lanhf(3)

NAME
lanhf - lan{hf,sf}: Hermitian/symmetric matrix, RFP

SYNOPSIS

  Functions
real function clanhf (norm, transr, uplo, n, a, work)
CLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.
double precision function dlansf (norm, transr, uplo, n, a, work)
DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.
real function slansf (norm, transr, uplo, n, a, work)
SLANSF
double precision function zlanhf (norm, transr, uplo, n, a, work)
ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

Detailed Description

Function Documentation

  real function clanhf (character norm, character transr, character uplo, integer n, complex, dimension( 0: * ) a, real, dimension( 0: * ) work)

CLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

Purpose:

    !>
    !> CLANHF  returns the value of the one norm,  or the Frobenius norm, or
    !> the  infinity norm,  or the  element of  largest absolute value  of a
    !> complex Hermitian matrix A in RFP format.
    !>

Returns
    CLANHF

    !>
    !>    CLANHF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
    !>             (
    !>             ( norm1(A),         NORM = '1', 'O' or 'o'
    !>             (
    !>             ( normI(A),         NORM = 'I' or 'i'
    !>             (
    !>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
    !>
    !> where  norm1  denotes the  one norm of a matrix (maximum column sum),
    !> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
    !> normF  denotes the  Frobenius norm of a matrix (square root of sum of
    !> squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.
    !>

Parameters
    NORM

    !>          NORM is CHARACTER
    !>            Specifies the value to be returned in CLANHF as described
    !>            above.
    !>

    TRANSR

    !>          TRANSR is CHARACTER
    !>            Specifies whether the RFP format of A is normal or
    !>            conjugate-transposed format.
    !>            = 'N':  RFP format is Normal
    !>            = 'C':  RFP format is Conjugate-transposed
    !>

    UPLO

    !>          UPLO is CHARACTER
    !>            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
    !>

    N

    !>          N is INTEGER
    !>            The order of the matrix A.  N >= 0.  When N = 0, CLANHF is
    !>            set to zero.
    !>

    A

    !>          A is COMPLEX array, dimension ( 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 ( N*(N+1)/2 ) elements of upper packed A
    !>            either in normal or conjugate-transpose Format. If
    !>            UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) 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.
    !>

    WORK

    !>          WORK is REAL array, dimension (LWORK),
    !>            where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
    !>            WORK is not referenced.
    !>

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
    !>

Definition at line 245 of file clanhf.f.

  double precision function dlansf (character norm, character transr, character uplo, integer n, double precision, dimension( 0: * ) a, double precision, dimension( 0: * ) work)

DLANSF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a symmetric matrix in RFP format.

Purpose:

    !>
    !> DLANSF returns the value of the one norm, or the Frobenius norm, or
    !> the infinity norm, or the element of largest absolute value of a
    !> real symmetric matrix A in RFP format.
    !>

Returns
    DLANSF

    !>
    !>    DLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
    !>             (
    !>             ( norm1(A),         NORM = '1', 'O' or 'o'
    !>             (
    !>             ( normI(A),         NORM = 'I' or 'i'
    !>             (
    !>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
    !>
    !> where  norm1  denotes the  one norm of a matrix (maximum column sum),
    !> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
    !> normF  denotes the  Frobenius norm of a matrix (square root of sum of
    !> squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.
    !>

Parameters
    NORM

    !>          NORM is CHARACTER*1
    !>          Specifies the value to be returned in DLANSF as described
    !>          above.
    !>

    TRANSR

    !>          TRANSR is CHARACTER*1
    !>          Specifies whether the RFP format of A is normal or
    !>          transposed format.
    !>          = 'N':  RFP format is Normal;
    !>          = 'T':  RFP format is Transpose.
    !>

    UPLO

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

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0. When N = 0, DLANSF is
    !>          set to zero.
    !>

    A

    !>          A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 );
    !>          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
    !>          part of the symmetric matrix A stored in RFP format. See the
    !>           below for more details.
    !>          Unchanged on exit.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)),
    !>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
    !>          WORK is not referenced.
    !>

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
    !>

Definition at line 208 of file dlansf.f.

  real function slansf (character norm, character transr, character uplo, integer n, real, dimension( 0: * ) a, real, dimension( 0: * ) work)

SLANSF

Purpose:

    !>
    !> SLANSF returns the value of the one norm, or the Frobenius norm, or
    !> the infinity norm, or the element of largest absolute value of a
    !> real symmetric matrix A in RFP format.
    !>

Returns
    SLANSF

    !>
    !>    SLANSF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
    !>             (
    !>             ( norm1(A),         NORM = '1', 'O' or 'o'
    !>             (
    !>             ( normI(A),         NORM = 'I' or 'i'
    !>             (
    !>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
    !>
    !> where  norm1  denotes the  one norm of a matrix (maximum column sum),
    !> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
    !> normF  denotes the  Frobenius norm of a matrix (square root of sum of
    !> squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.
    !>

Parameters
    NORM

    !>          NORM is CHARACTER*1
    !>          Specifies the value to be returned in SLANSF as described
    !>          above.
    !>

    TRANSR

    !>          TRANSR is CHARACTER*1
    !>          Specifies whether the RFP format of A is normal or
    !>          transposed format.
    !>          = 'N':  RFP format is Normal;
    !>          = 'T':  RFP format is Transpose.
    !>

    UPLO

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

    N

    !>          N is INTEGER
    !>          The order of the matrix A. N >= 0. When N = 0, SLANSF is
    !>          set to zero.
    !>

    A

    !>          A is REAL array, dimension ( N*(N+1)/2 );
    !>          On entry, the upper (if UPLO = 'U') or lower (if UPLO = 'L')
    !>          part of the symmetric matrix A stored in RFP format. See the
    !>           below for more details.
    !>          Unchanged on exit.
    !>

    WORK

    !>          WORK is REAL array, dimension (MAX(1,LWORK)),
    !>          where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
    !>          WORK is not referenced.
    !>

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
    !>

Definition at line 208 of file slansf.f.

  double precision function zlanhf (character norm, character transr, character uplo, integer n, complex*16, dimension( 0: * ) a, double precision, dimension( 0: * ) work)

ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format.

Purpose:

    !>
    !> ZLANHF  returns the value of the one norm,  or the Frobenius norm, or
    !> the  infinity norm,  or the  element of  largest absolute value  of a
    !> complex Hermitian matrix A in RFP format.
    !>

Returns
    ZLANHF

    !>
    !>    ZLANHF = ( max(abs(A(i,j))), NORM = 'M' or 'm'
    !>             (
    !>             ( norm1(A),         NORM = '1', 'O' or 'o'
    !>             (
    !>             ( normI(A),         NORM = 'I' or 'i'
    !>             (
    !>             ( normF(A),         NORM = 'F', 'f', 'E' or 'e'
    !>
    !> where  norm1  denotes the  one norm of a matrix (maximum column sum),
    !> normI  denotes the  infinity norm  of a matrix  (maximum row sum) and
    !> normF  denotes the  Frobenius norm of a matrix (square root of sum of
    !> squares).  Note that  max(abs(A(i,j)))  is not a  matrix norm.
    !>

Parameters
    NORM

    !>          NORM is CHARACTER
    !>            Specifies the value to be returned in ZLANHF as described
    !>            above.
    !>

    TRANSR

    !>          TRANSR is CHARACTER
    !>            Specifies whether the RFP format of A is normal or
    !>            conjugate-transposed format.
    !>            = 'N':  RFP format is Normal
    !>            = 'C':  RFP format is Conjugate-transposed
    !>

    UPLO

    !>          UPLO is CHARACTER
    !>            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
    !>

    N

    !>          N is INTEGER
    !>            The order of the matrix A.  N >= 0.  When N = 0, ZLANHF is
    !>            set to zero.
    !>

    A

    !>          A is COMPLEX*16 array, dimension ( 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 ( N*(N+1)/2 ) elements of upper packed A
    !>            either in normal or conjugate-transpose Format. If
    !>            UPLO = 'L' the RFP A contains the ( N*(N+1) /2 ) 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.
    !>

    WORK

    !>          WORK is DOUBLE PRECISION array, dimension (LWORK),
    !>            where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise,
    !>            WORK is not referenced.
    !>

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
    !>

Definition at line 245 of file zlanhf.f.

Author

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