00001 SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO ) 00002 * 00003 * -- LAPACK 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 INFO, LDA, N 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ) 00014 * .. 00015 * 00016 * Purpose 00017 * ======= 00018 * 00019 * DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal 00020 * form T by an orthogonal similarity transformation: Q' * A * Q = T. 00021 * 00022 * Arguments 00023 * ========= 00024 * 00025 * UPLO (input) CHARACTER*1 00026 * Specifies whether the upper or lower triangular part of the 00027 * symmetric matrix A is stored: 00028 * = 'U': Upper triangular 00029 * = 'L': Lower triangular 00030 * 00031 * N (input) INTEGER 00032 * The order of the matrix A. N >= 0. 00033 * 00034 * A (input/output) DOUBLE PRECISION array, dimension (LDA,N) 00035 * On entry, the symmetric matrix A. If UPLO = 'U', the leading 00036 * n-by-n upper triangular part of A contains the upper 00037 * triangular part of the matrix A, and the strictly lower 00038 * triangular part of A is not referenced. If UPLO = 'L', the 00039 * leading n-by-n lower triangular part of A contains the lower 00040 * triangular part of the matrix A, and the strictly upper 00041 * triangular part of A is not referenced. 00042 * On exit, if UPLO = 'U', the diagonal and first superdiagonal 00043 * of A are overwritten by the corresponding elements of the 00044 * tridiagonal matrix T, and the elements above the first 00045 * superdiagonal, with the array TAU, represent the orthogonal 00046 * matrix Q as a product of elementary reflectors; if UPLO 00047 * = 'L', the diagonal and first subdiagonal of A are over- 00048 * written by the corresponding elements of the tridiagonal 00049 * matrix T, and the elements below the first subdiagonal, with 00050 * the array TAU, represent the orthogonal matrix Q as a product 00051 * of elementary reflectors. See Further Details. 00052 * 00053 * LDA (input) INTEGER 00054 * The leading dimension of the array A. LDA >= max(1,N). 00055 * 00056 * D (output) DOUBLE PRECISION array, dimension (N) 00057 * The diagonal elements of the tridiagonal matrix T: 00058 * D(i) = A(i,i). 00059 * 00060 * E (output) DOUBLE PRECISION array, dimension (N-1) 00061 * The off-diagonal elements of the tridiagonal matrix T: 00062 * E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 00063 * 00064 * TAU (output) DOUBLE PRECISION array, dimension (N-1) 00065 * The scalar factors of the elementary reflectors (see Further 00066 * Details). 00067 * 00068 * INFO (output) INTEGER 00069 * = 0: successful exit 00070 * < 0: if INFO = -i, the i-th argument had an illegal value. 00071 * 00072 * Further Details 00073 * =============== 00074 * 00075 * If UPLO = 'U', the matrix Q is represented as a product of elementary 00076 * reflectors 00077 * 00078 * Q = H(n-1) . . . H(2) H(1). 00079 * 00080 * Each H(i) has the form 00081 * 00082 * H(i) = I - tau * v * v' 00083 * 00084 * where tau is a real scalar, and v is a real vector with 00085 * v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in 00086 * A(1:i-1,i+1), and tau in TAU(i). 00087 * 00088 * If UPLO = 'L', the matrix Q is represented as a product of elementary 00089 * reflectors 00090 * 00091 * Q = H(1) H(2) . . . H(n-1). 00092 * 00093 * Each H(i) has the form 00094 * 00095 * H(i) = I - tau * v * v' 00096 * 00097 * where tau is a real scalar, and v is a real vector with 00098 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), 00099 * and tau in TAU(i). 00100 * 00101 * The contents of A on exit are illustrated by the following examples 00102 * with n = 5: 00103 * 00104 * if UPLO = 'U': if UPLO = 'L': 00105 * 00106 * ( d e v2 v3 v4 ) ( d ) 00107 * ( d e v3 v4 ) ( e d ) 00108 * ( d e v4 ) ( v1 e d ) 00109 * ( d e ) ( v1 v2 e d ) 00110 * ( d ) ( v1 v2 v3 e d ) 00111 * 00112 * where d and e denote diagonal and off-diagonal elements of T, and vi 00113 * denotes an element of the vector defining H(i). 00114 * 00115 * ===================================================================== 00116 * 00117 * .. Parameters .. 00118 DOUBLE PRECISION ONE, ZERO, HALF 00119 PARAMETER ( ONE = 1.0D0, ZERO = 0.0D0, 00120 $ HALF = 1.0D0 / 2.0D0 ) 00121 * .. 00122 * .. Local Scalars .. 00123 LOGICAL UPPER 00124 INTEGER I 00125 DOUBLE PRECISION ALPHA, TAUI 00126 * .. 00127 * .. External Subroutines .. 00128 EXTERNAL DAXPY, DLARFG, DSYMV, DSYR2, XERBLA 00129 * .. 00130 * .. External Functions .. 00131 LOGICAL LSAME 00132 DOUBLE PRECISION DDOT 00133 EXTERNAL LSAME, DDOT 00134 * .. 00135 * .. Intrinsic Functions .. 00136 INTRINSIC MAX, MIN 00137 * .. 00138 * .. Executable Statements .. 00139 * 00140 * Test the input parameters 00141 * 00142 INFO = 0 00143 UPPER = LSAME( UPLO, 'U' ) 00144 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00145 INFO = -1 00146 ELSE IF( N.LT.0 ) THEN 00147 INFO = -2 00148 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00149 INFO = -4 00150 END IF 00151 IF( INFO.NE.0 ) THEN 00152 CALL XERBLA( 'DSYTD2', -INFO ) 00153 RETURN 00154 END IF 00155 * 00156 * Quick return if possible 00157 * 00158 IF( N.LE.0 ) 00159 $ RETURN 00160 * 00161 IF( UPPER ) THEN 00162 * 00163 * Reduce the upper triangle of A 00164 * 00165 DO 10 I = N - 1, 1, -1 00166 * 00167 * Generate elementary reflector H(i) = I - tau * v * v' 00168 * to annihilate A(1:i-1,i+1) 00169 * 00170 CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI ) 00171 E( I ) = A( I, I+1 ) 00172 * 00173 IF( TAUI.NE.ZERO ) THEN 00174 * 00175 * Apply H(i) from both sides to A(1:i,1:i) 00176 * 00177 A( I, I+1 ) = ONE 00178 * 00179 * Compute x := tau * A * v storing x in TAU(1:i) 00180 * 00181 CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO, 00182 $ TAU, 1 ) 00183 * 00184 * Compute w := x - 1/2 * tau * (x'*v) * v 00185 * 00186 ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 ) 00187 CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 ) 00188 * 00189 * Apply the transformation as a rank-2 update: 00190 * A := A - v * w' - w * v' 00191 * 00192 CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A, 00193 $ LDA ) 00194 * 00195 A( I, I+1 ) = E( I ) 00196 END IF 00197 D( I+1 ) = A( I+1, I+1 ) 00198 TAU( I ) = TAUI 00199 10 CONTINUE 00200 D( 1 ) = A( 1, 1 ) 00201 ELSE 00202 * 00203 * Reduce the lower triangle of A 00204 * 00205 DO 20 I = 1, N - 1 00206 * 00207 * Generate elementary reflector H(i) = I - tau * v * v' 00208 * to annihilate A(i+2:n,i) 00209 * 00210 CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, 00211 $ TAUI ) 00212 E( I ) = A( I+1, I ) 00213 * 00214 IF( TAUI.NE.ZERO ) THEN 00215 * 00216 * Apply H(i) from both sides to A(i+1:n,i+1:n) 00217 * 00218 A( I+1, I ) = ONE 00219 * 00220 * Compute x := tau * A * v storing y in TAU(i:n-1) 00221 * 00222 CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA, 00223 $ A( I+1, I ), 1, ZERO, TAU( I ), 1 ) 00224 * 00225 * Compute w := x - 1/2 * tau * (x'*v) * v 00226 * 00227 ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ), 00228 $ 1 ) 00229 CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 ) 00230 * 00231 * Apply the transformation as a rank-2 update: 00232 * A := A - v * w' - w * v' 00233 * 00234 CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1, 00235 $ A( I+1, I+1 ), LDA ) 00236 * 00237 A( I+1, I ) = E( I ) 00238 END IF 00239 D( I ) = A( I, I ) 00240 TAU( I ) = TAUI 00241 20 CONTINUE 00242 D( N ) = A( N, N ) 00243 END IF 00244 * 00245 RETURN 00246 * 00247 * End of DSYTD2 00248 * 00249 END
1.7.2