1 SUBROUTINE SSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO ) 2* 3* -- LAPACK routine (version 3.0) -- 4* Univ. of Tennessee, Univ. of California Berkeley, NAG Ltd., 5* Courant Institute, Argonne National Lab, and Rice University 6* June 30, 1999 7* 8* .. Scalar Arguments .. 9 CHARACTER UPLO 10 INTEGER INFO, LDA, LWORK, N 11* .. 12* .. Array Arguments .. 13 REAL A( LDA, * ), D( * ), E( * ), TAU( * ), 14 $ WORK( * ) 15* .. 16* 17* Purpose 18* ======= 19* 20* SSYTRD reduces a real symmetric matrix A to real symmetric 21* tridiagonal form T by an orthogonal similarity transformation: 22* Q**T * A * Q = T. 23* 24* Arguments 25* ========= 26* 27* UPLO (input) CHARACTER*1 28* = 'U': Upper triangle of A is stored; 29* = 'L': Lower triangle of A is stored. 30* 31* N (input) INTEGER 32* The order of the matrix A. N >= 0. 33* 34* A (input/output) REAL array, dimension (LDA,N) 35* On entry, the symmetric matrix A. If UPLO = 'U', the leading 36* N-by-N upper triangular part of A contains the upper 37* triangular part of the matrix A, and the strictly lower 38* triangular part of A is not referenced. If UPLO = 'L', the 39* leading N-by-N lower triangular part of A contains the lower 40* triangular part of the matrix A, and the strictly upper 41* triangular part of A is not referenced. 42* On exit, if UPLO = 'U', the diagonal and first superdiagonal 43* of A are overwritten by the corresponding elements of the 44* tridiagonal matrix T, and the elements above the first 45* superdiagonal, with the array TAU, represent the orthogonal 46* matrix Q as a product of elementary reflectors; if UPLO 47* = 'L', the diagonal and first subdiagonal of A are over- 48* written by the corresponding elements of the tridiagonal 49* matrix T, and the elements below the first subdiagonal, with 50* the array TAU, represent the orthogonal matrix Q as a product 51* of elementary reflectors. See Further Details. 52* 53* LDA (input) INTEGER 54* The leading dimension of the array A. LDA >= max(1,N). 55* 56* D (output) REAL array, dimension (N) 57* The diagonal elements of the tridiagonal matrix T: 58* D(i) = A(i,i). 59* 60* E (output) REAL array, dimension (N-1) 61* The off-diagonal elements of the tridiagonal matrix T: 62* E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'. 63* 64* TAU (output) REAL array, dimension (N-1) 65* The scalar factors of the elementary reflectors (see Further 66* Details). 67* 68* WORK (workspace/output) REAL array, dimension (LWORK) 69* On exit, if INFO = 0, WORK(1) returns the optimal LWORK. 70* 71* LWORK (input) INTEGER 72* The dimension of the array WORK. LWORK >= 1. 73* For optimum performance LWORK >= N*NB, where NB is the 74* optimal blocksize. 75* 76* If LWORK = -1, then a workspace query is assumed; the routine 77* only calculates the optimal size of the WORK array, returns 78* this value as the first entry of the WORK array, and no error 79* message related to LWORK is issued by XERBLA. 80* 81* INFO (output) INTEGER 82* = 0: successful exit 83* < 0: if INFO = -i, the i-th argument had an illegal value 84* 85* Further Details 86* =============== 87* 88* If UPLO = 'U', the matrix Q is represented as a product of elementary 89* reflectors 90* 91* Q = H(n-1) . . . H(2) H(1). 92* 93* Each H(i) has the form 94* 95* H(i) = I - tau * v * v' 96* 97* where tau is a real scalar, and v is a real vector with 98* v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in 99* A(1:i-1,i+1), and tau in TAU(i). 100* 101* If UPLO = 'L', the matrix Q is represented as a product of elementary 102* reflectors 103* 104* Q = H(1) H(2) . . . H(n-1). 105* 106* Each H(i) has the form 107* 108* H(i) = I - tau * v * v' 109* 110* where tau is a real scalar, and v is a real vector with 111* v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i), 112* and tau in TAU(i). 113* 114* The contents of A on exit are illustrated by the following examples 115* with n = 5: 116* 117* if UPLO = 'U': if UPLO = 'L': 118* 119* ( d e v2 v3 v4 ) ( d ) 120* ( d e v3 v4 ) ( e d ) 121* ( d e v4 ) ( v1 e d ) 122* ( d e ) ( v1 v2 e d ) 123* ( d ) ( v1 v2 v3 e d ) 124* 125* where d and e denote diagonal and off-diagonal elements of T, and vi 126* denotes an element of the vector defining H(i). 127* 128* ===================================================================== 129* 130* .. Parameters .. 131 REAL ONE 132 PARAMETER ( ONE = 1.0E+0 ) 133* .. 134* .. Local Scalars .. 135 LOGICAL LQUERY, UPPER 136 INTEGER I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB, 137 $ NBMIN, NX 138* .. 139* .. External Subroutines .. 140 EXTERNAL SLATRD, SSYR2K, SSYTD2, XERBLA 141* .. 142* .. Intrinsic Functions .. 143 INTRINSIC MAX 144* .. 145* .. External Functions .. 146 LOGICAL LSAME 147 INTEGER ILAENV 148 EXTERNAL LSAME, ILAENV 149* .. 150* .. Executable Statements .. 151* 152* Test the input parameters 153* 154 INFO = 0 155 UPPER = LSAME( UPLO, 'U' ) 156 LQUERY = ( LWORK.EQ.-1 ) 157 IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN 158 INFO = -1 159 ELSE IF( N.LT.0 ) THEN 160 INFO = -2 161 ELSE IF( LDA.LT.MAX( 1, N ) ) THEN 162 INFO = -4 163 ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN 164 INFO = -9 165 END IF 166* 167 IF( INFO.EQ.0 ) THEN 168* 169* Determine the block size. 170* 171 NB = ILAENV( 1, 'SSYTRD', UPLO, N, -1, -1, -1 ) 172 LWKOPT = N*NB 173 WORK( 1 ) = LWKOPT 174 END IF 175* 176 IF( INFO.NE.0 ) THEN 177 CALL XERBLA( 'SSYTRD', -INFO ) 178 RETURN 179 ELSE IF( LQUERY ) THEN 180 RETURN 181 END IF 182* 183* Quick return if possible 184* 185 IF( N.EQ.0 ) THEN 186 WORK( 1 ) = 1 187 RETURN 188 END IF 189* 190 NX = N 191 IWS = 1 192 IF( NB.GT.1 .AND. NB.LT.N ) THEN 193* 194* Determine when to cross over from blocked to unblocked code 195* (last block is always handled by unblocked code). 196* 197 NX = MAX( NB, ILAENV( 3, 'SSYTRD', UPLO, N, -1, -1, -1 ) ) 198 IF( NX.LT.N ) THEN 199* 200* Determine if workspace is large enough for blocked code. 201* 202 LDWORK = N 203 IWS = LDWORK*NB 204 IF( LWORK.LT.IWS ) THEN 205* 206* Not enough workspace to use optimal NB: determine the 207* minimum value of NB, and reduce NB or force use of 208* unblocked code by setting NX = N. 209* 210 NB = MAX( LWORK / LDWORK, 1 ) 211 NBMIN = ILAENV( 2, 'SSYTRD', UPLO, N, -1, -1, -1 ) 212 IF( NB.LT.NBMIN ) 213 $ NX = N 214 END IF 215 ELSE 216 NX = N 217 END IF 218 ELSE 219 NB = 1 220 END IF 221* 222 IF( UPPER ) THEN 223* 224* Reduce the upper triangle of A. 225* Columns 1:kk are handled by the unblocked method. 226* 227 KK = N - ( ( N-NX+NB-1 ) / NB )*NB 228 DO 20 I = N - NB + 1, KK + 1, -NB 229* 230* Reduce columns i:i+nb-1 to tridiagonal form and form the 231* matrix W which is needed to update the unreduced part of 232* the matrix 233* 234 CALL SLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK, 235 $ LDWORK ) 236* 237* Update the unreduced submatrix A(1:i-1,1:i-1), using an 238* update of the form: A := A - V*W' - W*V' 239* 240 CALL SSYR2K( UPLO, 'No transpose', I-1, NB, -ONE, A( 1, I ), 241 $ LDA, WORK, LDWORK, ONE, A, LDA ) 242* 243* Copy superdiagonal elements back into A, and diagonal 244* elements into D 245* 246 DO 10 J = I, I + NB - 1 247 A( J-1, J ) = E( J-1 ) 248 D( J ) = A( J, J ) 249 10 CONTINUE 250 20 CONTINUE 251* 252* Use unblocked code to reduce the last or only block 253* 254 CALL SSYTD2( UPLO, KK, A, LDA, D, E, TAU, IINFO ) 255 ELSE 256* 257* Reduce the lower triangle of A 258* 259 DO 40 I = 1, N - NX, NB 260* 261* Reduce columns i:i+nb-1 to tridiagonal form and form the 262* matrix W which is needed to update the unreduced part of 263* the matrix 264* 265 CALL SLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ), 266 $ TAU( I ), WORK, LDWORK ) 267* 268* Update the unreduced submatrix A(i+ib:n,i+ib:n), using 269* an update of the form: A := A - V*W' - W*V' 270* 271 CALL SSYR2K( UPLO, 'No transpose', N-I-NB+1, NB, -ONE, 272 $ A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE, 273 $ A( I+NB, I+NB ), LDA ) 274* 275* Copy subdiagonal elements back into A, and diagonal 276* elements into D 277* 278 DO 30 J = I, I + NB - 1 279 A( J+1, J ) = E( J ) 280 D( J ) = A( J, J ) 281 30 CONTINUE 282 40 CONTINUE 283* 284* Use unblocked code to reduce the last or only block 285* 286 CALL SSYTD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ), 287 $ TAU( I ), IINFO ) 288 END IF 289* 290 WORK( 1 ) = LWKOPT 291 RETURN 292* 293* End of SSYTRD 294* 295 END 296