# 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**

## Detailed Description

## 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**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:**

Definition at line **187** of file **ztplqt.f**.

## Author

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