# 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

## 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

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

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

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

```          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*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

```          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*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

```          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*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

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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## Info

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