SRC/zlanhf.f(3)

📦

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🌸 NAME 🌸

okay so bestie, welcome to the SRC/zlanhf.f(3) page from the Library Functions Manual ✨. The name on the cover is SRC/zlanhf.f — yes, that’s literally it, and we are about to get into the whole ZLANHF situation together 💅.

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

double precision function zlanhf (norm, transr, uplo, n, a, work) — ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format 🫶.

💖 Function/Subroutine Documentation 💖

double precision function zlanhf (character norm, character transr,
    character uplo, integer n, complex*16, dimension( 0: * ) a, double
    precision, dimension( 0: * ) work)

ZLANHF returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the element of largest absolute value of a Hermitian matrix in RFP format 🌈.

🎀 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 💖.

🦋 Returns 🦋

ZLANHF gives you this whole menu:

    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, bestie — it’s just the biggest absolute element vibe check 😭.

🍓 Parameters 🍓

NORM

NORM is CHARACTER. It specifies the value to be returned in ZLANHF as described above 💅.

TRANSR

TRANSR is CHARACTER. It 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

UPLO is 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

N is INTEGER. It’s the order of the matrix A. N >= 0. When N = 0, ZLANHF is set to zero 🎀.

A

A is COMPLEX*16 array, dimension ( N*(N+1)/2 ); 🫶

On entry, it’s 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

WORK is DOUBLE PRECISION array, dimension (LWORK), where LWORK >= N when NORM = 'I' or '1' or 'O'; otherwise, WORK is not referenced 😌.

👑 Author 👑

Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

🔥 Further Details 🔥

okay bestie, here’s the full visual tea ☕. First we 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

Definition at line 245 of file zlanhf.f 📌.

📎 Author (the Doxygen one)

Generated automatically by Doxygen for LAPACK from the source code 🤖.

Kisses bestie 💜 — this cutie is from LAPACK, Version 3.12.0, and the page is SRC/zlanhf.f(3). Periodt ✨.

📦 where this page comes from

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