# lahr2 - Man Page

lahr2: step in gehrd

## Synopsis

### Functions

subroutine clahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
CLAHR2 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.
subroutine dlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
DLAHR2 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.
subroutine slahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
SLAHR2 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.
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 Documentation

### subroutine clahr2 (integer n, integer k, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( nb ) tau, complex, dimension( ldt, nb ) t, integer ldt, complex, dimension( ldy, nb ) y, integer ldy)

CLAHR2 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:

``` CLAHR2 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 CGEHRD.```
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 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 array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.```

T

```          T is COMPLEX 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 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

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 CLAHRD
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 CLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's CLAHRD.)```
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 clahr2.f.

### subroutine dlahr2 (integer n, integer k, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( nb ) tau, double precision, dimension( ldt, nb ) t, integer ldt, double precision, dimension( ldy, nb ) y, integer ldy)

DLAHR2 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:

``` DLAHR2 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.

This is an auxiliary routine called by DGEHRD.```
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.```

T

```          T is DOUBLE PRECISION 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 DOUBLE PRECISION 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

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   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 DLAHRD
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 DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)```
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 dlahr2.f.

### subroutine slahr2 (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)

SLAHR2 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:

``` SLAHR2 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.

This is an auxiliary routine called by SGEHRD.```
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 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

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   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 SLAHRD
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 SLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's SLAHRD.)```
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 slahr2.f.

### 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

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.

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