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LAPACK 3.3.0
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LAPACK 3.3.0
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00001 SUBROUTINE ZHETRD( 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 D( * ), E( * ) 00014 COMPLEX*16 A( LDA, * ), TAU( * ), WORK( * ) 00015 * .. 00016 * 00017 * Purpose 00018 * ======= 00019 * 00020 * ZHETRD reduces a complex Hermitian matrix A to real symmetric 00021 * tridiagonal form T by a unitary similarity transformation: 00022 * Q**H * 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) COMPLEX*16 array, dimension (LDA,N) 00035 * On entry, the Hermitian 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 unitary 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 unitary 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) COMPLEX*16 array, dimension (N-1) 00065 * The scalar factors of the elementary reflectors (see Further 00066 * Details). 00067 * 00068 * WORK (workspace/output) COMPLEX*16 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 complex scalar, and v is a complex 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 complex scalar, and v is a complex 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 COMPLEX*16 CONE 00134 PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ) ) 00135 * .. 00136 * .. Local Scalars .. 00137 LOGICAL LQUERY, UPPER 00138 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, 00139 $ NBMIN, NX 00140 * .. 00141 * .. External Subroutines .. 00142 EXTERNAL XERBLA, ZHER2K, ZHETD2, ZLATRD 00143 * .. 00144 * .. Intrinsic Functions .. 00145 INTRINSIC MAX 00146 * .. 00147 * .. External Functions .. 00148 LOGICAL LSAME 00149 INTEGER ILAENV 00150 EXTERNAL LSAME, ILAENV 00151 * .. 00152 * .. Executable Statements .. 00153 * 00154 * Test the input parameters 00155 * 00156 INFO = 0 00157 UPPER = LSAME( UPLO, 'U' ) 00158 LQUERY = ( LWORK.EQ.-1 ) 00159 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 00160 INFO = -1 00161 ELSE IF( N.LT.0 ) THEN 00162 INFO = -2 00163 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 00164 INFO = -4 00165 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN 00166 INFO = -9 00167 END IF 00168 * 00169 IF( INFO.EQ.0 ) THEN 00170 * 00171 * Determine the block size. 00172 * 00173 NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 ) 00174 LWKOPT = N*NB 00175 WORK( 1 ) = LWKOPT 00176 END IF 00177 * 00178 IF( INFO.NE.0 ) THEN 00179 CALL XERBLA( 'ZHETRD', -INFO ) 00180 RETURN 00181 ELSE IF( LQUERY ) THEN 00182 RETURN 00183 END IF 00184 * 00185 * Quick return if possible 00186 * 00187 IF( N.EQ.0 ) THEN 00188 WORK( 1 ) = 1 00189 RETURN 00190 END IF 00191 * 00192 NX = N 00193 IWS = 1 00194 IF( NB.GT.1 .AND. NB.LT.N ) THEN 00195 * 00196 * Determine when to cross over from blocked to unblocked code 00197 * (last block is always handled by unblocked code). 00198 * 00199 NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) ) 00200 IF( NX.LT.N ) THEN 00201 * 00202 * Determine if workspace is large enough for blocked code. 00203 * 00204 LDWORK = N 00205 IWS = LDWORK*NB 00206 IF( LWORK.LT.IWS ) THEN 00207 * 00208 * Not enough workspace to use optimal NB: determine the 00209 * minimum value of NB, and reduce NB or force use of 00210 * unblocked code by setting NX = N. 00211 * 00212 NB = MAX( LWORK / LDWORK, 1 ) 00213 NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 ) 00214 IF( NB.LT.NBMIN ) 00215 $ NX = N 00216 END IF 00217 ELSE 00218 NX = N 00219 END IF 00220 ELSE 00221 NB = 1 00222 END IF 00223 * 00224 IF( UPPER ) THEN 00225 * 00226 * Reduce the upper triangle of A. 00227 * Columns 1:kk are handled by the unblocked method. 00228 * 00229 KK = N - ( ( N-NX+NB-1 ) / NB )*NB 00230 DO 20 I = N - NB + 1, KK + 1, -NB 00231 * 00232 * Reduce columns i:i+nb-1 to tridiagonal form and form the 00233 * matrix W which is needed to update the unreduced part of 00234 * the matrix 00235 * 00236 CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, 00237 $ LDWORK ) 00238 * 00239 * Update the unreduced submatrix A(1:i-1,1:i-1), using an 00240 * update of the form: A := A - V*W' - W*V' 00241 * 00242 CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE, 00243 $ A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA ) 00244 * 00245 * Copy superdiagonal elements back into A, and diagonal 00246 * elements into D 00247 * 00248 DO 10 J = I, I + NB - 1 00249 A( J-1, J ) = E( J-1 ) 00250 D( J ) = A( J, J ) 00251 10 CONTINUE 00252 20 CONTINUE 00253 * 00254 * Use unblocked code to reduce the last or only block 00255 * 00256 CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) 00257 ELSE 00258 * 00259 * Reduce the lower triangle of A 00260 * 00261 DO 40 I = 1, N - NX, NB 00262 * 00263 * Reduce columns i:i+nb-1 to tridiagonal form and form the 00264 * matrix W which is needed to update the unreduced part of 00265 * the matrix 00266 * 00267 CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), 00268 $ TAU( I ), WORK, LDWORK ) 00269 * 00270 * Update the unreduced submatrix A(i+nb:n,i+nb:n), using 00271 * an update of the form: A := A - V*W' - W*V' 00272 * 00273 CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE, 00274 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, 00275 $ A( I+NB, I+NB ), LDA ) 00276 * 00277 * Copy subdiagonal elements back into A, and diagonal 00278 * elements into D 00279 * 00280 DO 30 J = I, I + NB - 1 00281 A( J+1, J ) = E( J ) 00282 D( J ) = A( J, J ) 00283 30 CONTINUE 00284 40 CONTINUE 00285 * 00286 * Use unblocked code to reduce the last or only block 00287 * 00288 CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), 00289 $ TAU( I ), IINFO ) 00290 END IF 00291 * 00292 WORK( 1 ) = LWKOPT 00293 RETURN 00294 * 00295 * End of ZHETRD 00296 * 00297 END
1.7.3 
