00001 SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, 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, LWORK, N 00011 * .. 00012 * .. Array Arguments .. 00013 DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ), 00014 $ WORK( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * DSYTRD reduces a real symmetric matrix A to real symmetric 00021 * tridiagonal form T by an orthogonal similarity transformation: 00022 * Q**T * A * Q = T. 00023 * 00024 * Arguments 00025 * ========= 00026 * 00027 * UPLO (input) CHARACTER*1 00028 * = 'U': Upper triangle of A is stored; 00029 * = 'L': Lower triangle of A is stored. 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 * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK)) 00069 * On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 00070 * 00071 * LWORK (input) INTEGER 00072 * The dimension of the array WORK. LWORK >= 1. 00073 * For optimum performance LWORK >= N*NB, where NB is the 00074 * optimal blocksize. 00075 * 00076 * If LWORK = -1, then a workspace query is assumed; the routine 00077 * only calculates the optimal size of the WORK array, returns 00078 * this value as the first entry of the WORK array, and no error 00079 * message related to LWORK is issued by XERBLA. 00080 * 00081 * INFO (output) INTEGER 00082 * = 0: successful exit 00083 * < 0: if INFO = -i, the i-th argument had an illegal value 00084 * 00085 * Further Details 00086 * =============== 00087 * 00088 * If UPLO = 'U', the matrix Q is represented as a product of elementary 00089 * reflectors 00090 * 00091 * Q = H(n-1) . . . H(2) H(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(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in 00099 * A(1:i-1,i+1), and tau in TAU(i). 00100 * 00101 * If UPLO = 'L', the matrix Q is represented as a product of elementary 00102 * reflectors 00103 * 00104 * Q = H(1) H(2) . . . H(n-1). 00105 * 00106 * Each H(i) has the form 00107 * 00108 * H(i) = I - tau * v * v' 00109 * 00110 * where tau is a real scalar, and v is a real vector with 00111 * v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), 00112 * and tau in TAU(i). 00113 * 00114 * The contents of A on exit are illustrated by the following examples 00115 * with n = 5: 00116 * 00117 * if UPLO = 'U': if UPLO = 'L': 00118 * 00119 * ( d e v2 v3 v4 ) ( d ) 00120 * ( d e v3 v4 ) ( e d ) 00121 * ( d e v4 ) ( v1 e d ) 00122 * ( d e ) ( v1 v2 e d ) 00123 * ( d ) ( v1 v2 v3 e d ) 00124 * 00125 * where d and e denote diagonal and off-diagonal elements of T, and vi 00126 * denotes an element of the vector defining H(i). 00127 * 00128 * ===================================================================== 00129 * 00130 * .. Parameters .. 00131 DOUBLE PRECISION ONE 00132 PARAMETER ( ONE = 1.0D+0 ) 00133 * .. 00134 * .. Local Scalars .. 00135 LOGICAL LQUERY, UPPER 00136 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, 00137 $ NBMIN, NX 00138 * .. 00139 * .. External Subroutines .. 00140 EXTERNAL DLATRD, DSYR2K, DSYTD2, XERBLA 00141 * .. 00142 * .. Intrinsic Functions .. 00143 INTRINSIC MAX 00144 * .. 00145 * .. External Functions .. 00146 LOGICAL LSAME 00147 INTEGER ILAENV 00148 EXTERNAL LSAME, ILAENV 00149 * .. 00150 * .. Executable Statements .. 00151 * 00152 * Test the input parameters 00153 * 00154 INFO = 0 00155 UPPER = LSAME( UPLO, 'U' ) 00156 LQUERY = ( LWORK.EQ.-1 ) 00157 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00158 INFO = -1 00159 ELSE IF( N.LT.0 ) THEN 00160 INFO = -2 00161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00162 INFO = -4 00163 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN 00164 INFO = -9 00165 END IF 00166 * 00167 IF( INFO.EQ.0 ) THEN 00168 * 00169 * Determine the block size. 00170 * 00171 NB = ILAENV( 1, 'DSYTRD', UPLO, N, -1, -1, -1 ) 00172 LWKOPT = N*NB 00173 WORK( 1 ) = LWKOPT 00174 END IF 00175 * 00176 IF( INFO.NE.0 ) THEN 00177 CALL XERBLA( 'DSYTRD', -INFO ) 00178 RETURN 00179 ELSE IF( LQUERY ) THEN 00180 RETURN 00181 END IF 00182 * 00183 * Quick return if possible 00184 * 00185 IF( N.EQ.0 ) THEN 00186 WORK( 1 ) = 1 00187 RETURN 00188 END IF 00189 * 00190 NX = N 00191 IWS = 1 00192 IF( NB.GT.1 .AND. NB.LT.N ) THEN 00193 * 00194 * Determine when to cross over from blocked to unblocked code 00195 * (last block is always handled by unblocked code). 00196 * 00197 NX = MAX( NB, ILAENV( 3, 'DSYTRD', UPLO, N, -1, -1, -1 ) ) 00198 IF( NX.LT.N ) THEN 00199 * 00200 * Determine if workspace is large enough for blocked code. 00201 * 00202 LDWORK = N 00203 IWS = LDWORK*NB 00204 IF( LWORK.LT.IWS ) THEN 00205 * 00206 * Not enough workspace to use optimal NB: determine the 00207 * minimum value of NB, and reduce NB or force use of 00208 * unblocked code by setting NX = N. 00209 * 00210 NB = MAX( LWORK / LDWORK, 1 ) 00211 NBMIN = ILAENV( 2, 'DSYTRD', UPLO, N, -1, -1, -1 ) 00212 IF( NB.LT.NBMIN ) 00213 $ NX = N 00214 END IF 00215 ELSE 00216 NX = N 00217 END IF 00218 ELSE 00219 NB = 1 00220 END IF 00221 * 00222 IF( UPPER ) THEN 00223 * 00224 * Reduce the upper triangle of A. 00225 * Columns 1:kk are handled by the unblocked method. 00226 * 00227 KK = N - ( ( N-NX+NB-1 ) / NB )*NB 00228 DO 20 I = N - NB + 1, KK + 1, -NB 00229 * 00230 * Reduce columns i:i+nb-1 to tridiagonal form and form the 00231 * matrix W which is needed to update the unreduced part of 00232 * the matrix 00233 * 00234 CALL DLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, 00235 $ LDWORK ) 00236 * 00237 * Update the unreduced submatrix A(1:i-1,1:i-1), using an 00238 * update of the form: A := A - V*W' - W*V' 00239 * 00240 CALL DSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), 00241 $ LDA, WORK, LDWORK, ONE, A, LDA ) 00242 * 00243 * Copy superdiagonal elements back into A, and diagonal 00244 * elements into D 00245 * 00246 DO 10 J = I, I + NB - 1 00247 A( J-1, J ) = E( J-1 ) 00248 D( J ) = A( J, J ) 00249 10 CONTINUE 00250 20 CONTINUE 00251 * 00252 * Use unblocked code to reduce the last or only block 00253 * 00254 CALL DSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) 00255 ELSE 00256 * 00257 * Reduce the lower triangle of A 00258 * 00259 DO 40 I = 1, N - NX, NB 00260 * 00261 * Reduce columns i:i+nb-1 to tridiagonal form and form the 00262 * matrix W which is needed to update the unreduced part of 00263 * the matrix 00264 * 00265 CALL DLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), 00266 $ TAU( I ), WORK, LDWORK ) 00267 * 00268 * Update the unreduced submatrix A(i+ib:n,i+ib:n), using 00269 * an update of the form: A := A - V*W' - W*V' 00270 * 00271 CALL DSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, 00272 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, 00273 $ A( I+NB, I+NB ), LDA ) 00274 * 00275 * Copy subdiagonal elements back into A, and diagonal 00276 * elements into D 00277 * 00278 DO 30 J = I, I + NB - 1 00279 A( J+1, J ) = E( J ) 00280 D( J ) = A( J, J ) 00281 30 CONTINUE 00282 40 CONTINUE 00283 * 00284 * Use unblocked code to reduce the last or only block 00285 * 00286 CALL DSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), 00287 $ TAU( I ), IINFO ) 00288 END IF 00289 * 00290 WORK( 1 ) = LWKOPT 00291 RETURN 00292 * 00293 * End of DSYTRD 00294 * 00295 END
1.7.2