# tpqrt2 - Man Page

tpqrt2: QR factor, level 2

## Synopsis

### Functions

subroutine ctpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
CTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine dtpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
DTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine stpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
STPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.
subroutine ztpqrt2 (m, n, l, a, lda, b, ldb, t, ldt, info)
ZTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

## Function Documentation

### subroutine ctpqrt2 (integer m, integer n, integer l, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldt, * ) t, integer ldt, integer info)

CTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

``` CTPQRT2 computes a QR 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 total number of rows of the matrix B.
M >= 0.```

N

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

L

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

A

```          A is COMPLEX array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.```

LDA

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

B

```          B is COMPLEX array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first M-L rows
are rectangular, and the last L rows are upper 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 N-by-N upper triangular factor T of the block reflector.
See Further Details.```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= max(1,N)```

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:

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

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
[ B2 ]  <-     L-by-N upper trapezoidal.

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

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ]  <- upper triangular N-by-N
[ B ]  <- M-by-N pentagonal

so that W can be represented as

W = [ I ]  <- identity, N-by-N
[ 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 ] <- (M-L)-by-N rectangular
[ V2 ] <-     L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W * T * W**H

where W**H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.```

Definition at line 172 of file ctpqrt2.f.

### subroutine dtpqrt2 (integer m, integer n, integer l, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( ldt, * ) t, integer ldt, integer info)

DTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

``` DTPQRT2 computes a QR 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 total number of rows of the matrix B.
M >= 0.```

N

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

L

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

A

```          A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.```

LDA

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

B

```          B is DOUBLE PRECISION array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first M-L rows
are rectangular, and the last L rows are upper 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 N-by-N upper triangular factor T of the block reflector.
See Further Details.```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= max(1,N)```

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:

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

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
[ B2 ]  <-     L-by-N upper trapezoidal.

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

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ]  <- upper triangular N-by-N
[ B ]  <- M-by-N pentagonal

so that W can be represented as

W = [ I ]  <- identity, N-by-N
[ 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 ] <- (M-L)-by-N rectangular
[ V2 ] <-     L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W * T * W**T

where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.```

Definition at line 172 of file dtpqrt2.f.

### subroutine stpqrt2 (integer m, integer n, integer l, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldt, * ) t, integer ldt, integer info)

STPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

``` STPQRT2 computes a QR 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 total number of rows of the matrix B.
M >= 0.```

N

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

L

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

A

```          A is REAL array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.```

LDA

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

B

```          B is REAL array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first M-L rows
are rectangular, and the last L rows are upper 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 N-by-N upper triangular factor T of the block reflector.
See Further Details.```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= max(1,N)```

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:

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

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
[ B2 ]  <-     L-by-N upper trapezoidal.

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

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ]  <- upper triangular N-by-N
[ B ]  <- M-by-N pentagonal

so that W can be represented as

W = [ I ]  <- identity, N-by-N
[ 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 ] <- (M-L)-by-N rectangular
[ V2 ] <-     L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W * T * W^H

where W^H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.```

Definition at line 172 of file stpqrt2.f.

### subroutine ztpqrt2 (integer m, integer n, integer l, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldt, * ) t, integer ldt, integer info)

ZTPQRT2 computes a QR factorization of a real or complex 'triangular-pentagonal' matrix, which is composed of a triangular block and a pentagonal block, using the compact WY representation for Q.

Purpose:

``` ZTPQRT2 computes a QR 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 total number of rows of the matrix B.
M >= 0.```

N

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

L

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

A

```          A is COMPLEX*16 array, dimension (LDA,N)
On entry, the upper triangular N-by-N matrix A.
On exit, the elements on and above the diagonal of the array
contain the upper triangular matrix R.```

LDA

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

B

```          B is COMPLEX*16 array, dimension (LDB,N)
On entry, the pentagonal M-by-N matrix B.  The first M-L rows
are rectangular, and the last L rows are upper 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 N-by-N upper triangular factor T of the block reflector.
See Further Details.```

LDT

```          LDT is INTEGER
The leading dimension of the array T.  LDT >= max(1,N)```

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:

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

C = [ A ]
[ B ]

where A is an upper triangular N-by-N matrix, and B is M-by-N pentagonal
matrix consisting of a (M-L)-by-N rectangular matrix B1 on top of a L-by-N
upper trapezoidal matrix B2:

B = [ B1 ]  <- (M-L)-by-N rectangular
[ B2 ]  <-     L-by-N upper trapezoidal.

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

The matrix W stores the elementary reflectors H(i) in the i-th column
below the diagonal (of A) in the (N+M)-by-N input matrix C

C = [ A ]  <- upper triangular N-by-N
[ B ]  <- M-by-N pentagonal

so that W can be represented as

W = [ I ]  <- identity, N-by-N
[ 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 ] <- (M-L)-by-N rectangular
[ V2 ] <-     L-by-N upper trapezoidal.

The columns of V represent the vectors which define the H(i)'s.
The (M+N)-by-(M+N) block reflector H is then given by

H = I - W * T * W**H

where W**H is the conjugate transpose of W and T is the upper triangular
factor of the block reflector.```

Definition at line 172 of file ztpqrt2.f.

## Author

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## Info

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