# tplqt - Man Page

tplqt: QR factor

## Synopsis

### Functions

subroutine ctplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
CTPLQT
subroutine dtplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
DTPLQT
subroutine stplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
STPLQT
subroutine ztplqt (m, n, l, mb, a, lda, b, ldb, t, ldt, work, info)
ZTPLQT

## Function Documentation

### subroutine ctplqt (integer m, integer n, integer l, integer mb, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( * ) work, integer info)

CTPLQT

Purpose:

``` CTPLQT computes a blocked LQ factorization of a complex
'triangular-pentagonal' matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.```

N

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

L

```          L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0.  See Further Details.```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.  M >= MB >= 1.```

A

```          A is COMPLEX array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.```

LDA

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

B

```          B is COMPLEX array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V.  See Further Details.```

LDB

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

T

```          T is COMPLEX array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See Further Details.```

LDT

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

WORK

`          WORK is COMPLEX array, dimension (MB*M)`

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

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ]  <- M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
[ C ] = [ A ] [ B ]
[ A ]  <- lower triangular M-by-M
[ B ]  <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ]  <- identity, M-by-M
[ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above.  Note that V has the same form as B; that is,
[ V ] = [ V1 ] [ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <-     M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)'s.

The number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Definition at line 172 of file ctplqt.f.

### subroutine dtplqt (integer m, integer n, integer l, integer mb, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( * ) work, integer info)

DTPLQT

Purpose:

``` DTPLQT computes a blocked LQ factorization of a real
'triangular-pentagonal' matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.```

N

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

L

```          L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0.  See Further Details.```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.  M >= MB >= 1.```

A

```          A is DOUBLE PRECISION array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V.  See Further Details.```

LDB

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

T

```          T is DOUBLE PRECISION array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See Further Details.```

LDT

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

WORK

`          WORK is DOUBLE PRECISION array, dimension (MB*M)`

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

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ]  <- M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
[ C ] = [ A ] [ B ]
[ A ]  <- lower triangular M-by-M
[ B ]  <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ]  <- identity, M-by-M
[ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above.  Note that V has the same form as B; that is,
[ V ] = [ V1 ] [ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <-     M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)'s.

The number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Definition at line 187 of file dtplqt.f.

### subroutine stplqt (integer m, integer n, integer l, integer mb, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, real, dimension( * ) work, integer info)

STPLQT

Purpose:

``` STPLQT computes a blocked LQ factorization of a real
'triangular-pentagonal' matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.```

N

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

L

```          L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0.  See Further Details.```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.  M >= MB >= 1.```

A

```          A is REAL array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.```

LDA

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

B

```          B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V.  See Further Details.```

LDB

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

T

```          T is REAL array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See Further Details.```

LDT

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

WORK

`          WORK is REAL array, dimension (MB*M)`

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

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ]  <- M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
[ C ] = [ A ] [ B ]
[ A ]  <- lower triangular M-by-M
[ B ]  <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ]  <- identity, M-by-M
[ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above.  Note that V has the same form as B; that is,
[ V ] = [ V1 ] [ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <-     M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)'s.

The number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Definition at line 187 of file stplqt.f.

### subroutine ztplqt (integer m, integer n, integer l, integer mb, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( * ) work, integer info)

ZTPLQT

Purpose:

``` ZTPLQT computes a blocked LQ factorization of a complex
'triangular-pentagonal' matrix C, which is composed of a
triangular block A and pentagonal block B, using the compact
WY representation for Q.```
Parameters

M

```          M is INTEGER
The number of rows of the matrix B, and the order of the
triangular matrix A.
M >= 0.```

N

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

L

```          L is INTEGER
The number of rows of the lower trapezoidal part of B.
MIN(M,N) >= L >= 0.  See Further Details.```

MB

```          MB is INTEGER
The block size to be used in the blocked QR.  M >= MB >= 1.```

A

```          A is COMPLEX*16 array, dimension (LDA,M)
On entry, the lower triangular M-by-M matrix A.
On exit, the elements on and below the diagonal of the array
contain the lower triangular matrix L.```

LDA

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

B

```          B is COMPLEX*16 array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first N-L columns
are rectangular, and the last L columns are lower trapezoidal.
On exit, B contains the pentagonal matrix V.  See Further Details.```

LDB

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

T

```          T is COMPLEX*16 array, dimension (LDT,N)
The lower triangular block reflectors stored in compact form
as a sequence of upper triangular blocks.  See Further Details.```

LDT

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

WORK

`          WORK is COMPLEX*16 array, dimension (MB*M)`

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

Univ. of Colorado Denver

NAG Ltd.

Further Details:

```  The input matrix C is a M-by-(M+N) matrix

C = [ A ] [ B ]

where A is an lower triangular M-by-M matrix, and B is M-by-N pentagonal
matrix consisting of a M-by-(N-L) rectangular matrix B1 on left of a M-by-L
upper trapezoidal matrix B2:
[ B ] = [ B1 ] [ B2 ]
[ B1 ]  <- M-by-(N-L) rectangular
[ B2 ]  <-     M-by-L lower trapezoidal.

The lower trapezoidal matrix B2 consists of the first L columns of a
M-by-M lower triangular matrix, where 0 <= L <= MIN(M,N).  If L=0,
B is rectangular M-by-N; if M=L=N, B is lower triangular.

The matrix W stores the elementary reflectors H(i) in the i-th row
above the diagonal (of A) in the M-by-(M+N) input matrix C
[ C ] = [ A ] [ B ]
[ A ]  <- lower triangular M-by-M
[ B ]  <- M-by-N pentagonal

so that W can be represented as
[ W ] = [ I ] [ V ]
[ I ]  <- identity, M-by-M
[ V ]  <- M-by-N, same form as B.

Thus, all of information needed for W is contained on exit in B, which
we call V above.  Note that V has the same form as B; that is,
[ V ] = [ V1 ] [ V2 ]
[ V1 ] <- M-by-(N-L) rectangular
[ V2 ] <-     M-by-L lower trapezoidal.

The rows of V represent the vectors which define the H(i)'s.

The number of blocks is B = ceiling(M/MB), where each
block is of order MB except for the last block, which is of order
IB = M - (M-1)*MB.  For each of the B blocks, a upper triangular block
reflector factor is computed: T1, T2, ..., TB.  The MB-by-MB (and IB-by-IB
for the last block) T's are stored in the MB-by-N matrix T as

T = [T1 T2 ... TB].```

Definition at line 187 of file ztplqt.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Info

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