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---
:name: clahr2
:md5sum: 830e928c511e3ec5d35e891b95ae225c
:category: :subroutine
:arguments:
- n:
:type: integer
:intent: input
- k:
:type: integer
:intent: input
- nb:
:type: integer
:intent: input
- a:
:type: complex
:intent: input/output
:dims:
- lda
- n-k+1
- lda:
:type: integer
:intent: input
- tau:
:type: complex
:intent: output
:dims:
- MAX(1,nb)
- t:
:type: complex
:intent: output
:dims:
- ldt
- MAX(1,nb)
- ldt:
:type: integer
:intent: input
- y:
:type: complex
:intent: output
:dims:
- ldy
- MAX(1,nb)
- ldy:
:type: integer
:intent: input
:substitutions:
ldy: n
ldt: nb
:fortran_help: " SUBROUTINE CLAHR2( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )\n\n\
* Purpose\n\
* =======\n\
*\n\
* CLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)\n\
* matrix A so that elements below the k-th subdiagonal are zero. The\n\
* reduction is performed by an unitary similarity transformation\n\
* Q' * A * Q. The routine returns the matrices V and T which determine\n\
* Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T.\n\
*\n\
* This is an auxiliary routine called by CGEHRD.\n\
*\n\n\
* Arguments\n\
* =========\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrix A.\n\
*\n\
* K (input) INTEGER\n\
* The offset for the reduction. Elements below the k-th\n\
* subdiagonal in the first NB columns are reduced to zero.\n\
* K < N.\n\
*\n\
* NB (input) INTEGER\n\
* The number of columns to be reduced.\n\
*\n\
* A (input/output) COMPLEX array, dimension (LDA,N-K+1)\n\
* On entry, the n-by-(n-k+1) general matrix A.\n\
* On exit, the elements on and above the k-th subdiagonal in\n\
* the first NB columns are overwritten with the corresponding\n\
* elements of the reduced matrix; the elements below the k-th\n\
* subdiagonal, with the array TAU, represent the matrix Q as a\n\
* product of elementary reflectors. The other columns of A are\n\
* unchanged. See Further Details.\n\
*\n\
* LDA (input) INTEGER\n\
* The leading dimension of the array A. LDA >= max(1,N).\n\
*\n\
* TAU (output) COMPLEX array, dimension (NB)\n\
* The scalar factors of the elementary reflectors. See Further\n\
* Details.\n\
*\n\
* T (output) COMPLEX array, dimension (LDT,NB)\n\
* The upper triangular matrix T.\n\
*\n\
* LDT (input) INTEGER\n\
* The leading dimension of the array T. LDT >= NB.\n\
*\n\
* Y (output) COMPLEX array, dimension (LDY,NB)\n\
* The n-by-nb matrix Y.\n\
*\n\
* LDY (input) INTEGER\n\
* The leading dimension of the array Y. LDY >= N.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* The matrix Q is represented as a product of nb elementary reflectors\n\
*\n\
* Q = H(1) H(2) . . . H(nb).\n\
*\n\
* Each H(i) has the form\n\
*\n\
* H(i) = I - tau * v * v'\n\
*\n\
* where tau is a complex scalar, and v is a complex vector with\n\
* v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in\n\
* A(i+k+1:n,i), and tau in TAU(i).\n\
*\n\
* The elements of the vectors v together form the (n-k+1)-by-nb matrix\n\
* V which is needed, with T and Y, to apply the transformation to the\n\
* unreduced part of the matrix, using an update of the form:\n\
* A := (I - V*T*V') * (A - Y*V').\n\
*\n\
* The contents of A on exit are illustrated by the following example\n\
* with n = 7, k = 3 and nb = 2:\n\
*\n\
* ( a a a a a )\n\
* ( a a a a a )\n\
* ( a a a a a )\n\
* ( h h a a a )\n\
* ( v1 h a a a )\n\
* ( v1 v2 a a a )\n\
* ( v1 v2 a a a )\n\
*\n\
* where a denotes an element of the original matrix A, h denotes a\n\
* modified element of the upper Hessenberg matrix H, and vi denotes an\n\
* element of the vector defining H(i).\n\
*\n\
* This subroutine is a slight modification of LAPACK-3.0's DLAHRD\n\
* incorporating improvements proposed by Quintana-Orti and Van de\n\
* Gejin. Note that the entries of A(1:K,2:NB) differ from those\n\
* returned by the original LAPACK-3.0's DLAHRD routine. (This\n\
* subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)\n\
*\n\
* References\n\
* ==========\n\
*\n\
* Gregorio Quintana-Orti and Robert van de Geijn, \"Improving the\n\
* performance of reduction to Hessenberg form,\" ACM Transactions on\n\
* Mathematical Software, 32(2):180-194, June 2006.\n\
*\n\
* =====================================================================\n\
*\n"
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