NAG CL Interfacef16ukc (zhf_​norm)

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1Purpose

f16ukc returns the value of the $1$-norm, the $\infty$-norm, the Frobenius norm, or the maximum absolute value of the elements of a complex Hermitian matrix $A$ stored in Rectangular Full Packed (RFP) format.

2Specification

 #include
 void f16ukc (Nag_OrderType order, Nag_NormType norm, Nag_RFP_Store transr, Nag_UploType uplo, Integer n, const Complex ar[], double *r, NagError *fail)
The function may be called by the names: f16ukc, nag_blast_zhf_norm or nag_zhf_norm.

3Description

Given a complex $n×n$ symmetric matrix, $A$, f16ukc calculates one of the values given by
 ${‖A‖}_{1}=\underset{j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{i=1}^{n}|{a}_{ij}|$ (the $1$-norm of $A$), ${‖A‖}_{\infty }=\underset{i}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}\sum _{j=1}^{n}|{a}_{ij}|$ (the $\infty$-norm of $A$), ${‖A‖}_{F}={\left(\sum _{i=1}^{n}\sum _{j=1}^{n}{|{a}_{ij}|}^{2}\right)}^{1/2}$ (the Frobenius norm of $A$),   or $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (the maximum absolute element value of $A$).
$A$ is stored in compact form using the RFP format. The RFP storage format is described in Section 3.4.3 in the F07 Chapter Introduction.

4References

Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee https://www.netlib.org/blas/blast-forum/blas-report.pdf
Gustavson F G, Waśniewski J, Dongarra J J and Langou J (2010) Rectangular full packed format for Cholesky's algorithm: factorization, solution, and inversion ACM Trans. Math. Software 37, 2

5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{norm}$Nag_NormType Input
On entry: specifies the value to be returned.
${\mathbf{norm}}=\mathrm{Nag_OneNorm}$
The $1$-norm.
${\mathbf{norm}}=\mathrm{Nag_InfNorm}$
The $\infty$-norm.
${\mathbf{norm}}=\mathrm{Nag_FrobeniusNorm}$
The Frobenius (or Euclidean) norm.
${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$
The value $\underset{i,j}{\mathrm{max}}\phantom{\rule{0.25em}{0ex}}|{a}_{ij}|$ (not a norm).
Constraint: ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, $\mathrm{Nag_InfNorm}$, $\mathrm{Nag_FrobeniusNorm}$ or $\mathrm{Nag_MaxNorm}$.
3: $\mathbf{transr}$Nag_RFP_Store Input
On entry: specifies whether the normal RFP representation of $A$ or its conjugate transpose is stored.
${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$
The matrix $A$ is stored in normal RFP format.
${\mathbf{transr}}=\mathrm{Nag_RFP_ConjTrans}$
The conjugate transpose of the RFP representation of the matrix $A$ is stored.
Constraint: ${\mathbf{transr}}=\mathrm{Nag_RFP_Normal}$ or $\mathrm{Nag_RFP_ConjTrans}$.
4: $\mathbf{uplo}$Nag_UploType Input
On entry: specifies whether the upper or lower triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Upper}$
The upper triangular part of $A$ is stored.
${\mathbf{uplo}}=\mathrm{Nag_Lower}$
The lower triangular part of $A$ is stored.
Constraint: ${\mathbf{uplo}}=\mathrm{Nag_Upper}$ or $\mathrm{Nag_Lower}$.
5: $\mathbf{n}$Integer Input
On entry: $n$, the order of the matrix $A$.
If $n=0$, f16ukc returns immediately.
Constraint: ${\mathbf{n}}\ge 0$.
6: $\mathbf{ar}\left[{\mathbf{n}}×\left({\mathbf{n}}+1\right)/2\right]$const Complex Input
On entry: the upper or lower triangular part (as specified by uplo) of the $n×n$ Hermitian matrix $A$, in either normal or transposed RFP format (as specified by transr). The storage format is described in detail in Section 3.4.3 in the F07 Chapter Introduction.
7: $\mathbf{r}$double * Output
On exit: the value of the norm specified by norm.
8: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_INT
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 0$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.

7Accuracy

The BLAS standard requires accurate implementations which avoid unnecessary over/underflow (see Section 2.7 of Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001)).

8Parallelism and Performance

f16ukc is not threaded in any implementation.

None.

10Example

This example reads in the lower triangular part of a symmetric matrix, converts this to RFP format, then calculates the norm of the matrix for each of the available norm types.

10.1Program Text

Program Text (f16ukce.c)

10.2Program Data

Program Data (f16ukce.d)

10.3Program Results

Program Results (f16ukce.r)