# tfsm - Man Page

tfsm: triangular-matrix solve, RFP format

## Synopsis

### Functions

subroutine ctfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
CTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine dtfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine stfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).
subroutine ztfsm (transr, side, uplo, trans, diag, m, n, alpha, a, b, ldb)
ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

## Function Documentation

### subroutine ctfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, complex alpha, complex, dimension( 0: * ) a, complex, dimension( 0: ldb-1, 0: * ) b, integer ldb)

CTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

``` Level 3 BLAS like routine for A in RFP Format.

CTFSM solves the matrix equation

op( A )*X = alpha*B  or  X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit,  upper or lower triangular matrix  and  op( A )  is one  of

op( A ) = A   or   op( A ) = A**H.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.```
Parameters

TRANSR

```          TRANSR is CHARACTER*1
= 'N':  The Normal Form of RFP A is stored;
= 'C':  The Conjugate-transpose Form of RFP A is stored.```

SIDE

```          SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:

SIDE = 'L' or 'l'   op( A )*X = alpha*B.

SIDE = 'R' or 'r'   X*op( A ) = alpha*B.

Unchanged on exit.```

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix

Unchanged on exit.```

TRANS

```          TRANS is CHARACTER*1
On entry, TRANS  specifies the form of op( A ) to be used
in the matrix multiplication as follows:

TRANS  = 'N' or 'n'   op( A ) = A.

TRANS  = 'C' or 'c'   op( A ) = conjg( A' ).

Unchanged on exit.```

DIAG

```          DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:

DIAG = 'U' or 'u'   A is assumed to be unit triangular.

DIAG = 'N' or 'n'   A is not assumed to be unit
triangular.

Unchanged on exit.```

M

```          M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.```

N

```          N is INTEGER
On entry, N specifies the number of columns of B.  N must be
at least zero.
Unchanged on exit.```

ALPHA

```          ALPHA is COMPLEX
On entry,  ALPHA specifies the scalar  alpha. When  alpha is
zero then  A is not referenced and  B need not be set before
entry.
Unchanged on exit.```

A

```          A is COMPLEX array, dimension (N*(N+1)/2)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as
defined when TRANSR = 'N'. The contents of RFP A are defined
by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
elements of upper packed A either in normal or
conjugate-transpose Format. If UPLO = 'L' the RFP A contains
the NT elements of lower packed A either in normal or
conjugate-transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.```

B

```          B is COMPLEX array, dimension (LDB,N)
Before entry,  the leading  m by n part of the array  B must
contain  the  right-hand  side  matrix  B,  and  on exit  is
overwritten by the solution matrix  X.```

LDB

```          LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in  the  calling  (sub)  program.   LDB  must  be  at  least
max( 1, m ).
Unchanged on exit.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  We first consider Standard Packed Format when N is even.
We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N even and TRANSR = 'N'.

RFP A                   RFP A

-- -- --
03 04 05                33 43 53
-- --
13 14 15                00 44 54
--
23 24 25                10 11 55

33 34 35                20 21 22
--
00 44 45                30 31 32
-- --
01 11 55                40 41 42
-- -- --
02 12 22                50 51 52

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- -- --                -- -- -- -- -- --
03 13 23 33 00 01 02    33 00 10 20 30 40 50
-- -- -- -- --                -- -- -- -- --
04 14 24 34 44 11 12    43 44 11 21 31 41 51
-- -- -- -- -- --                -- -- -- --
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We next  consider Standard Packed Format when N is odd.
We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two   columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two   columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N odd  and TRANSR = 'N'.

RFP A                   RFP A

-- --
02 03 04                00 33 43
--
12 13 14                10 11 44

22 23 24                20 21 22
--
00 33 34                30 31 32
-- --
01 11 44                40 41 42

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- --                   -- -- -- -- -- --
02 12 22 00 01             00 10 20 30 40 50
-- -- -- --                   -- -- -- -- --
03 13 23 33 11             33 11 21 31 41 51
-- -- -- -- --                   -- -- -- --
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 296 of file ctfsm.f.

### subroutine dtfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, double precision alpha, double precision, dimension( 0: * ) a, double precision, dimension( 0: ldb-1, 0: * ) b, integer ldb)

DTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

``` Level 3 BLAS like routine for A in RFP Format.

DTFSM  solves the matrix equation

op( A )*X = alpha*B  or  X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit,  upper or lower triangular matrix  and  op( A )  is one  of

op( A ) = A   or   op( A ) = A**T.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.```
Parameters

TRANSR

```          TRANSR is CHARACTER*1
= 'N':  The Normal Form of RFP A is stored;
= 'T':  The Transpose Form of RFP A is stored.```

SIDE

```          SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:

SIDE = 'L' or 'l'   op( A )*X = alpha*B.

SIDE = 'R' or 'r'   X*op( A ) = alpha*B.

Unchanged on exit.```

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix

Unchanged on exit.```

TRANS

```          TRANS is CHARACTER*1
On entry, TRANS  specifies the form of op( A ) to be used
in the matrix multiplication as follows:

TRANS  = 'N' or 'n'   op( A ) = A.

TRANS  = 'T' or 't'   op( A ) = A'.

Unchanged on exit.```

DIAG

```          DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:

DIAG = 'U' or 'u'   A is assumed to be unit triangular.

DIAG = 'N' or 'n'   A is not assumed to be unit
triangular.

Unchanged on exit.```

M

```          M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.```

N

```          N is INTEGER
On entry, N specifies the number of columns of B.  N must be
at least zero.
Unchanged on exit.```

ALPHA

```          ALPHA is DOUBLE PRECISION
On entry,  ALPHA specifies the scalar  alpha. When  alpha is
zero then  A is not referenced and  B need not be set before
entry.
Unchanged on exit.```

A

```          A is DOUBLE PRECISION array, dimension (NT)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = 'T' then RFP is the transpose of RFP A as
defined when TRANSR = 'N'. The contents of RFP A are defined
by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
elements of upper packed A either in normal or
transpose Format. If UPLO = 'L' the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.```

B

```          B is DOUBLE PRECISION array, dimension (LDB,N)
Before entry,  the leading  m by n part of the array  B must
contain  the  right-hand  side  matrix  B,  and  on exit  is
overwritten by the solution matrix  X.```

LDB

```          LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in  the  calling  (sub)  program.   LDB  must  be  at  least
max( 1, m ).
Unchanged on exit.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.

RFP A                   RFP A

03 04 05                33 43 53
13 14 15                00 44 54
23 24 25                10 11 55
33 34 35                20 21 22
00 44 45                30 31 32
01 11 55                40 41 42
02 12 22                50 51 52

Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

03 13 23 33 00 01 02    33 00 10 20 30 40 50
04 14 24 34 44 11 12    43 44 11 21 31 41 51
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.

RFP A                   RFP A

02 03 04                00 33 43
12 13 14                10 11 44
22 23 24                20 21 22
00 33 34                30 31 32
01 11 44                40 41 42

Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

02 12 22 00 01             00 10 20 30 40 50
03 13 23 33 11             33 11 21 31 41 51
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 275 of file dtfsm.f.

### subroutine stfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, real alpha, real, dimension( 0: * ) a, real, dimension( 0: ldb-1, 0: * ) b, integer ldb)

STFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

``` Level 3 BLAS like routine for A in RFP Format.

STFSM  solves the matrix equation

op( A )*X = alpha*B  or  X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit,  upper or lower triangular matrix  and  op( A )  is one  of

op( A ) = A   or   op( A ) = A**T.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.```
Parameters

TRANSR

```          TRANSR is CHARACTER*1
= 'N':  The Normal Form of RFP A is stored;
= 'T':  The Transpose Form of RFP A is stored.```

SIDE

```          SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:

SIDE = 'L' or 'l'   op( A )*X = alpha*B.

SIDE = 'R' or 'r'   X*op( A ) = alpha*B.

Unchanged on exit.```

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix

Unchanged on exit.```

TRANS

```          TRANS is CHARACTER*1
On entry, TRANS  specifies the form of op( A ) to be used
in the matrix multiplication as follows:

TRANS  = 'N' or 'n'   op( A ) = A.

TRANS  = 'T' or 't'   op( A ) = A'.

Unchanged on exit.```

DIAG

```          DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:

DIAG = 'U' or 'u'   A is assumed to be unit triangular.

DIAG = 'N' or 'n'   A is not assumed to be unit
triangular.

Unchanged on exit.```

M

```          M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.```

N

```          N is INTEGER
On entry, N specifies the number of columns of B.  N must be
at least zero.
Unchanged on exit.```

ALPHA

```          ALPHA is REAL
On entry,  ALPHA specifies the scalar  alpha. When  alpha is
zero then  A is not referenced and  B need not be set before
entry.
Unchanged on exit.```

A

```          A is REAL array, dimension (NT)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = 'T' then RFP is the transpose of RFP A as
defined when TRANSR = 'N'. The contents of RFP A are defined
by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
elements of upper packed A either in normal or
transpose Format. If UPLO = 'L' the RFP A contains
the NT elements of lower packed A either in normal or
transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = 'T'. When TRANSR is 'N' the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.```

B

```          B is REAL array, dimension (LDB,N)
Before entry,  the leading  m by n part of the array  B must
contain  the  right-hand  side  matrix  B,  and  on exit  is
overwritten by the solution matrix  X.```

LDB

```          LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in  the  calling  (sub)  program.   LDB  must  be  at  least
max( 1, m ).
Unchanged on exit.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  We first consider Rectangular Full Packed (RFP) Format when N is
even. We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
the transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
the transpose of the last three columns of AP lower.
This covers the case N even and TRANSR = 'N'.

RFP A                   RFP A

03 04 05                33 43 53
13 14 15                00 44 54
23 24 25                10 11 55
33 34 35                20 21 22
00 44 45                30 31 32
01 11 55                40 41 42
02 12 22                50 51 52

Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

03 13 23 33 00 01 02    33 00 10 20 30 40 50
04 14 24 34 44 11 12    43 44 11 21 31 41 51
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We then consider Rectangular Full Packed (RFP) Format when N is
odd. We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
the transpose of the first two columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
the transpose of the last two columns of AP lower.
This covers the case N odd and TRANSR = 'N'.

RFP A                   RFP A

02 03 04                00 33 43
12 13 14                10 11 44
22 23 24                20 21 22
00 33 34                30 31 32
01 11 44                40 41 42

Now let TRANSR = 'T'. RFP A in both UPLO cases is just the
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

02 12 22 00 01             00 10 20 30 40 50
03 13 23 33 11             33 11 21 31 41 51
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 275 of file stfsm.f.

### subroutine ztfsm (character transr, character side, character uplo, character trans, character diag, integer m, integer n, complex*16 alpha, complex*16, dimension( 0: * ) a, complex*16, dimension( 0: ldb-1, 0: * ) b, integer ldb)

ZTFSM solves a matrix equation (one operand is a triangular matrix in RFP format).

Purpose:

``` Level 3 BLAS like routine for A in RFP Format.

ZTFSM  solves the matrix equation

op( A )*X = alpha*B  or  X*op( A ) = alpha*B

where alpha is a scalar, X and B are m by n matrices, A is a unit, or
non-unit,  upper or lower triangular matrix  and  op( A )  is one  of

op( A ) = A   or   op( A ) = A**H.

A is in Rectangular Full Packed (RFP) Format.

The matrix X is overwritten on B.```
Parameters

TRANSR

```          TRANSR is CHARACTER*1
= 'N':  The Normal Form of RFP A is stored;
= 'C':  The Conjugate-transpose Form of RFP A is stored.```

SIDE

```          SIDE is CHARACTER*1
On entry, SIDE specifies whether op( A ) appears on the left
or right of X as follows:

SIDE = 'L' or 'l'   op( A )*X = alpha*B.

SIDE = 'R' or 'r'   X*op( A ) = alpha*B.

Unchanged on exit.```

UPLO

```          UPLO is CHARACTER*1
On entry, UPLO specifies whether the RFP matrix A came from
an upper or lower triangular matrix as follows:
UPLO = 'U' or 'u' RFP A came from an upper triangular matrix
UPLO = 'L' or 'l' RFP A came from a  lower triangular matrix

Unchanged on exit.```

TRANS

```          TRANS is CHARACTER*1
On entry, TRANS  specifies the form of op( A ) to be used
in the matrix multiplication as follows:

TRANS  = 'N' or 'n'   op( A ) = A.

TRANS  = 'C' or 'c'   op( A ) = conjg( A' ).

Unchanged on exit.```

DIAG

```          DIAG is CHARACTER*1
On entry, DIAG specifies whether or not RFP A is unit
triangular as follows:

DIAG = 'U' or 'u'   A is assumed to be unit triangular.

DIAG = 'N' or 'n'   A is not assumed to be unit
triangular.

Unchanged on exit.```

M

```          M is INTEGER
On entry, M specifies the number of rows of B. M must be at
least zero.
Unchanged on exit.```

N

```          N is INTEGER
On entry, N specifies the number of columns of B.  N must be
at least zero.
Unchanged on exit.```

ALPHA

```          ALPHA is COMPLEX*16
On entry,  ALPHA specifies the scalar  alpha. When  alpha is
zero then  A is not referenced and  B need not be set before
entry.
Unchanged on exit.```

A

```          A is COMPLEX*16 array, dimension (N*(N+1)/2)
NT = N*(N+1)/2. On entry, the matrix A in RFP Format.
RFP Format is described by TRANSR, UPLO and N as follows:
If TRANSR='N' then RFP A is (0:N,0:K-1) when N is even;
K=N/2. RFP A is (0:N-1,0:K) when N is odd; K=N/2. If
TRANSR = 'C' then RFP is the Conjugate-transpose of RFP A as
defined when TRANSR = 'N'. The contents of RFP A are defined
by UPLO as follows: If UPLO = 'U' the RFP A contains the NT
elements of upper packed A either in normal or
conjugate-transpose Format. If UPLO = 'L' the RFP A contains
the NT elements of lower packed A either in normal or
conjugate-transpose Format. The LDA of RFP A is (N+1)/2 when
TRANSR = 'C'. When TRANSR is 'N' the LDA is N+1 when N is
even and is N when is odd.
See the Note below for more details. Unchanged on exit.```

B

```          B is COMPLEX*16 array, dimension (LDB,N)
Before entry,  the leading  m by n part of the array  B must
contain  the  right-hand  side  matrix  B,  and  on exit  is
overwritten by the solution matrix  X.```

LDB

```          LDB is INTEGER
On entry, LDB specifies the first dimension of B as declared
in  the  calling  (sub)  program.   LDB  must  be  at  least
max( 1, m ).
Unchanged on exit.```
Author

Univ. of Tennessee

Univ. of California Berkeley

NAG Ltd.

Further Details:

```  We first consider Standard Packed Format when N is even.
We give an example where N = 6.

AP is Upper             AP is Lower

00 01 02 03 04 05       00
11 12 13 14 15       10 11
22 23 24 25       20 21 22
33 34 35       30 31 32 33
44 45       40 41 42 43 44
55       50 51 52 53 54 55

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last
three columns of AP upper. The lower triangle A(4:6,0:2) consists of
conjugate-transpose of the first three columns of AP upper.
For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:2,0:2) consists of
conjugate-transpose of the last three columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N even and TRANSR = 'N'.

RFP A                   RFP A

-- -- --
03 04 05                33 43 53
-- --
13 14 15                00 44 54
--
23 24 25                10 11 55

33 34 35                20 21 22
--
00 44 45                30 31 32
-- --
01 11 55                40 41 42
-- -- --
02 12 22                50 51 52

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- -- --                -- -- -- -- -- --
03 13 23 33 00 01 02    33 00 10 20 30 40 50
-- -- -- -- --                -- -- -- -- --
04 14 24 34 44 11 12    43 44 11 21 31 41 51
-- -- -- -- -- --                -- -- -- --
05 15 25 35 45 55 22    53 54 55 22 32 42 52

We next  consider Standard Packed Format when N is odd.
We give an example where N = 5.

AP is Upper                 AP is Lower

00 01 02 03 04              00
11 12 13 14              10 11
22 23 24              20 21 22
33 34              30 31 32 33
44              40 41 42 43 44

Let TRANSR = 'N'. RFP holds AP as follows:
For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last
three columns of AP upper. The lower triangle A(3:4,0:1) consists of
conjugate-transpose of the first two   columns of AP upper.
For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first
three columns of AP lower. The upper triangle A(0:1,1:2) consists of
conjugate-transpose of the last two   columns of AP lower.
To denote conjugate we place -- above the element. This covers the
case N odd  and TRANSR = 'N'.

RFP A                   RFP A

-- --
02 03 04                00 33 43
--
12 13 14                10 11 44

22 23 24                20 21 22
--
00 33 34                30 31 32
-- --
01 11 44                40 41 42

Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
transpose of RFP A above. One therefore gets:

RFP A                   RFP A

-- -- --                   -- -- -- -- -- --
02 12 22 00 01             00 10 20 30 40 50
-- -- -- --                   -- -- -- -- --
03 13 23 33 11             33 11 21 31 41 51
-- -- -- -- --                   -- -- -- --
04 14 24 34 44             43 44 22 32 42 52```

Definition at line 296 of file ztfsm.f.

## Author

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## Info

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