--- name: "clanhf" section: "3lapack" platform: "linux" package: "liblapack-doc" version: "3.2.1-8" release: "Debian 6.0 (Squeeze)" platforms: ["linux"] url: "https://new.carta.tech/clanhf/3lapack" --- # clanhf(3lapack) 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 = '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. ## 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. = 'N': RFP format is Normal\ \= 'C': 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 = 'U' or 'u' RFP A came from an upper triangular matrix UPLO = 'L' or 'l' 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='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 (workspace) REAL array, dimension (LWORK), where LWORK \>= N when NORM = 'I' or '1' or 'O'; 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 = '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\