SUBROUTINE DLATRD( UPLO, N, NB, A, LDA, E, TAU, W, LDW ) ! ! -- LAPACK auxiliary routine (version 3.1) -- ! Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. ! November 2006 ! ! .. Scalar Arguments .. CHARACTER UPLO INTEGER LDA, LDW, N, NB ! .. ! .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), E( * ), TAU( * ), W( LDW, * ) ! .. ! ! Purpose ! ======= ! ! DLATRD reduces NB rows and columns of a real symmetric matrix A to ! symmetric tridiagonal form by an orthogonal similarity ! transformation Q' * A * Q, and returns the matrices V and W which are ! needed to apply the transformation to the unreduced part of A. ! ! If UPLO = 'U', DLATRD reduces the last NB rows and columns of a ! matrix, of which the upper triangle is supplied; ! if UPLO = 'L', DLATRD reduces the first NB rows and columns of a ! matrix, of which the lower triangle is supplied. ! ! This is an auxiliary routine called by DSYTRD. ! ! Arguments ! ========= ! ! UPLO (input) CHARACTER*1 ! Specifies whether the upper or lower triangular part of the ! symmetric matrix A is stored: ! = 'U': Upper triangular ! = 'L': Lower triangular ! ! N (input) INTEGER ! The order of the matrix A. ! ! NB (input) INTEGER ! The number of rows and columns to be reduced. ! ! A (input/output) DOUBLE PRECISION array, dimension (LDA,N) ! On entry, the symmetric matrix A. If UPLO = 'U', the leading ! n-by-n upper triangular part of A contains the upper ! triangular part of the matrix A, and the strictly lower ! triangular part of A is not referenced. If UPLO = 'L', the ! leading n-by-n lower triangular part of A contains the lower ! triangular part of the matrix A, and the strictly upper ! triangular part of A is not referenced. ! On exit: ! if UPLO = 'U', the last NB columns have been reduced to ! tridiagonal form, with the diagonal elements overwriting ! the diagonal elements of A; the elements above the diagonal ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors; ! if UPLO = 'L', the first NB columns have been reduced to ! tridiagonal form, with the diagonal elements overwriting ! the diagonal elements of A; the elements below the diagonal ! with the array TAU, represent the orthogonal matrix Q as a ! product of elementary reflectors. ! See Further Details. ! ! LDA (input) INTEGER ! The leading dimension of the array A. LDA >= (1,N). ! ! E (output) DOUBLE PRECISION array, dimension (N-1) ! If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal ! elements of the last NB columns of the reduced matrix; ! if UPLO = 'L', E(1:nb) contains the subdiagonal elements of ! the first NB columns of the reduced matrix. ! ! TAU (output) DOUBLE PRECISION array, dimension (N-1) ! The scalar factors of the elementary reflectors, stored in ! TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'. ! See Further Details. ! ! W (output) DOUBLE PRECISION array, dimension (LDW,NB) ! The n-by-nb matrix W required to update the unreduced part ! of A. ! ! LDW (input) INTEGER ! The leading dimension of the array W. LDW >= max(1,N). ! ! Further Details ! =============== ! ! If UPLO = 'U', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(n) H(n-1) . . . H(n-nb+1). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i), ! and tau in TAU(i-1). ! ! If UPLO = 'L', the matrix Q is represented as a product of elementary ! reflectors ! ! Q = H(1) H(2) . . . H(nb). ! ! Each H(i) has the form ! ! H(i) = I - tau * v * v' ! ! where tau is a real scalar, and v is a real vector with ! v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i), ! and tau in TAU(i). ! ! The elements of the vectors v together form the n-by-nb matrix V ! which is needed, with W, to apply the transformation to the unreduced ! part of the matrix, using a symmetric rank-2k update of the form: ! A := A - V*W' - W*V'. ! ! The contents of A on exit are illustrated by the following examples ! with n = 5 and nb = 2: ! ! if UPLO = 'U': if UPLO = 'L': ! ! ( a a a v4 v5 ) ( d ) ! ( a a v4 v5 ) ( 1 d ) ! ( a 1 v5 ) ( v1 1 a ) ! ( d 1 ) ( v1 v2 a a ) ! ( d ) ( v1 v2 a a a ) ! ! where d denotes a diagonal element of the reduced matrix, a denotes ! an element of the original matrix that is unchanged, and vi denotes ! an element of the vector defining H(i). ! ! ===================================================================== ! ! .. Parameters .. DOUBLE PRECISION ZERO, ONE, HALF PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0, HALF = 0.5D+0 ) ! .. ! .. Local Scalars .. INTEGER I, IW DOUBLE PRECISION ALPHA ! .. ! .. External Subroutines .. ! EXTERNAL DAXPY, DGEMV, DLARFG, DSCAL, DSYMV ! .. ! .. External Functions .. ! LOGICAL LSAME ! DOUBLE PRECISION DDOT ! EXTERNAL LSAME, DDOT ! .. ! .. Intrinsic Functions .. INTRINSIC MIN ! .. ! .. Executable Statements .. ! ! Quick return if possible ! IF( N.LE.0 ) & RETURN ! IF( LSAME( UPLO, 'U' ) ) THEN ! ! Reduce last NB columns of upper triangle ! DO 10 I = N, N - NB + 1, -1 IW = I - N + NB IF( I.LT.N ) THEN ! ! Update A(1:i,i) ! CALL DGEMV( 'No transpose', I, N-I, -ONE, A( 1, I+1 ), & LDA, W( I, IW+1 ), LDW, ONE, A( 1, I ), 1 ) CALL DGEMV( 'No transpose', I, N-I, -ONE, W( 1, IW+1 ), & LDW, A( I, I+1 ), LDA, ONE, A( 1, I ), 1 ) END IF IF( I.GT.1 ) THEN ! ! Generate elementary reflector H(i) to annihilate ! A(1:i-2,i) ! CALL DLARFG( I-1, A( I-1, I ), A( 1, I ), 1, TAU( I-1 ) ) E( I-1 ) = A( I-1, I ) A( I-1, I ) = ONE ! ! Compute W(1:i-1,i) ! CALL DSYMV( 'Upper', I-1, ONE, A, LDA, A( 1, I ), 1, & ZERO, W( 1, IW ), 1 ) IF( I.LT.N ) THEN CALL DGEMV( 'Transpose', I-1, N-I, ONE, W( 1, IW+1 ), & LDW, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, -ONE, & A( 1, I+1 ), LDA, W( I+1, IW ), 1, ONE, & W( 1, IW ), 1 ) CALL DGEMV( 'Transpose', I-1, N-I, ONE, A( 1, I+1 ), & LDA, A( 1, I ), 1, ZERO, W( I+1, IW ), 1 ) CALL DGEMV( 'No transpose', I-1, N-I, -ONE, & W( 1, IW+1 ), LDW, W( I+1, IW ), 1, ONE, & W( 1, IW ), 1 ) END IF CALL DSCAL( I-1, TAU( I-1 ), W( 1, IW ), 1 ) ALPHA = -HALF*TAU( I-1 )*DDOT( I-1, W( 1, IW ), 1, & A( 1, I ), 1 ) CALL DAXPY( I-1, ALPHA, A( 1, I ), 1, W( 1, IW ), 1 ) END IF ! 10 CONTINUE ELSE ! ! Reduce first NB columns of lower triangle ! DO 20 I = 1, NB ! ! Update A(i:n,i) ! CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, A( I, 1 ), & LDA, W( I, 1 ), LDW, ONE, A( I, I ), 1 ) CALL DGEMV( 'No transpose', N-I+1, I-1, -ONE, W( I, 1 ), & LDW, A( I, 1 ), LDA, ONE, A( I, I ), 1 ) IF( I.LT.N ) THEN ! ! Generate elementary reflector H(i) to annihilate ! A(i+2:n,i) ! CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1, & TAU( I ) ) E( I ) = A( I+1, I ) A( I+1, I ) = ONE ! ! Compute W(i+1:n,i) ! CALL DSYMV( 'Lower', N-I, ONE, A( I+1, I+1 ), LDA, & A( I+1, I ), 1, ZERO, W( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I-1, ONE, W( I+1, 1 ), LDW, & A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, A( I+1, 1 ), & LDA, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL DGEMV( 'Transpose', N-I, I-1, ONE, A( I+1, 1 ), LDA, & A( I+1, I ), 1, ZERO, W( 1, I ), 1 ) CALL DGEMV( 'No transpose', N-I, I-1, -ONE, W( I+1, 1 ), & LDW, W( 1, I ), 1, ONE, W( I+1, I ), 1 ) CALL DSCAL( N-I, TAU( I ), W( I+1, I ), 1 ) ALPHA = -HALF*TAU( I )*DDOT( N-I, W( I+1, I ), 1, & A( I+1, I ), 1 ) CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, W( I+1, I ), 1 ) END IF ! 20 CONTINUE END IF ! RETURN ! ! End of DLATRD ! END SUBROUTINE DLATRD