Man page - tfsm(3)

Packages contains this manual

Manual

tfsm

NAME
SYNOPSIS
Functions
Detailed Description
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)
subroutine dtfsm (character transr, character side, character uplo,character trans, character diag, integer m, integer n, double precisionalpha, double precision, dimension( 0: * ) a, double precision,dimension( 0: ldb-1, 0: * ) b, integer ldb)
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, integerldb)
subroutine ztfsm (character transr, character side, character uplo,character trans, character diag, integer m, integer n, complex*16alpha, complex*16, dimension( 0: * ) a, complex*16, dimension( 0:ldb-1, 0: * ) b, integer ldb)
Author

NAME

tfsm - 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).

Detailed Description

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 (NT)
NT = N*(N+1)/2 if SIDE=’R’ and NT = M*(M+1)/2 otherwise.
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

Univ. of Colorado Denver

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

subroutine dtfsm (character transr, character side, character uplo,character trans, character diag, integer m, integer n, double precisionalpha, 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 if SIDE=’R’ and NT = M*(M+1)/2 otherwise.
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

Univ. of Colorado Denver

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

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, integerldb)

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 if SIDE=’R’ and NT = M*(M+1)/2 otherwise.
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

Univ. of Colorado Denver

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

subroutine ztfsm (character transr, character side, character uplo,character trans, character diag, integer m, integer n, complex*16alpha, 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 if SIDE=’R’ and NT = M*(M+1)/2 otherwise.
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

Univ. of Colorado Denver

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

Author

Generated automatically by Doxygen for LAPACK from the source code.