# NAG Library Chapter Introduction

## 1Scope of the Chapter

This chapter is concerned with basic linear algebra functions which perform elementary algebraic operations involving scalars, vectors and matrices. Most functions for such operations conform either to the specifications of the BLAS (Basic Linear Algebra Subprograms) or to the specifications of the BLAST (Basic Linear Algebra Subprograms Technical) Forum. This chapter includes functions conforming to both specifications. Two additional functions for such operations are available in Chapter f06.

## 2Background to the Problems

Most of the functions in this chapter meet the specification of the BLAS as described in Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001), Lawson et al. (1979), Dongarra et al. (1988) and Dongarra et al. (1990).
They are called extensively by functions in other chapters of the NAG C Library, especially in the linear algebra chapters. They are intended to be useful building-blocks for users of the Library who are developing their own applications. The functions fall into four main groups (following the definitions introduced by the BLAS):
• Level 1: vector operations;
• Level 2: matrix-vector operations and matrix operations which includes single matrix operations;
• Level 3: matrix-matrix operations.
The terminology reflects the number of operations involved, so for example a Level 2 function involves $\mathit{O}\left({n}^{2}\right)$ operations, for vectors and matrices of order $n$.
In many implementations of the NAG C Library, the BLAS functions in this chapter serve as interfaces to an efficient machine-specific implementation of the BLAS, usually provided by the vendor of the machine. Such implementations are stringently tested before being used with the NAG C Library, to ensure that they correctly meet the specifications of the BLAS, and that they return the desired accuracy.

### 2.1The Use of BLAS Names

Many of the functions in other chapters of the Library call the functions in this chapter, and in particular a number of the BLAS are called. These functions are usually called by the BLAS name and so, for correct operation of the Library, it is essential that you do not attempt to link your own versions of these functions. If you are in any doubt about how to avoid this, please consult the NAG Technical Support Service.
The BLAS names are used in order to make use of efficient implementations of the functions when these exist. Such implementations are stringently tested before being used, to ensure that they correctly meet the specification of the BLAS, and that they return the desired accuracy (see, for example, Dodson et al. (1991), Dongarra et al. (1988) and Dongarra et al. (1990)).

## 3Recommendations on Choice and Use of Available Functions

### 3.1Naming Scheme

#### 3.1.1NAG names

Table 1 shows the naming scheme for the functions in this chapter.
 Level-1 Level-2 Level-3 integer Chapter f16 function f16d_c – – ‘real’ BLAS function – f16p_c f16y_c ‘real’ Chapter f16 function f16f_c f16q_c – f16r_c ‘real’ BLAST function f16e_c f16q_c – f16r_c ‘complex’ BLAS function – f16s_c f16z_c ‘complex’ Chapter f16 function f16h_c f16t_c – f16u_c ‘complex’ BLAST function f16g_c f16t_c – f16u_c ‘mixed type’ BLAS function f16j_c – –
Table 1
The heading ‘mixed type’ is for functions where a mixture of data types is involved, such as a function that returns the real norm of a complex vector. In future marks of the Library, functions may be included in categories that are currently empty and further categories may be introduced.

### 3.2The Level-1 Vector Functions

The Level-1 functions perform operations either on a single vector or on a pair of vectors.

### 3.3The Level-2 Matrix-vector and Matrix Functions

The Level-2 functions perform operations involving either a matrix on its own, or a matrix and one or more vectors.

### 3.4The Level-3 Matrix-matrix Functions

The Level-3 functions perform operations involving matrix-matrix products.

### 3.5Vector Arguments

Vector arguments are represented by a one-dimensional array, immediately followed by an increment argument whose name consists of the three characters INC followed by the name of the array. For example, a vector $x$ is represented by the two arguments x and incx. The length of the vector, $n$ say, is passed as a separate argument, n.
The increment argument is the spacing (stride) in the array between the elements of the vector. For instance, if ${\mathbf{incx}}=2$, then the elements of $x$ are in locations $x\left(1\right),x\left(3\right),\dots ,x\left(2n-1\right)$ of the array x and the intermediate locations $x\left(2\right),x\left(4\right),\dots ,x\left(2n-2\right)$ are not referenced.
When ${\mathbf{incx}}>0$, the vector element ${x}_{i}$ is in the array element ${\mathbf{x}}\left(1+\left(i-1\right)×{\mathbf{incx}}\right)$. When ${\mathbf{incx}}\le 0$, the elements are stored in the reverse order so that the vector element ${x}_{i}$ is in the array element ${\mathbf{x}}\left(1-\left(n-i\right)×{\mathbf{incx}}\right)$ and hence, in particular, the element ${x}_{n}$ is in ${\mathbf{x}}\left(1\right)$. The declared length of the array x in the calling function must be at least $\left(1+\left({\mathbf{n}}-1\right)×\left|{\mathbf{incx}}\right|\right)$.
Negative increments are permitted only for:
• Level-1 functions which have more than one vector argument;
• Level-2 BLAS functions (but not for other Level-2 functions)
Zero increments are formally permitted for Level-1 functions with more than one argument (in which case the element ${\mathbf{x}}\left(1\right)$ is accessed repeatedly), but their use is strongly discouraged since the effect may be implementation-dependent. There is usually an alternative function in this chapter, with a simplified argument list, to achieve the required purpose. Zero increments are not permitted in the Level-2 BLAS.

### 3.6Matrix Arguments and Storage Schemes

In this chapter the following different storage schemes are used for matrices:
• conventional storage in a two-dimensional array;
• packed and RFP storage for symmetric, Hermitian or triangular matrices;
• band storage for band matrices;
These storage schemes are compatible with those used in Chapters f07 and f08. (Different schemes for packed or band storage are used in a few older functions in Chapters f01, f02, f03 and f04.)
Chapter f01 provides some utility functions for conversion between storage schemes.

#### 3.6.1Conventional storage

Please see Section 3.3.1 in the f07 Chapter Introduction for full details.

#### 3.6.2Packed storage

Please see Section 3.3.2 in the f07 Chapter Introduction for full details.

#### 3.6.3Rectangular Full Packed (RFP) storage

Please see Section 3.3.3 in the f07 Chapter Introduction for full details.

#### 3.6.4Band storage

Please see Section 3.3.4 in the f07 Chapter Introduction for full details.

#### 3.6.5Unit triangular matrices

Please see Section 3.3.5 in the f07 Chapter Introduction for full details.

#### 3.6.6Real diagonal elements of complex Hermitian matrices

Please see Section 3.3.6 in the f07 Chapter Introduction for full details.

### 3.7Option Arguments

In addition to the order argument of type Nag_OrderType, most functions in this Chapter have one or more option arguments of various types; only options of the correct type may be supplied.
The following option arguments are used in this chapter:
• If ${\mathbf{trans}}=\mathrm{NoTranspose}$, operate with the matrix (Not transposed);
• if ${\mathbf{trans}}=\mathrm{Transpose}$, operate with the Transpose of the matrix;
• if ${\mathbf{trans}}=\mathrm{ConjugateTranspose}$, operate with the Conjugate transpose of the matrix.
• If ${\mathbf{uplo}}=\mathrm{Nag_Upper}$, upper triangle or trapezoid of matrix;
• if ${\mathbf{uplo}}=\mathrm{Nag_Lower}$, lower triangle or trapezoid of matrix.
• If ${\mathbf{diag}}=\mathrm{Nag_UnitDiag}$, unit triangular;
• if ${\mathbf{diag}}=\mathrm{NotUnitTriangular}$, nonunit triangular.
• If ${\mathbf{side}}=\mathrm{LeftSide}$, operate from the left-hand side;
• if ${\mathbf{side}}=\mathrm{RightSide}$, operate from the right-hand side.
• If ${\mathbf{norm}}=\mathrm{Nag_OneNorm}$, $1$-norm of a matrix;
• if ${\mathbf{norm}}=\mathrm{Nag_InfNorm}$, $\infty$-norm of a matrix;
• if ${\mathbf{norm}}=\mathrm{Nag_FrobeniusNorm}$, Frobenius or Euclidean norm of a matrix;
• if ${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$, maximum absolute value of the elements of a matrix (not strictly a norm).

#### 3.7.1Matrix norms

The option argument norm specifies different matrix norms whose definitions are given here for reference (for a general $m$ by $n$ matrix $A$):
• One-norm (${\mathbf{norm}}=\mathrm{Nag_OneNorm}$):
 $A1=maxj∑i=1maij;$
• Infinity-norm (${\mathbf{norm}}=\mathrm{Nag_InfNorm}$):
 $A∞=maxi∑j=1naij;$
• Frobenius or Euclidean norm (${\mathbf{norm}}=\mathrm{Nag_FrobeniusNorm}$):
 $AF= ∑i=1m∑j=1naij2 1/2.$
If $A$ is symmetric or Hermitian, ${‖A‖}_{1}={‖A‖}_{\infty }$.
The argument norm can also be used to specify the maximum absolute value ${\mathrm{max}}_{i,j}\left|{a}_{ij}\right|$ (if ${\mathbf{norm}}=\mathrm{Nag_MaxNorm}$), but this is not a norm in the strict mathematical sense.

### 3.8Error Handling

Functions in this chapter use the usual NAG C Library error-handling.

## 4Functionality Index

 Matrix operations,
 complex matrices,
 matrix copy,
 rectangular matrix nag_zge_copy (f16tfc)
 triangular matrix nag_ztr_copy (f16tec)
 matrix initialization,
 matrix-matrix product,
 one matrix Hermitian nag_zhemm (f16zcc)
 one matrix symmetric nag_zsymm (f16ztc)
 one matrix triangular nag_ztrmm (f16zfc)
 rectangular matrices nag_zgemm (f16zac)
 rank-2k update,
 of a Hermitian matrix nag_zher2k (f16zrc)
 of a symmetric matrix nag_zsyr2k (f16zwc)
 rank-k update,
 of a Hermitian matrix nag_zherk (f16zpc)
 of a Hermitian matrix, RFP format nag_zhfrk (f16zqc)
 of a symmetric matrix nag_zsyrk (f16zuc)
 solution of triangular systems of equations nag_ztrsm (f16zjc)
 solution of triangular systems of equations, RFP format nag_ztfsm (f16zlc)
 real matrices,
 matrix copy,
 rectangular matrix nag_dge_copy (f16qfc)
 triangular matrix nag_dtr_copy (f16qec)
 matrix initialization,
 matrix-matrix product,
 one matrix symmetric nag_dsymm (f16ycc)
 one matrix triangular nag_dtrmm (f16yfc)
 rectangular matrices nag_dgemm (f16yac)
 rank-2k update of a symmetric matrix nag_dsyr2k (f16yrc)
 rank-k update,
 of a symmetric matrix nag_dsyrk (f16ypc)
 of a symmetric matrix, RFP format nag_dsfrk (f16yqc)
 solution of triangular systems of equations nag_dtrsm (f16yjc)
 solution of triangular systems of equations, RFP format nag_dtfsm (f16ylc)
 Matrix-vector operations,
 complex matrix and vector(s),
 compute a norm or the element of largest absolute value,
 band matrix nag_zgb_norm (f16ubc)
 general matrix nag_zge_norm (f16uac)
 Hermitian band matrix nag_zhb_norm (f16uec)
 Hermitian matrix nag_zhe_norm (f16ucc)
 Hermitian matrix, RFP format nag_zhf_norm (f16ukc)
 Hermitian packed matrix nag_zhp_norm (f16udc)
 symmetric matrix nag_zsy_norm (f16ufc)
 symmetric packed matrix nag_zsp_norm (f16ugc)
 matrix-vector product,
 Hermitian band matrix nag_zhbmv (f16sdc)
 Hermitian matrix nag_zhemv (f16scc)
 Hermitian packed matrix nag_zhpmv (f16sec)
 rectangular band matrix nag_zgbmv (f16sbc)
 rectangular matrix nag_zgemv (f16sac)
 symmetric matrix nag_zsymv (f16tac)
 symmetric packed matrix nag_zspmv (f16tcc)
 triangular band matrix nag_ztbmv (f16sgc)
 triangular matrix nag_ztrmv (f16sfc)
 triangular packed matrix nag_ztpmv (f16shc)
 rank-1 update,
 Hermitian matrix nag_zher (f16spc)
 Hermitian packed matrix nag_zhpr (f16sqc)
 rectangular matrix, unconjugated vector nag_zger (f16smc)
 rank-2 update,
 Hermitian matrix nag_zher2 (f16src)
 Hermitian packed matrix nag_zhpr2 (f16ssc)
 solution of a system of equations,
 triangular band matrix nag_ztbsv (f16skc)
 triangular matrix nag_ztrsv (f16sjc)
 triangular packed matrix nag_ztpsv (f16slc)
 real matrix and vector(s),
 compute a norm or the element of largest absolute value,
 band matrix nag_dgb_norm (f16rbc)
 general matrix nag_dge_norm (f16rac)
 symmetric band matrix nag_dsb_norm (f16rec)
 symmetric matrix nag_dsy_norm (f16rcc)
 symmetric matrix, RFP format nag_dsf_norm (f16rkc)
 symmetric packed matrix nag_dsp_norm (f16rdc)
 matrix-vector product,
 rectangular band matrix nag_dgbmv (f16pbc)
 rectangular matrix nag_dgemv (f16pac)
 symmetric band matrix nag_dsbmv (f16pdc)
 symmetric matrix nag_dsymv (f16pcc)
 symmetric packed matrix nag_dspmv (f16pec)
 triangular band matrix nag_dtbmv (f16pgc)
 triangular matrix nag_dtrmv (f16pfc)
 triangular packed matrix nag_dtpmv (f16phc)
 rank-1 update,
 rectangular matrix nag_dger (f16pmc)
 symmetric matrix nag_dsyr (f16ppc)
 symmetric packed matrix nag_dspr (f16pqc)
 rank-2 update,
 symmetric matrix nag_dsyr2 (f16prc)
 symmetric packed matrix nag_dspr2 (f16psc)
 solution of a system of equations,
 triangular band matrix nag_dtbsv (f16pkc)
 triangular matrix nag_dtrsv (f16pjc)
 triangular packed matrix nag_dtpsv (f16plc)
 Scalar and vector operations,
 complex vector(s),
 maximum absolute value and location nag_zamax_val (f16jsc)
 minimum absolute value and location nag_zamin_val (f16jtc)
 sum of elements nag_zsum (f16glc)
 sum of two scaled vectors nag_zaxpby (f16gcc)
 sum of two scaled vectors preserving input nag_zwaxpby (f16ghc)
 integer vector(s),
 maximum absolute value and location nag_iamax_val (f16dqc)
 maximum value and location nag_imax_val (f16dnc)
 minimum absolute value and location nag_iamin_val (f16drc)
 minimum value and location nag_imin_val (f16dpc)
 sum of elements nag_isum (f16dlc)
 real vector(s),
 dot product of two vectors with optional scaling and accumulation nag_ddot (f16eac)
 maximum absolute value and location nag_damax_val (f16jqc)
 maximum value and location nag_dmax_val (f16jnc)
 minimum absolute value and location nag_damin_val (f16jrc)
 minimum value and location nag_dmin_val (f16jpc)
 sum of elements nag_dsum (f16elc)
 sum of two scaled vectors nag_daxpby (f16ecc)
 sum of two scaled vectors preserving input nag_dwaxpby (f16ehc)

None.

## 6Functions Withdrawn or Scheduled for Withdrawal

None.
Basic Linear Algebra Subprograms Technical (BLAST) Forum (2001) Basic Linear Algebra Subprograms Technical (BLAST) Forum Standard University of Tennessee, Knoxville, Tennessee http://www.netlib.org/blas/blast-forum/blas-report.pdf
Dodson D S and Grimes R G (1982) Remark on Algorithm 539 ACM Trans. Math. Software 8 403–404
Dongarra J J, Du Croz J J, Duff I S and Hammarling S (1990) A set of Level 3 basic linear algebra subprograms ACM Trans. Math. Software 16 1–28
Dongarra J J, Du Croz J J, Hammarling S and Hanson R J (1988) An extended set of FORTRAN basic linear algebra subprograms ACM Trans. Math. Software 14 1–32
Dongarra J J, Moler C B, Bunch J R and Stewart G W (1979) LINPACK Users' Guide SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Lawson C L, Hanson R J, Kincaid D R and Krogh F T (1979) Basic linear algebra supbrograms for Fortran usage ACM Trans. Math. Software 5 308–325
© The Numerical Algorithms Group Ltd, Oxford, UK. 2017