---
name: "zlahr2.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/zlahr2.f"
---

# zlahr2.f(3)

## NAME

SRC/zlahr2.f

## SYNOPSIS

### Functions/Subroutines

subroutine **zlahr2** (n, k, nb, a, lda, tau, t, ldt, y, ldy)\
 **ZLAHR2** reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

## Function/Subroutine Documentation

### subroutine zlahr2 (integer n, integer k, integer nb, complex\*16, dimension( lda, \* ) a, integer lda, complex\*16, dimension( nb ) tau, complex\*16, dimension( ldt, nb ) t, integer ldt, complex\*16, dimension( ldy, nb ) y, integer ldy)

**ZLAHR2** reduces the specified number of first columns of a general rectangular matrix A so that elements below the specified subdiagonal are zero, and returns auxiliary matrices which are needed to apply the transformation to the unreduced part of A.

**Purpose:**

```
!>
!> ZLAHR2 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 an unitary similarity transformation
!> Q**H * A * Q. The routine returns the matrices V and T which determine
!> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
!>
!> This is an auxiliary routine called by ZGEHRD.
!>
```

**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.
!>          K < N.
!>
```

*NB*

```
!>          NB is INTEGER
!>          The number of columns to be reduced.
!>
```

*A*

```
!>          A is 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*

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

*TAU*

```
!>          TAU is COMPLEX*16 array, dimension (NB)
!>          The scalar factors of the elementary reflectors. See Further
!>          Details.
!>
```

*T*

```
!>          T is COMPLEX*16 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 COMPLEX*16 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**H
!>
!>  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**H) * (A - Y*V**H).
!>
!>  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 ZLAHRD
!>  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 ZLAHRD routine. (This
!>  subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
!>
```

**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.

Definition at line **180** of file **zlahr2.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.