---
name: "tplqt2"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/tplqt2"
---

# tplqt2(3)

tplqt2: QR factor, level 2

## NAME

tplqt2 - tplqt2: QR factor, level 2

## SYNOPSIS

### Functions

subroutine **ctplqt2** (m, n, l, a, lda, b, ldb, t, ldt, info)\
 **CTPLQT2**  subroutine **dtplqt2** (m, n, l, a, lda, b, ldb, t, ldt, info)\
 **DTPLQT2** computes a LQ 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 **stplqt2** (m, n, l, a, lda, b, ldb, t, ldt, info)\
 **STPLQT2** computes a LQ 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 **ztplqt2** (m, n, l, a, lda, b, ldb, t, ldt, info)\
 **ZTPLQT2** computes a LQ 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.

## Detailed Description

## Function Documentation

### subroutine ctplqt2 (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)

**CTPLQT2**

**Purpose:**

```
!>
!> CTPLQT2 computes a LQ a factorization of a complex
!> 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 lower trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>
```

*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,M)
!>          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,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 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
!>  N-by-N 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,
!>
!>               W = [ 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 (M+N)-by-(M+N) block reflector H is then given by
!>
!>               H = I - W**T * T * W
!>
!>  where W^H is the conjugate transpose of W and T is the upper triangular
!>  factor of the block reflector.
!>
```

Definition at line **161** of file **ctplqt2.f**.

### subroutine dtplqt2 (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)

**DTPLQT2** computes a LQ 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:**

```
!>
!> DTPLQT2 computes a LQ a factorization of a real
!> 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 lower trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>
```

*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,M)
!>          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,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 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
!>  N-by-N 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,
!>
!>               W = [ 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 (M+N)-by-(M+N) block reflector H is then given by
!>
!>               H = I - W**T * T * W
!>
!>  where W^H is the conjugate transpose of W and T is the upper triangular
!>  factor of the block reflector.
!>
```

Definition at line **176** of file **dtplqt2.f**.

### subroutine stplqt2 (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)

**STPLQT2** computes a LQ 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:**

```
!>
!> STPLQT2 computes a LQ a factorization of a real
!> 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 lower trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>
```

*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,M)
!>          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,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 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
!>  N-by-N 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,
!>
!>               W = [ 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 (M+N)-by-(M+N) block reflector H is then given by
!>
!>               H = I - W**T * T * W
!>
!>  where W^H is the conjugate transpose of W and T is the upper triangular
!>  factor of the block reflector.
!>
```

Definition at line **176** of file **stplqt2.f**.

### subroutine ztplqt2 (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)

**ZTPLQT2** computes a LQ 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:**

```
!>
!> ZTPLQT2 computes a LQ a factorization of a complex
!> 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 lower trapezoidal part of B.
!>          MIN(M,N) >= L >= 0.  See Further Details.
!>
```

*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,M)
!>          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,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 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
!>  N-by-N 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,
!>
!>               W = [ 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 (M+N)-by-(M+N) block reflector H is then given by
!>
!>               H = I - W**T * T * W
!>
!>  where W^H is the conjugate transpose of W and T is the upper triangular
!>  factor of the block reflector.
!>
```

Definition at line **176** of file **ztplqt2.f**.

## Author

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