--- name: "slansf" section: "3lapack" platform: "linux" package: "liblapack-doc" version: "3.2.1-8" release: "Debian 6.0 (Squeeze)" platforms: ["linux"] url: "https://new.carta.tech/slansf/3lapack" --- # slansf(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 real symmetric matrix A in RFP format ## NAME 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 ## SYNOPSIS - REAL FUNCTION SLANSF( NORM, TRANSR, UPLO, N, A, WORK ) - CHARACTER NORM, TRANSR, UPLO - INTEGER N - REAL A( 0: \* ), WORK( 0: \* ) ## 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. ## DESCRIPTION SLANSF returns the value\ \ 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. ## ARGUMENTS - NORM (input) CHARACTER Specifies the value to be returned in SLANSF as described above. - TRANSR (input) CHARACTER Specifies whether the RFP format of A is normal or transposed format. = 'N': RFP format is Normal;\ \= 'T': RFP format is Transpose. - UPLO (input) CHARACTER 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 (input) INTEGER The order of the matrix A. N \>= 0. When N = 0, SLANSF is set to zero. - A (input) 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 (workspace) REAL array, dimension (MAX(1,LWORK)), where LWORK \>= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced. ## 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 first 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\ Reference\ \=========\