input) INTEGER 00087 * The leading dimension of the array W. LDW >= max(1,N). 00088 * 00089 * Further Details 00090 * =============== 00091 * 00092 * If UPLO = 'U', the matrix Q is represented as a product of elementary 00093 * reflectors 00094 * 00095 * Q = H(n) H(n-1) . . . H(n-nb+1). 00096 * 00097 * Each H(i) has the form 00098 * 00099 * H(i) = I - tau * v * v' 00100 * 00101 * where tau is a complex scalar, and v is a complex vector with 00102 * v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), 00103 * and tau in TAU(i-1). 00104 * 00105 * If UPLO = 'L', the matrix Q is represented as a product of elementary 00106 * reflectors 00107 * 00108 * Q = H(1) H(2) . . . H(nb). 00109 * 00110 * Each H(i) has the form 00111 * 00112 * H(i) = I - tau * v * v' 00113 * 00114 * where tau is a complex scalar, and v is a complex vector with 00115 * v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), 00116 * and tau in TAU(i). 00117 * 00118 * The elements of the vectors v together form the n-by-nb matrix V 00119 * which is needed, with W, to apply the transformation to the unreduced 00120 * part of the matrix, using a Hermitian rank-2k update of the form: 00121 * A := A - V*W' - W*V'. 00122 * 00123 * The contents of A on exit are illustrated by the following examples 00124 * with n = 5 and nb = 2: 00125 * 00126 * if UPLO = 'U': if UPLO = 'L': 00127 * 00128 * ( a a a v4 v5 ) ( d ) 00129 * ( a a v4 v5 ) ( 1 d ) 00130 * ( a 1 v5 ) ( v1 1 a ) 00131 * ( d 1 ) ( v1 v2 a a ) 00132 * ( d ) ( v1 v2 a a a ) 00133 * 00134 * where d denotes a diagonal element of the reduced matrix, a denotes 00135 * an element of the original matrix that is unchanged, and vi denotes 00136 * an element of the vector defining H(i). 00137 * 00138 * ===================================================================== 00139 * 00140 * .. Parameters .. 00141 COMPLEX ZERO, ONE, HALF 00142 PARAMETER ( ZERO = ( 0.0E+0, 0.0E+0 ), 00143 $ ONE = ( 1.0E+0, 0.0E+0 ), 00144 $ HALF = ( 0.5E+0, 0.0E+0 ) ) 00145 * .. 00146 * .. Local Scalars .. 00147 INTEGER I, IW 00148 COMPLEX ALPHA 00149 * .. 00150 * .. External Subroutines .. 00151 EXTERNAL CAXPY, CGEMV, CHEMV, CLACGV, CLARFG, CSCAL 00152 * .. 00153 * .. External Functions .. 00154 LOGICAL LSAME 00155 COMPLEX CDOTC 00156 EXTERNAL LSAME, CDOTC 00157 * .. 00158 * .. Intrinsic Functions .. 00159 INTRINSIC MIN, REAL 00160 * .. 00161 * .. Executable Statements .. 00162 * 00163 * Quick return if possible 00164 * 00165 IF( N.LE.0 ) 00166 $ RETURN 00167 * 00168 IF( LSAME( UPLO, 'U' ) ) THEN 00169 * 00170 * Reduce last NB columns of upper triangle 00171 * 00172 DO 10 I = N, N - NB + 1, -1 00173 IW = I - N + NB 00174 IF( I.LT.N ) THEN 00175 * 00176 * Update A(1:i,i) 00177 * 00178 A( I, I ) = REAL( A( I, I ) ) 00179 CALL CLACGV( N-I, W( I, IW+1 ), LDW ) 00180 CALL CGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), 00181 $ LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) 00182 CALL CLACGV( N-I, W( I, IW+1 ), LDW ) 00183 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 00184 CALL CGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), 00185 $ LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) 00186 CALL CLACGV( N-I, A( I, I+1 ), LDA ) 00187 A( I, I ) = REAL( A( I, I ) ) 00188 END IF 00189 IF( I.GT.1 ) THEN 00190 * 00191 * Generate elementary reflector H(i) to annihilate 00192 * A(1:i-2,i) 00193 * 00194 ALPHA = A( I-1, I ) 00195 CALL CLARFG( I-1, ALPHA, A( 1, I ), 1, TAU( I-1 ) ) 00196 E( I-1 ) = ALPHA 00197 A( I-1, I ) = ONE 00198 * 00199 * Compute W(1:i-1,i) 00200 * 00201 CALL CHEMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, 00202 $ ZERO, W( 1, IW ), 1 ) 00203 IF( I.LT.N ) THEN 00204 CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, 00205 $ W( 1, IW+1 ), LDW, A( 1, I ), 1, ZERO, 00206 $ W( I+1, IW ), 1 ) 00207 CALL CGEMV( 'No transpose', I-1, N-I, -ONE, 00208 $ A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, 00209 $ W( 1, IW ), 1 ) 00210 CALL CGEMV( 'Conjugate transpose', I-1, N-I, ONE, 00211 $ A( 1, I+1 ), LDA, A( 1, I ), 1, ZERO, 00212 $ W( I+1, IW ), 1 ) 00213 CALL CGEMV( 'No transpose', I-1, N-I, -ONE, 00214 $ W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, 00215 $ W( 1, IW ), 1 ) 00216 END IF 00217 CALL CSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) 00218 ALPHA = -HALF*TAU( I-1 )*CDOTC( I-1, W( 1, IW ), 1, 00219 $ A( 1, I ), 1 ) 00220 CALL CAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) 00221 END IF 00222 * 00223 10 CONTINUE 00224 ELSE 00225 * 00226 * Reduce first NB columns of lower triangle 00227 * 00228 DO 20 I = 1, NB 00229 * 00230 * Update A(i:n,i) 00231 * 00232 A( I, I ) = REAL( A( I, I ) ) 00233 CALL CLACGV( I-1, W( I, 1 ), LDW ) 00234 CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), 00235 $ LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) 00236 CALL CLACGV( I-1, W( I, 1 ), LDW ) 00237 CALL CLACGV( I-1, A( I, 1 ), LDA ) 00238 CALL CGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), 00239 $ LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) 00240 CALL CLACGV( I-1, A( I, 1 ), LDA ) 00241 A( I, I ) = REAL( A( I, I ) ) 00242 IF( I.LT.N ) THEN 00243 * 00244 * Generate elementary reflector H(i) to annihilate 00245 * A(i+2:n,i) 00246 * 00247 ALPHA = A( I+1, I ) 00248 CALL CLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, 00249 $ TAU( I ) ) 00250 E( I ) = ALPHA 00251 A( I+1, I ) = ONE 00252 * 00253 * Compute W(i+1:n,i) 00254 * 00255 CALL CHEMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, 00256 $ A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) 00257 CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, 00258 $ W( I+1, 1 ), LDW, A( I+1, I ), 1, ZERO, 00259 $ W( 1, I ), 1 ) 00260 CALL CGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), 00261 $ LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 00262 CALL CGEMV( 'Conjugate transpose', N-I, I-1, ONE, 00263 $ A( I+1, 1 ), LDA, A( I+1, I ), 1, ZERO, 00264 $ W( 1, I ), 1 ) 00265 CALL CGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), 00266 $ LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) 00267 CALL CSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) 00268 ALPHA = -HALF*TAU( I )*CDOTC( N-I, W( I+1, I ), 1, 00269 $ A( I+1, I ), 1 ) 00270 CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) 00271 END IF 00272 * 00273 20 CONTINUE 00274 END IF 00275 * 00276 RETURN 00277 * 00278 * End of CLATRD 00279 * 00280 END
1.7.2