---
name: "pclahrd"
section: "3"
platform: "linux"
package: "scalapack-doc"
version: "1.5-11"
release: "Debian 11 (Bullseye)"
platforms: ["linux"]
url: "https://new.carta.tech/pclahrd"
---

# pclahrd(3)

reduce the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero

## NAME

PCLAHRD - reduce the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero

## SYNOPSIS

- SUBROUTINE PCLAHRD(

  N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK )

- INTEGER

  IA, IY, JA, JY, K, N, NB

- INTEGER

  DESCA( \* ), DESCY( \* )

- COMPLEX

  A( \* ), T( \* ), TAU( \* ), WORK( \* ), Y( \* )

## PURPOSE

PCLAHRD reduces the first NB columns of a complex general N-by-(N-K+1) distributed matrix A(IA:IA+N-1,JA:JA+N-K) so that elements below the k-th subdiagonal are zero. The reduction is performed by an 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 auxiliary routine called by PCGEHRD. In the following comments sub( A ) denotes A(IA:IA+N-1,JA:JA+N-1).\

## ARGUMENTS

- N (global input) INTEGER

  The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N \>= 0.

- K (global input) INTEGER

  The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero.

- NB (global input) INTEGER

  The number of columns to be reduced.

- A (local input/local output) COMPLEX pointer into

  the local memory to an array of dimension (LLD\_A, LOCc(JA+N-K)). On entry, this array contains the the local pieces of the N-by-(N-K+1) general distributed matrix A(IA:IA+N-1,JA:JA+N-K). 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 distributed 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(IA:IA+N-1,JA:JA+N-K) are unchanged. See Further Details. IA      (global input) INTEGER The row index in the global array A indicating the first row of sub( A ).

- JA (global input) INTEGER

  The column index in the global array A indicating the first column of sub( A ).

- DESCA (global and local input) INTEGER array of dimension DLEN\_.

  The array descriptor for the distributed matrix A.

- TAU (local output) COMPLEX array, dimension LOCc(JA+N-2)

  The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A.

- T (local output) COMPLEX array, dimension (NB\_A,NB\_A)

  The upper triangular matrix T.

- Y (local output) COMPLEX pointer into the local memory

  to an array of dimension (LLD\_Y,NB\_A). On exit, this array contains the local pieces of the N-by-NB distributed matrix Y. LLD\_Y \>= LOCr(IA+N-1).

- IY (global input) INTEGER

  The row index in the global array Y indicating the first row of sub( Y ).

- JY (global input) INTEGER

  The column index in the global array Y indicating the first column of sub( Y ).

- DESCY (global and local input) INTEGER array of dimension DLEN\_.

  The array descriptor for the distributed matrix Y.

- WORK (local workspace) COMPLEX array, dimension (NB)

  ## 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(ia+i+k:ia+n-1,ja+i-1), and tau in TAU(ja+i-1).\

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(ia:ia+n-1,ja:ja+n-k) := (I-V\*T\*V')\*(A(ia:ia+n-1,ja:ja+n-k)-Y\*V').

The contents of A(ia:ia+n-1,ja:ja+n-k) 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(ia:ia+n-1,ja:ja+n-k), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).\