# lasr - Man Page

lasr: apply series of plane rotations

## Synopsis

### Functions

subroutine clasr (side, pivot, direct, m, n, c, s, a, lda)
CLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine dlasr (side, pivot, direct, m, n, c, s, a, lda)
DLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine slasr (side, pivot, direct, m, n, c, s, a, lda)
SLASR applies a sequence of plane rotations to a general rectangular matrix.
subroutine zlasr (side, pivot, direct, m, n, c, s, a, lda)
ZLASR applies a sequence of plane rotations to a general rectangular matrix.

## Function Documentation

### subroutine clasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, complex, dimension( lda, * ) a, integer lda)

CLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

``` CLASR applies a sequence of real plane rotations to a complex matrix
A, from either the left or the right.

When SIDE = 'L', the transformation takes the form

A := P*A

and when SIDE = 'R', the transformation takes the form

A := A*P**T

where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.

When DIRECT = 'F' (Forward sequence), then

P = P(z-1) * ... * P(2) * P(1)

and when DIRECT = 'B' (Backward sequence), then

P = P(1) * P(2) * ... * P(z-1)

where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

R(k) = (  c(k)  s(k) )
= ( -s(k)  c(k) ).

When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form

P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )

where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.

When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form

P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )

where R(k) appears in rows and columns 1 and k+1.

Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form

P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )

where R(k) appears in rows and columns k and z.  The rotations are
performed without ever forming P(k) explicitly.```
Parameters

SIDE

```          SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L':  Left, compute A := P*A
= 'R':  Right, compute A:= A*P**T```

PIVOT

```          PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V':  Variable pivot, the plane (k,k+1)
= 'T':  Top pivot, the plane (1,k+1)
= 'B':  Bottom pivot, the plane (k,z)```

DIRECT

```          DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
= 'B':  Backward, P = P(1)*P(2)*...*P(z-1)```

M

```          M is INTEGER
The number of rows of the matrix A.  If m <= 1, an immediate
return is effected.```

N

```          N is INTEGER
The number of columns of the matrix A.  If n <= 1, an
immediate return is effected.```

C

```          C is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.```

S

```          S is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations.  The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = (  c(k)  s(k) )
( -s(k)  c(k) ).```

A

```          A is COMPLEX array, dimension (LDA,N)
The M-by-N matrix A.  On exit, A is overwritten by P*A if
SIDE = 'R' or by A*P**T if SIDE = 'L'.```

LDA

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

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 199 of file clasr.f.

### subroutine dlasr (character side, character pivot, character direct, integer m, integer n, double precision, dimension( * ) c, double precision, dimension( * ) s, double precision, dimension( lda, * ) a, integer lda)

DLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

``` DLASR applies a sequence of plane rotations to a real matrix A,
from either the left or the right.

When SIDE = 'L', the transformation takes the form

A := P*A

and when SIDE = 'R', the transformation takes the form

A := A*P**T

where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.

When DIRECT = 'F' (Forward sequence), then

P = P(z-1) * ... * P(2) * P(1)

and when DIRECT = 'B' (Backward sequence), then

P = P(1) * P(2) * ... * P(z-1)

where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

R(k) = (  c(k)  s(k) )
= ( -s(k)  c(k) ).

When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form

P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )

where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.

When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form

P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )

where R(k) appears in rows and columns 1 and k+1.

Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form

P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )

where R(k) appears in rows and columns k and z.  The rotations are
performed without ever forming P(k) explicitly.```
Parameters

SIDE

```          SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L':  Left, compute A := P*A
= 'R':  Right, compute A:= A*P**T```

PIVOT

```          PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V':  Variable pivot, the plane (k,k+1)
= 'T':  Top pivot, the plane (1,k+1)
= 'B':  Bottom pivot, the plane (k,z)```

DIRECT

```          DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
= 'B':  Backward, P = P(1)*P(2)*...*P(z-1)```

M

```          M is INTEGER
The number of rows of the matrix A.  If m <= 1, an immediate
return is effected.```

N

```          N is INTEGER
The number of columns of the matrix A.  If n <= 1, an
immediate return is effected.```

C

```          C is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.```

S

```          S is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations.  The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = (  c(k)  s(k) )
( -s(k)  c(k) ).```

A

```          A is DOUBLE PRECISION array, dimension (LDA,N)
The M-by-N matrix A.  On exit, A is overwritten by P*A if
SIDE = 'L' or by A*P**T if SIDE = 'R'.```

LDA

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

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 198 of file dlasr.f.

### subroutine slasr (character side, character pivot, character direct, integer m, integer n, real, dimension( * ) c, real, dimension( * ) s, real, dimension( lda, * ) a, integer lda)

SLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

``` SLASR applies a sequence of plane rotations to a real matrix A,
from either the left or the right.

When SIDE = 'L', the transformation takes the form

A := P*A

and when SIDE = 'R', the transformation takes the form

A := A*P**T

where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.

When DIRECT = 'F' (Forward sequence), then

P = P(z-1) * ... * P(2) * P(1)

and when DIRECT = 'B' (Backward sequence), then

P = P(1) * P(2) * ... * P(z-1)

where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

R(k) = (  c(k)  s(k) )
= ( -s(k)  c(k) ).

When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form

P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )

where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.

When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form

P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )

where R(k) appears in rows and columns 1 and k+1.

Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form

P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )

where R(k) appears in rows and columns k and z.  The rotations are
performed without ever forming P(k) explicitly.```
Parameters

SIDE

```          SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L':  Left, compute A := P*A
= 'R':  Right, compute A:= A*P**T```

PIVOT

```          PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V':  Variable pivot, the plane (k,k+1)
= 'T':  Top pivot, the plane (1,k+1)
= 'B':  Bottom pivot, the plane (k,z)```

DIRECT

```          DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
= 'B':  Backward, P = P(1)*P(2)*...*P(z-1)```

M

```          M is INTEGER
The number of rows of the matrix A.  If m <= 1, an immediate
return is effected.```

N

```          N is INTEGER
The number of columns of the matrix A.  If n <= 1, an
immediate return is effected.```

C

```          C is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.```

S

```          S is REAL array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations.  The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = (  c(k)  s(k) )
( -s(k)  c(k) ).```

A

```          A is REAL array, dimension (LDA,N)
The M-by-N matrix A.  On exit, A is overwritten by P*A if
SIDE = 'R' or by A*P**T if SIDE = 'L'.```

LDA

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

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Definition at line 198 of file slasr.f.

### subroutine zlasr (character side, character pivot, character direct, integer m, integer n, double precision, dimension( * ) c, double precision, dimension( * ) s, complex*16, dimension( lda, * ) a, integer lda)

ZLASR applies a sequence of plane rotations to a general rectangular matrix.

Purpose:

``` ZLASR applies a sequence of real plane rotations to a complex matrix
A, from either the left or the right.

When SIDE = 'L', the transformation takes the form

A := P*A

and when SIDE = 'R', the transformation takes the form

A := A*P**T

where P is an orthogonal matrix consisting of a sequence of z plane
rotations, with z = M when SIDE = 'L' and z = N when SIDE = 'R',
and P**T is the transpose of P.

When DIRECT = 'F' (Forward sequence), then

P = P(z-1) * ... * P(2) * P(1)

and when DIRECT = 'B' (Backward sequence), then

P = P(1) * P(2) * ... * P(z-1)

where P(k) is a plane rotation matrix defined by the 2-by-2 rotation

R(k) = (  c(k)  s(k) )
= ( -s(k)  c(k) ).

When PIVOT = 'V' (Variable pivot), the rotation is performed
for the plane (k,k+1), i.e., P(k) has the form

P(k) = (  1                                            )
(       ...                                     )
(              1                                )
(                   c(k)  s(k)                  )
(                  -s(k)  c(k)                  )
(                                1              )
(                                     ...       )
(                                            1  )

where R(k) appears as a rank-2 modification to the identity matrix in
rows and columns k and k+1.

When PIVOT = 'T' (Top pivot), the rotation is performed for the
plane (1,k+1), so P(k) has the form

P(k) = (  c(k)                    s(k)                 )
(         1                                     )
(              ...                              )
(                     1                         )
( -s(k)                    c(k)                 )
(                                 1             )
(                                      ...      )
(                                             1 )

where R(k) appears in rows and columns 1 and k+1.

Similarly, when PIVOT = 'B' (Bottom pivot), the rotation is
performed for the plane (k,z), giving P(k) the form

P(k) = ( 1                                             )
(      ...                                      )
(             1                                 )
(                  c(k)                    s(k) )
(                         1                     )
(                              ...              )
(                                     1         )
(                 -s(k)                    c(k) )

where R(k) appears in rows and columns k and z.  The rotations are
performed without ever forming P(k) explicitly.```
Parameters

SIDE

```          SIDE is CHARACTER*1
Specifies whether the plane rotation matrix P is applied to
A on the left or the right.
= 'L':  Left, compute A := P*A
= 'R':  Right, compute A:= A*P**T```

PIVOT

```          PIVOT is CHARACTER*1
Specifies the plane for which P(k) is a plane rotation
matrix.
= 'V':  Variable pivot, the plane (k,k+1)
= 'T':  Top pivot, the plane (1,k+1)
= 'B':  Bottom pivot, the plane (k,z)```

DIRECT

```          DIRECT is CHARACTER*1
Specifies whether P is a forward or backward sequence of
plane rotations.
= 'F':  Forward, P = P(z-1)*...*P(2)*P(1)
= 'B':  Backward, P = P(1)*P(2)*...*P(z-1)```

M

```          M is INTEGER
The number of rows of the matrix A.  If m <= 1, an immediate
return is effected.```

N

```          N is INTEGER
The number of columns of the matrix A.  If n <= 1, an
immediate return is effected.```

C

```          C is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The cosines c(k) of the plane rotations.```

S

```          S is DOUBLE PRECISION array, dimension
(M-1) if SIDE = 'L'
(N-1) if SIDE = 'R'
The sines s(k) of the plane rotations.  The 2-by-2 plane
rotation part of the matrix P(k), R(k), has the form
R(k) = (  c(k)  s(k) )
( -s(k)  c(k) ).```

A

```          A is COMPLEX*16 array, dimension (LDA,N)
The M-by-N matrix A.  On exit, A is overwritten by P*A if
SIDE = 'R' or by A*P**T if SIDE = 'L'.```

LDA

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

Univ. of Tennessee

Univ. of California Berkeley