# tpmqrt - Man Page

tpmqrt: applies Q

## Synopsis

### Functions

subroutine **ctpmqrt** (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)**CTPMQRT**

subroutine **dtpmqrt** (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)**DTPMQRT**

subroutine **stpmqrt** (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)**STPMQRT**

subroutine **ztpmqrt** (side, trans, m, n, k, l, nb, v, ldv, t, ldt, a, lda, b, ldb, work, info)**ZTPMQRT**

## Detailed Description

## Function Documentation

### subroutine ctpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, complex, dimension( ldv, * ) v, integer ldv, complex, dimension( ldt, * ) t, integer ldt, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( * ) work, integer info)

**CTPMQRT**

**Purpose:**

CTPMQRT applies a complex orthogonal matrix Q obtained from a 'triangular-pentagonal' complex block reflector H to a general complex matrix C, which consists of two blocks A and B.

**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 B. M >= 0.

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

*K*K is INTEGER The number of elementary reflectors whose product defines the matrix Q.

*L*L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.

*NB*NB is INTEGER The block size used for the storage of T. K >= NB >= 1. This must be the same value of NB used to generate T in CTPQRT.

*V*V is COMPLEX array, dimension (LDV,K) The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPQRT in B. See Further Details.

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

*T*T is COMPLEX array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPQRT, stored as a NB-by-K matrix.

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

*A*A is COMPLEX array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

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

*B*B is COMPLEX array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

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

*WORK*WORK is COMPLEX array. The dimension of WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.

*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 columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2: V = [V1] [V2]. The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular. If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K. [B] If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K. The complex orthogonal matrix Q is formed from V and T. If TRANS='N' and SIDE='L', C is on exit replaced with Q * C. If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C. If TRANS='N' and SIDE='R', C is on exit replaced with C * Q. If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

Definition at line **214** of file **ctpmqrt.f**.

### subroutine dtpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, double precision, dimension( ldv, * ) v, integer ldv, double precision, dimension( ldt, * ) t, integer ldt, double precision, dimension( lda, * ) a, integer lda, double precision, dimension( ldb, * ) b, integer ldb, double precision, dimension( * ) work, integer info)

**DTPMQRT**

**Purpose:**

DTPMQRT applies a real orthogonal matrix Q obtained from a 'triangular-pentagonal' real block reflector H to a general real matrix C, which consists of two blocks A and B.

**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 B. M >= 0.

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

*K*K is INTEGER The number of elementary reflectors whose product defines the matrix Q.

*L*L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.

*NB*NB is INTEGER The block size used for the storage of T. K >= NB >= 1. This must be the same value of NB used to generate T in CTPQRT.

*V*V is DOUBLE PRECISION array, dimension (LDV,K) The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPQRT in B. See Further Details.

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

*T*T is DOUBLE PRECISION array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPQRT, stored as a NB-by-K matrix.

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

*A*A is DOUBLE PRECISION array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.

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

*B*B is DOUBLE PRECISION array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q**T*C or C*Q or C*Q**T. See Further Details.

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

*WORK*WORK is DOUBLE PRECISION array. The dimension of WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.

*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 columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2: V = [V1] [V2]. The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular. If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K. [B] If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K. The real orthogonal matrix Q is formed from V and T. If TRANS='N' and SIDE='L', C is on exit replaced with Q * C. If TRANS='T' and SIDE='L', C is on exit replaced with Q**T * C. If TRANS='N' and SIDE='R', C is on exit replaced with C * Q. If TRANS='T' and SIDE='R', C is on exit replaced with C * Q**T.

Definition at line **214** of file **dtpmqrt.f**.

### subroutine stpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, real, dimension( ldv, * ) v, integer ldv, real, dimension( ldt, * ) t, integer ldt, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( * ) work, integer info)

**STPMQRT**

**Purpose:**

STPMQRT applies a real orthogonal matrix Q obtained from a 'triangular-pentagonal' real block reflector H to a general real matrix C, which consists of two blocks A and B.

**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 B. M >= 0.

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

*K*K is INTEGER The number of elementary reflectors whose product defines the matrix Q.

*L*L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.

*NB*NB is INTEGER The block size used for the storage of T. K >= NB >= 1. This must be the same value of NB used to generate T in CTPQRT.

*V*V is REAL array, dimension (LDV,K) The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPQRT in B. See Further Details.

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

*T*T is REAL array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPQRT, stored as a NB-by-K matrix.

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

*A*A is REAL array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.

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

*B*B is REAL array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q^T*C or C*Q or C*Q^T. See Further Details.

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

*WORK*WORK is REAL array. The dimension of WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.

*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 columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2: V = [V1] [V2]. The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular. If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K. [B] If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K. The real orthogonal matrix Q is formed from V and T. If TRANS='N' and SIDE='L', C is on exit replaced with Q * C. If TRANS='T' and SIDE='L', C is on exit replaced with Q^T * C. If TRANS='N' and SIDE='R', C is on exit replaced with C * Q. If TRANS='T' and SIDE='R', C is on exit replaced with C * Q^T.

Definition at line **214** of file **stpmqrt.f**.

### subroutine ztpmqrt (character side, character trans, integer m, integer n, integer k, integer l, integer nb, complex*16, dimension( ldv, * ) v, integer ldv, complex*16, dimension( ldt, * ) t, integer ldt, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( * ) work, integer info)

**ZTPMQRT**

**Purpose:**

ZTPMQRT applies a complex orthogonal matrix Q obtained from a 'triangular-pentagonal' complex block reflector H to a general complex matrix C, which consists of two blocks A and B.

**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 B. M >= 0.

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

*K*K is INTEGER The number of elementary reflectors whose product defines the matrix Q.

*L*L is INTEGER The order of the trapezoidal part of V. K >= L >= 0. See Further Details.

*NB**V*V is COMPLEX*16 array, dimension (LDV,K) The i-th column must contain the vector which defines the elementary reflector H(i), for i = 1,2,...,k, as returned by CTPQRT in B. See Further Details.

*LDV**T*T is COMPLEX*16 array, dimension (LDT,K) The upper triangular factors of the block reflectors as returned by CTPQRT, stored as a NB-by-K matrix.

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

*A*A is COMPLEX*16 array, dimension (LDA,N) if SIDE = 'L' or (LDA,K) if SIDE = 'R' On entry, the K-by-N or M-by-K matrix A. On exit, A is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

*LDA**B*B is COMPLEX*16 array, dimension (LDB,N) On entry, the M-by-N matrix B. On exit, B is overwritten by the corresponding block of Q*C or Q**H*C or C*Q or C*Q**H. See Further Details.

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

*WORK*WORK is COMPLEX*16 array. The dimension of WORK is N*NB if SIDE = 'L', or M*NB if SIDE = 'R'.

*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 columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(K); V is composed of a rectangular block V1 and a trapezoidal block V2: V = [V1] [V2]. The size of the trapezoidal block V2 is determined by the parameter L, where 0 <= L <= K; V2 is upper trapezoidal, consisting of the first L rows of a K-by-K upper triangular matrix. If L=K, V2 is upper triangular; if L=0, there is no trapezoidal block, hence V = V1 is rectangular. If SIDE = 'L': C = [A] where A is K-by-N, B is M-by-N and V is M-by-K. [B] If SIDE = 'R': C = [A B] where A is M-by-K, B is M-by-N and V is N-by-K. The complex orthogonal matrix Q is formed from V and T. If TRANS='N' and SIDE='L', C is on exit replaced with Q * C. If TRANS='C' and SIDE='L', C is on exit replaced with Q**H * C. If TRANS='N' and SIDE='R', C is on exit replaced with C * Q. If TRANS='C' and SIDE='R', C is on exit replaced with C * Q**H.

Definition at line **214** of file **ztpmqrt.f**.

## Author

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