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