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
              'Notes' 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
              'Notes' 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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