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.
They are called extensively by functions in other chapters of the NAG 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 operations, for vectors and matrices of order .
In many implementations of the NAG 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 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
Table 1 shows the naming scheme for the functions in this chapter.
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.
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 is represented by the two arguments x and incx. The length of the vector, say, is passed as a separate argument,
The increment argument is the spacing (stride) in the array between the elements of the vector. For instance, if ,
then the elements of are in locations of the array
and the intermediate locations are not referenced.
When , the vector element is in the array element . When , the elements are stored in the reverse order so that the vector element is in the array element and hence, in particular, the element is in . The declared length of the array x in the calling function must be at least .
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 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.
Please see Section 3.4.1 in the F07 Chapter Introduction for full details.
Please see Section 3.4.2 in the F07 Chapter Introduction for full details.
3.6.3Rectangular Full Packed (RFP) storage
Please see Section 3.4.3 in the F07 Chapter Introduction for full details.
Please see Section 3.4.4 in the F07 Chapter Introduction for full details.
3.6.5Unit triangular matrices
Please see Section 3.4.5 in the F07 Chapter Introduction for full details.
3.6.6Real diagonal elements of complex Hermitian matrices
Please see Section 3.4.6 in the F07 Chapter Introduction for full details.
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 , operate with the matrix (Not transposed);
if , operate with the Transpose of the matrix;
if , operate with the Conjugate transpose of the matrix.
If , upper triangle or trapezoid of matrix;
if , lower triangle or trapezoid of matrix.
If , unit triangular;
if , nonunit triangular.
If , operate from the left-hand side;
if , operate from the right-hand side.
If , -norm of a matrix;
if , -norm of a matrix;
if , Frobenius or Euclidean norm of a matrix;
if , maximum absolute value of the elements of a matrix (not strictly a norm).
The option argument
specifies different matrix norms whose definitions are given here for reference (for a general by matrix ):
Frobenius or Euclidean norm ():
If is symmetric or Hermitian, .
can also be used to specify the maximum absolute value (if
but this is not a norm in the strict mathematical sense.
Functions in this chapter use the usual NAG Library error-handling.