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Diffstat (limited to 'SRC/zlahrd.f')
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diff --git a/SRC/zlahrd.f b/SRC/zlahrd.f new file mode 100644 index 00000000..e7eb9de9 --- /dev/null +++ b/SRC/zlahrd.f @@ -0,0 +1,213 @@ + SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) +* +* -- LAPACK auxiliary routine (version 3.1) -- +* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd.. +* November 2006 +* +* .. Scalar Arguments .. + INTEGER K, LDA, LDT, LDY, N, NB +* .. +* .. Array Arguments .. + COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), + $ Y( LDY, NB ) +* .. +* +* Purpose +* ======= +* +* ZLAHRD reduces the first NB columns of a complex general n-by-(n-k+1) +* matrix A so that elements below the k-th subdiagonal are zero. The +* reduction is performed by a unitary 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 OBSOLETE auxiliary routine. +* This routine will be 'deprecated' in a future release. +* Please use the new routine ZLAHR2 instead. +* +* 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. +* +* NB (input) INTEGER +* The number of columns to be reduced. +* +* A (input/output) COMPLEX*16 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) COMPLEX*16 array, dimension (NB) +* The scalar factors of the elementary reflectors. See Further +* Details. +* +* T (output) COMPLEX*16 array, dimension (LDT,NB) +* The upper triangular matrix T. +* +* LDT (input) INTEGER +* The leading dimension of the array T. LDT >= NB. +* +* Y (output) COMPLEX*16 array, dimension (LDY,NB) +* The n-by-nb matrix Y. +* +* LDY (input) INTEGER +* The leading dimension of the array Y. LDY >= max(1,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 complex scalar, and v is a complex 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 h a a a ) +* ( a h a a a ) +* ( a h 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). +* +* ===================================================================== +* +* .. Parameters .. + COMPLEX*16 ZERO, ONE + PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ), + $ ONE = ( 1.0D+0, 0.0D+0 ) ) +* .. +* .. Local Scalars .. + INTEGER I + COMPLEX*16 EI +* .. +* .. External Subroutines .. + EXTERNAL ZAXPY, ZCOPY, ZGEMV, ZLACGV, ZLARFG, ZSCAL, + $ ZTRMV +* .. +* .. 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(1:n,i) +* +* Compute i-th column of A - Y * V' +* + CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) + CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, + $ A( K+I-1, 1 ), LDA, ONE, A( 1, I ), 1 ) + CALL ZLACGV( I-1, A( K+I-1, 1 ), LDA ) +* +* 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 ZCOPY( I-1, A( K+1, I ), 1, T( 1, NB ), 1 ) + CALL ZTRMV( 'Lower', 'Conjugate transpose', 'Unit', I-1, + $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) +* +* w := w + V2'*b2 +* + CALL ZGEMV( 'Conjugate 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 ZTRMV( 'Upper', 'Conjugate transpose', 'Non-unit', I-1, + $ T, LDT, T( 1, NB ), 1 ) +* +* b2 := b2 - V2*w +* + CALL ZGEMV( '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 ZTRMV( 'Lower', 'No transpose', 'Unit', I-1, + $ A( K+1, 1 ), LDA, T( 1, NB ), 1 ) + CALL ZAXPY( 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) +* + EI = A( K+I, I ) + CALL ZLARFG( N-K-I+1, EI, A( MIN( K+I+1, N ), I ), 1, + $ TAU( I ) ) + A( K+I, I ) = ONE +* +* Compute Y(1:n,i) +* + CALL ZGEMV( 'No transpose', N, N-K-I+1, ONE, A( 1, I+1 ), LDA, + $ A( K+I, I ), 1, ZERO, Y( 1, I ), 1 ) + CALL ZGEMV( 'Conjugate transpose', N-K-I+1, I-1, ONE, + $ A( K+I, 1 ), LDA, A( K+I, I ), 1, ZERO, T( 1, I ), + $ 1 ) + CALL ZGEMV( 'No transpose', N, I-1, -ONE, Y, LDY, T( 1, I ), 1, + $ ONE, Y( 1, I ), 1 ) + CALL ZSCAL( N, TAU( I ), Y( 1, I ), 1 ) +* +* Compute T(1:i,i) +* + CALL ZSCAL( I-1, -TAU( I ), T( 1, I ), 1 ) + CALL ZTRMV( '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 +* + RETURN +* +* End of ZLAHRD +* + END |