SUBROUTINE DLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) * * -- LAPACK auxiliary routine (version 3.2.1) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * -- April 2009 -- * * .. Scalar Arguments .. INTEGER K, LDA, LDT, LDY, N, NB * .. * .. Array Arguments .. DOUBLE PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), \$ Y( LDY, NB ) * .. * * Purpose * ======= * * DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) * matrix A so that elements below the k-th subdiagonal are zero. The * reduction is performed by an orthogonal similarity transformation * Q' * A * Q. The routine returns the matrices V and T which determine * Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. * * This is an auxiliary routine called by DGEHRD. * * Arguments * ========= * * N (input) INTEGER * The order of the matrix A. * * K (input) INTEGER * The offset for the reduction. Elements below the k-th * subdiagonal in the first NB columns are reduced to zero. * K < N. * * NB (input) INTEGER * The number of columns to be reduced. * * A (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1) * On entry, the n-by-(n-k+1) general matrix A. * On exit, the elements on and above the k-th subdiagonal in * the first NB columns are overwritten with the corresponding * elements of the reduced matrix; the elements below the k-th * subdiagonal, with the array TAU, represent the matrix Q as a * product of elementary reflectors. The other columns of A are * unchanged. See Further Details. * * LDA (input) INTEGER * The leading dimension of the array A. LDA >= max(1,N). * * TAU (output) DOUBLE PRECISION array, dimension (NB) * The scalar factors of the elementary reflectors. See Further * Details. * * T (output) DOUBLE PRECISION array, dimension (LDT,NB) * The upper triangular matrix T. * * LDT (input) INTEGER * The leading dimension of the array T. LDT >= NB. * * Y (output) DOUBLE PRECISION array, dimension (LDY,NB) * The n-by-nb matrix Y. * * LDY (input) INTEGER * The leading dimension of the array Y. LDY >= N. * * Further Details * =============== * * The matrix Q is represented as a product of nb 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+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in * A(i+k+1:n,i), and tau in TAU(i). * * The elements of the vectors v together form the (n-k+1)-by-nb matrix * V which is needed, with T and Y, to apply the transformation to the * unreduced part of the matrix, using an update of the form: * A := (I - V*T*V') * (A - Y*V'). * * The contents of A on exit are illustrated by the following example * with n = 7, k = 3 and nb = 2: * * ( a a a a a ) * ( a a a a a ) * ( a a a a a ) * ( h h a a a ) * ( v1 h a a a ) * ( v1 v2 a a a ) * ( v1 v2 a a a ) * * where a denotes an element of the original matrix A, h denotes a * modified element of the upper Hessenberg matrix H, and vi denotes an * element of the vector defining H(i). * * This subroutine is a slight modification of LAPACK-3.0's DLAHRD * incorporating improvements proposed by Quintana-Orti and Van de * Gejin. Note that the entries of A(1:K,2:NB) differ from those * returned by the original LAPACK-3.0's DLAHRD routine. (This * subroutine is not backward compatible with LAPACK-3.0's DLAHRD.) * * References * ========== * * Gregorio Quintana-Orti and Robert van de Geijn, "Improving the * performance of reduction to Hessenberg form," ACM Transactions on * Mathematical Software, 32(2):180-194, June 2006. * * ===================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, \$ ONE = 1.0D+0 ) * .. * .. Local Scalars .. INTEGER I DOUBLE PRECISION EI * .. * .. External Subroutines .. EXTERNAL DAXPY, DCOPY, DGEMM, DGEMV, DLACPY, \$ DLARFG, DSCAL, DTRMM, DTRMV * .. * .. Intrinsic Functions .. INTRINSIC MIN * .. * .. Executable Statements .. * * Quick return if possible * IF( N.LE.1 ) \$ RETURN * DO 10 I = 1, NB IF( I.GT.1 ) THEN * * Update A(K+1:N,I) * * Update I-th column of A - Y * V' * CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, Y(K+1,1), LDY, \$ A( K+I-1, 1 ), LDA, ONE, A( K+1, I ), 1 ) * * Apply I - V * T' * V' to this column (call it b) from the * left, using the last column of T as workspace * * Let V = ( V1 ) and b = ( b1 ) (first I-1 rows) * ( V2 ) ( b2 ) * * where V1 is unit lower triangular * * w := V1' * b1 * CALL DCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) CALL DTRMV( 'Lower', 'Transpose', 'UNIT', \$ I-1, A( K+1, 1 ), \$ LDA, T( 1, NB ), 1 ) * * w := w + V2'*b2 * CALL DGEMV( 'Transpose', N-K-I+1, I-1, \$ ONE, A( K+I, 1 ), \$ LDA, A( K+I, I ), 1, ONE, T( 1, NB ), 1 ) * * w := T'*w * CALL DTRMV( 'Upper', 'Transpose', 'NON-UNIT', \$ I-1, T, LDT, \$ T( 1, NB ), 1 ) * * b2 := b2 - V2*w * CALL DGEMV( 'NO TRANSPOSE', N-K-I+1, I-1, -ONE, \$ A( K+I, 1 ), \$ LDA, T( 1, NB ), 1, ONE, A( K+I, I ), 1 ) * * b1 := b1 - V1*w * CALL DTRMV( 'Lower', 'NO TRANSPOSE', \$ 'UNIT', I-1, \$ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) CALL DAXPY( I-1, -ONE, T( 1, NB ), 1, A( K+1, I ), 1 ) * A( K+I-1, I-1 ) = EI END IF * * Generate the elementary reflector H(I) to annihilate * A(K+I+1:N,I) * CALL DLARFG( N-K-I+1, A( K+I, I ), A( MIN( K+I+1, N ), I ), 1, \$ TAU( I ) ) EI = A( K+I, I ) A( K+I, I ) = ONE * * Compute Y(K+1:N,I) * CALL DGEMV( 'NO TRANSPOSE', N-K, N-K-I+1, \$ ONE, A( K+1, I+1 ), \$ LDA, A( K+I, I ), 1, ZERO, Y( K+1, I ), 1 ) CALL DGEMV( 'Transpose', N-K-I+1, I-1, \$ ONE, A( K+I, 1 ), LDA, \$ A( K+I, I ), 1, ZERO, T( 1, I ), 1 ) CALL DGEMV( 'NO TRANSPOSE', N-K, I-1, -ONE, \$ Y( K+1, 1 ), LDY, \$ T( 1, I ), 1, ONE, Y( K+1, I ), 1 ) CALL DSCAL( N-K, TAU( I ), Y( K+1, I ), 1 ) * * Compute T(1:I,I) * CALL DSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) CALL DTRMV( 'Upper', 'No Transpose', 'NON-UNIT', \$ I-1, T, LDT, \$ T( 1, I ), 1 ) T( I, I ) = TAU( I ) * 10 CONTINUE A( K+NB, NB ) = EI * * Compute Y(1:K,1:NB) * CALL DLACPY( 'ALL', K, NB, A( 1, 2 ), LDA, Y, LDY ) CALL DTRMM( 'RIGHT', 'Lower', 'NO TRANSPOSE', \$ 'UNIT', K, NB, \$ ONE, A( K+1, 1 ), LDA, Y, LDY ) IF( N.GT.K+NB ) \$ CALL DGEMM( 'NO TRANSPOSE', 'NO TRANSPOSE', K, \$ NB, N-K-NB, ONE, \$ A( 1, 2+NB ), LDA, A( K+1+NB, 1 ), LDA, ONE, Y, \$ LDY ) CALL DTRMM( 'RIGHT', 'Upper', 'NO TRANSPOSE', \$ 'NON-UNIT', K, NB, \$ ONE, T, LDT, Y, LDY ) * RETURN * * End of DLAHR2 * END