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# LAPACK-3

## man page of LAPACK-3

### LAPACK-3: returns the value of the one norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute

```NAME
LAPACK-3 - 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
DOUBLE PRECISION FUNCTION ZLANHF( NORM, TRANSR, UPLO, N, A, WORK )

CHARACTER    NORM, TRANSR, UPLO

INTEGER      N

DOUBLE       PRECISION WORK( 0: * )

COMPLEX*16   A( 0: * )

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.

DESCRIPTION
ZLANHF returns the value
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.

ARGUMENTS
NORM      (input) CHARACTER
Specifies the value to be returned in ZLANHF 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, ZLANHF 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) DOUBLE PRECISION 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

LAPACK routine (version 3.2.1)   March 2011                       LAPACK-3(3)
```

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