slansf (l) - Linux Manuals

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

Command to display slansf manual in Linux: $ man l slansf

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 = 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 SLANSF as described above.
TRANSR (input) CHARACTER: Specifies whether the RFP format of A is normal or transposed format. = aqNaq: RFP format is Normal;
= aqTaq: 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:
= aqUaq: RFP A came from an upper triangular matrix;
= aqLaq: 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 = aqUaq) or lower (if UPLO = aqLaq) 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 = aqIaq or aq1aq or aqOaq; 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 = 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 the 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 the transpose of the last three columns of AP lower.
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 = aqTaq. 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 = 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 the 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 the transpose of the last two columns of AP lower.
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 = aqTaq. 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
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