clanhf (l) - Linux Manuals

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

NAME

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

SYNOPSIS

REAL FUNCTION

CLANHF( NORM, TRANSR, UPLO, N, A, WORK )

    

CHARACTER NORM, TRANSR, UPLO

    

INTEGER N

    

REAL WORK( 0: * )

    

COMPLEX A( 0: * )

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.

DESCRIPTION

CLANHF returns the value

CLANHF = ( max(abs(A(i,j))), NORM = aqMaq or aqmaq

      (

      ( norm1(A),         NORM = aq1aq, aqOaq or aqoaq

      (

      ( normI(A),         NORM = aqIaq or aqiaq

      (

      ( normF(A),         NORM = aqFaq, aqfaq, aqEaq or aqeaq 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.

ARGUMENTS

NORM (input) CHARACTER

Specifies the value to be returned in CLANHF as described above.

TRANSR (input) CHARACTER

Specifies whether the RFP format of A is normal or conjugate-transposed format. = aqNaq: RFP format is Normal
= aqCaq: RFP format is Conjugate-transposed

UPLO (input) CHARACTER

On entry, UPLO specifies whether the RFP matrix A came from an upper or lower triangular matrix as follows: UPLO = aqUaq or aquaq RFP A came from an upper triangular matrix UPLO = aqLaq or aqlaq RFP A came from a lower triangular matrix

N (input) INTEGER

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

A (input) 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=aqNaq 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 = aqCaq then RFP is the Conjugate-transpose of RFP A as defined when TRANSR = aqNaq. The contents of RFP A are defined by UPLO as follows: If UPLO = aqUaq the RFP A contains the ( N*(N+1)/2 ) elements of upper packed A either in normal or conjugate-transpose Format. If UPLO = aqLaq 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 = aqCaq. When TRANSR is aqNaq 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 (workspace) REAL array, dimension (LWORK),

where LWORK >= N when NORM = aqIaq or aq1aq or aqOaq; otherwise, WORK is not referenced.

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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

 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 = aqCaq. 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 = aqNaq. RFP holds AP as follows:
For UPLO = aqUaq 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 = aqLaq 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 = aqNaq.

 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 = aqCaq. 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