# lamtsqr - Man Page

lamtsqr: multiply by Q from latsqr

## Synopsis

### Functions

subroutine clamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
CLAMTSQR
subroutine dlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
DLAMTSQR
subroutine slamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
SLAMTSQR
subroutine zlamtsqr (side, trans, m, n, k, mb, nb, a, lda, t, ldt, c, ldc, work, lwork, info)
ZLAMTSQR

## Function Documentation

### subroutine clamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldt, * ) t, integer ldt, complex, dimension(ldc, * ) c, integer ldc, complex, dimension( * ) work, integer lwork, integer info)

CLAMTSQR

Purpose:

```      CLAMTSQR overwrites the general complex M-by-N matrix C with

SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'C':      Q**H * C       C * Q**H
where Q is a complex unitary matrix defined as the product
of blocked elementary reflectors computed by tall skinny
QR factorization (CLATSQR)```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'L': apply Q or Q**H from the Left;
= 'R': apply Q or Q**H from the Right.```

TRANS

```          TRANS is CHARACTER*1
= 'N':  No transpose, apply Q;
= 'C':  Conjugate Transpose, apply Q**H.```

M

```          M is INTEGER
The number of rows of the matrix A.  M >=0.```

N

```          N is INTEGER
The number of columns of the matrix C. N >= 0.```

K

```          K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0;```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.
MB > N. (must be the same as CLATSQR)```

NB

```          NB is INTEGER
The column block size to be used in the blocked QR.
N >= NB >= 1.```

A

```          A is COMPLEX array, dimension (LDA,K)
The i-th column must contain the vector which defines the
blockedelementary reflector H(i), for i = 1,2,...,k, as
returned by CLATSQR in the first k columns of
its array argument A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).```

T

```          T is COMPLEX array, dimension
( N * Number of blocks(CEIL(M-K/MB-K)),
The blocked upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See below
for further details.```

LDT

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

C

```          C is COMPLEX array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).```

WORK

`         (workspace) COMPLEX array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.

If SIDE = 'L', LWORK >= max(1,N)*NB;
if SIDE = 'R', LWORK >= max(1,MB)*NB.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

``` Tall-Skinny QR (TSQR) performs QR by a sequence of unitary transformations,
representing Q as a product of other unitary matrices
Q = Q(1) * Q(2) * . . . * Q(k)
where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
stored under the diagonal of rows 1:MB of A, and by upper triangular
block reflectors, stored in array T(1:LDT,1:N).

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
The last Q(k) may use fewer rows.

For more details of the overall algorithm, see the description of
Sequential TSQR in Section 2.2 of [1].

[1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
SIAM J. Sci. Comput, vol. 34, no. 1, 2012```

Definition at line 197 of file clamtsqr.f.

### subroutine dlamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension(ldc, * ) c, integer ldc, double precision, dimension( * ) work, integer lwork, integer info)

DLAMTSQR

Purpose:

```      DLAMTSQR overwrites the general real M-by-N matrix C with

SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'T':      Q**T * C       C * Q**T
where Q is a real orthogonal matrix defined as the product
of blocked elementary reflectors computed by tall skinny
QR factorization (DLATSQR)```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.```

TRANS

```          TRANS is CHARACTER*1
= 'N':  No transpose, apply Q;
= 'T':  Transpose, apply Q**T.```

M

```          M is INTEGER
The number of rows of the matrix A.  M >=0.```

N

```          N is INTEGER
The number of columns of the matrix C. N >= 0.```

K

```          K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0;```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.
MB > N. (must be the same as DLATSQR)```

NB

```          NB is INTEGER
The column block size to be used in the blocked QR.
N >= NB >= 1.```

A

```          A is DOUBLE PRECISION array, dimension (LDA,K)
The i-th column must contain the vector which defines the
blockedelementary reflector H(i), for i = 1,2,...,k, as
returned by DLATSQR in the first k columns of
its array argument A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).```

T

```          T is DOUBLE PRECISION array, dimension
( N * Number of blocks(CEIL(M-K/MB-K)),
The blocked upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See below
for further details.```

LDT

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

C

```          C is DOUBLE PRECISION array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).```

WORK

`         (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.

If SIDE = 'L', LWORK >= max(1,N)*NB;
if SIDE = 'R', LWORK >= max(1,MB)*NB.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

``` Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
representing Q as a product of other orthogonal matrices
Q = Q(1) * Q(2) * . . . * Q(k)
where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
stored under the diagonal of rows 1:MB of A, and by upper triangular
block reflectors, stored in array T(1:LDT,1:N).

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
The last Q(k) may use fewer rows.

For more details of the overall algorithm, see the description of
Sequential TSQR in Section 2.2 of [1].

[1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
SIAM J. Sci. Comput, vol. 34, no. 1, 2012```

Definition at line 197 of file dlamtsqr.f.

### subroutine slamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, real, dimension( lda, * ) a, integer lda, real, dimension( ldt, * ) t, integer ldt, real, dimension(ldc, * ) c, integer ldc, real, dimension( * ) work, integer lwork, integer info)

SLAMTSQR

Purpose:

```      SLAMTSQR overwrites the general real M-by-N matrix C with

SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'T':      Q**T * C       C * Q**T
where Q is a real orthogonal matrix defined as the product
of blocked elementary reflectors computed by tall skinny
QR factorization (SLATSQR)```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'L': apply Q or Q**T from the Left;
= 'R': apply Q or Q**T from the Right.```

TRANS

```          TRANS is CHARACTER*1
= 'N':  No transpose, apply Q;
= 'T':  Transpose, apply Q**T.```

M

```          M is INTEGER
The number of rows of the matrix A.  M >=0.```

N

```          N is INTEGER
The number of columns of the matrix C. N >= 0.```

K

```          K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0;```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.
MB > N. (must be the same as SLATSQR)```

NB

```          NB is INTEGER
The column block size to be used in the blocked QR.
N >= NB >= 1.```

A

```          A is REAL array, dimension (LDA,K)
The i-th column must contain the vector which defines the
blockedelementary reflector H(i), for i = 1,2,...,k, as
returned by SLATSQR in the first k columns of
its array argument A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).```

T

```          T is REAL array, dimension
( N * Number of blocks(CEIL(M-K/MB-K)),
The blocked upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See below
for further details.```

LDT

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

C

```          C is REAL array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**T*C or C*Q**T or C*Q.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).```

WORK

`         (workspace) REAL array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.

If SIDE = 'L', LWORK >= max(1,N)*NB;
if SIDE = 'R', LWORK >= max(1,MB)*NB.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

``` Tall-Skinny QR (TSQR) performs QR by a sequence of orthogonal transformations,
representing Q as a product of other orthogonal matrices
Q = Q(1) * Q(2) * . . . * Q(k)
where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
stored under the diagonal of rows 1:MB of A, and by upper triangular
block reflectors, stored in array T(1:LDT,1:N).

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
The last Q(k) may use fewer rows.

For more details of the overall algorithm, see the description of
Sequential TSQR in Section 2.2 of [1].

[1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
SIAM J. Sci. Comput, vol. 34, no. 1, 2012```

Definition at line 197 of file slamtsqr.f.

### subroutine zlamtsqr (character side, character trans, integer m, integer n, integer k, integer mb, integer nb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension(ldc, * ) c, integer ldc, complex*16, dimension( * ) work, integer lwork, integer info)

ZLAMTSQR

Purpose:

```      ZLAMTSQR overwrites the general complex M-by-N matrix C with

SIDE = 'L'     SIDE = 'R'
TRANS = 'N':      Q * C          C * Q
TRANS = 'C':      Q**H * C       C * Q**H
where Q is a complex unitary matrix defined as the product
of blocked elementary reflectors computed by tall skinny
QR factorization (ZLATSQR)```
Parameters

SIDE

```          SIDE is CHARACTER*1
= 'L': apply Q or Q**H from the Left;
= 'R': apply Q or Q**H from the Right.```

TRANS

```          TRANS is CHARACTER*1
= 'N':  No transpose, apply Q;
= 'C':  Conjugate Transpose, apply Q**H.```

M

```          M is INTEGER
The number of rows of the matrix A.  M >=0.```

N

```          N is INTEGER
The number of columns of the matrix C. N >= 0.```

K

```          K is INTEGER
The number of elementary reflectors whose product defines
the matrix Q. M >= K >= 0;```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.
MB > N. (must be the same as ZLATSQR)```

NB

```          NB is INTEGER
The column block size to be used in the blocked QR.
N >= NB >= 1.```

A

```          A is COMPLEX*16 array, dimension (LDA,K)
The i-th column must contain the vector which defines the
blockedelementary reflector H(i), for i = 1,2,...,k, as
returned by ZLATSQR in the first k columns of
its array argument A.```

LDA

```          LDA is INTEGER
The leading dimension of the array A.
If SIDE = 'L', LDA >= max(1,M);
if SIDE = 'R', LDA >= max(1,N).```

T

```          T is COMPLEX*16 array, dimension
( N * Number of blocks(CEIL(M-K/MB-K)),
The blocked upper triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See below
for further details.```

LDT

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

C

```          C is COMPLEX*16 array, dimension (LDC,N)
On entry, the M-by-N matrix C.
On exit, C is overwritten by Q*C or Q**H*C or C*Q**H or C*Q.```

LDC

```          LDC is INTEGER
The leading dimension of the array C. LDC >= max(1,M).```

WORK

`         (workspace) COMPLEX*16 array, dimension (MAX(1,LWORK))`

LWORK

```          LWORK is INTEGER
The dimension of the array WORK.

If SIDE = 'L', LWORK >= max(1,N)*NB;
if SIDE = 'R', LWORK >= max(1,MB)*NB.
If LWORK = -1, then a workspace query is assumed; the routine
only calculates the optimal size of the WORK array, returns
this value as the first entry of the WORK array, and no error
message related to LWORK is issued by XERBLA.```

INFO

```          INFO is INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal value```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

``` Tall-Skinny QR (TSQR) performs QR by a sequence of unitary transformations,
representing Q as a product of other unitary matrices
Q = Q(1) * Q(2) * . . . * Q(k)
where each Q(i) zeros out subdiagonal entries of a block of MB rows of A:
Q(1) zeros out the subdiagonal entries of rows 1:MB of A
Q(2) zeros out the bottom MB-N rows of rows [1:N,MB+1:2*MB-N] of A
Q(3) zeros out the bottom MB-N rows of rows [1:N,2*MB-N+1:3*MB-2*N] of A
. . .

Q(1) is computed by GEQRT, which represents Q(1) by Householder vectors
stored under the diagonal of rows 1:MB of A, and by upper triangular
block reflectors, stored in array T(1:LDT,1:N).

Q(i) for i>1 is computed by TPQRT, which represents Q(i) by Householder vectors
stored in rows [(i-1)*(MB-N)+N+1:i*(MB-N)+N] of A, and by upper triangular
block reflectors, stored in array T(1:LDT,(i-1)*N+1:i*N).
The last Q(k) may use fewer rows.

For more details of the overall algorithm, see the description of
Sequential TSQR in Section 2.2 of [1].

[1] “Communication-Optimal Parallel and Sequential QR and LU Factorizations,”
J. Demmel, L. Grigori, M. Hoemmen, J. Langou,
SIAM J. Sci. Comput, vol. 34, no. 1, 2012```

Definition at line 197 of file zlamtsqr.f.

## Author

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## Info

Tue Nov 28 2023 12:08:43 Version 3.12.0 LAPACK