# slahrd.f - Man Page

SRC/DEPRECATED/slahrd.f

## Synopsis

### Functions/Subroutines

subroutine **slahrd** (n, k, nb, a, lda, tau, t, ldt, y, ldy)**SLAHRD** reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

## Function/Subroutine Documentation

### subroutine slahrd (integer n, integer k, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( nb ) tau, real, dimension( ldt, nb ) t, integer ldt, real, dimension( ldy, nb ) y, integer ldy)

**SLAHRD** reduces the first nb columns of a general rectangular matrix A so that elements below the k-th subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

**Purpose:**

This routine is deprecated and has been replaced by routine SLAHR2. SLAHRD 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**T * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T.

**Parameters***N*N is INTEGER The order of the matrix A.

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

*NB*NB is INTEGER The number of columns to be reduced.

*A*A is REAL 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*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*TAU*TAU is REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details.

*T*T is REAL array, dimension (LDT,NB) The upper triangular matrix T.

*LDT*LDT is INTEGER The leading dimension of the array T. LDT >= NB.

*Y*Y is REAL array, dimension (LDY,NB) The n-by-nb matrix Y.

*LDY*LDY is INTEGER The leading dimension of the array Y. LDY >= N.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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**T 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**T) * (A - Y*V**T). 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).

Definition at line **166** of file **slahrd.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page slahrd(3) is an alias of slahrd.f(3).