Annotation of rpl/lapack/lapack/dsytrd.f, revision 1.1
1.1 ! bertrand 1: SUBROUTINE DSYTRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
! 2: *
! 3: * -- LAPACK routine (version 3.2) --
! 4: * -- LAPACK is a software package provided by Univ. of Tennessee, --
! 5: * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
! 6: * November 2006
! 7: *
! 8: * .. Scalar Arguments ..
! 9: CHARACTER UPLO
! 10: INTEGER INFO, LDA, LWORK, N
! 11: * ..
! 12: * .. Array Arguments ..
! 13: DOUBLE PRECISION A( LDA, * ), D( * ), E( * ), TAU( * ),
! 14: $ WORK( * )
! 15: * ..
! 16: *
! 17: * Purpose
! 18: * =======
! 19: *
! 20: * DSYTRD 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N)
! 57: * The diagonal elements of the tridiagonal matrix T:
! 58: * D(i) = A(i,i).
! 59: *
! 60: * E (output) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (N-1)
! 65: * The scalar factors of the elementary reflectors (see Further
! 66: * Details).
! 67: *
! 68: * WORK (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,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: DOUBLE PRECISION ONE
! 132: PARAMETER ( ONE = 1.0D+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 DLATRD, DSYR2K, DSYTD2, 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, 'DSYTRD', 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( 'DSYTRD', -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, 'DSYTRD', 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, 'DSYTRD', 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 DLATRD( 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 DSYR2K( 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 DSYTD2( 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 DLATRD( 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 DSYR2K( 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 DSYTD2( 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 DSYTRD
! 294: *
! 295: END
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