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F06 (Blas)
Linear Algebra Support Routines

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1 Scope of the Chapter

This chapter is concerned with basic linear algebra routines which perform elementary algebraic operations involving scalars, vectors and matrices. It includes routines which conform to the specifications of the BLAS (Basic Linear Algebra Subprograms).

2 Background to the Problems

A number of the routines in this chapter meet the specification of the Basic Linear Algebra Subprograms (BLAS) as described in Lawson et al. (1979), Dodson et al. (1991), Dongarra et al. (1988) and Dongarra et al. (1990). The first reference describes a set of routines concerned with operations on scalars and vectors: these will be referred to here as the Level-0 and the Level-1 BLAS; the second reference describes a set of routines concerned with operations on sparse vectors: these will be referred to here as the Level-1 Sparse BLAS; the third reference describes a set of routines concerned with matrix-vector operations: these will be referred to here as the Level-2 BLAS; and the fourth reference describes a set of routines concerned with matrix-matrix operations: these will be referred to here as the Level-3 BLAS.
More generally we refer to the scalar routines in the chapter as Level-0 routines, to the vector routines as Level-1 routines, to the matrix-vector and matrix routines as Level-2 routines, and to the matrix-matrix routines as Level-3 routines. The terminology reflects the number of operations involved. For example, a Level-2 routine involves O(n2) operations for an n×n matrix.

2.1 The Use of BLAS Names

Many of the routines in other chapters of the Library call the routines in this chapter, and in particular a number of the BLAS are called. These routines 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 routines. If you are in any doubt about how to avoid this, please consult your computer centre or the NAG Response Centre.
The BLAS names are used in order to make use of efficient implementations of the routines 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)).

2.2 Background Information

Most of the routines in this chapter implement straightforward scalar, vector and matrix operations that need no further explanation beyond a statement of the purpose of the routine. In this section we give some additional background information to those few cases where additional explanation may be necessary. A sub-section is devoted to each topic.

2.2.1 Real plane rotations

There are a number of routines in the chapter concerned with setting up and applying plane rotations. This section discusses the real case and the next section looks at the complex case. For further background information see Golub and Van Loan (1996).
A plane rotation matrix for the (i,j) plane, Rij, is an orthogonal matrix that is different from the unit matrix only in the elements rii, rjj, rij and rji. If we put
R=( rii rij rji rjj ) , (1)
then, in the real case, it is usual to choose Rij so that
R=( -c s -s c ) ,   c=cosθ,   s=sinθ.  
An exception is routine f06fpf which applies the so-called symmetric rotation for which
R=( c -s s -c ) . (2)
The application of plane rotations is straightforward and needs no further elaboration, so further comment is made only on the construction of plane rotations.
The most common use of plane rotations is to choose c and s so that for given a and b,
( -c s -s c ) ( a b )=( d 0 ) . (3)
In such an application the matrix R is often termed a Givens rotation matrix. There are two approaches to the construction of real Givens rotations in Chapter F06.
The BLAS routine f06aaf, see Lawson et al. (1979) and Dodson and Grimes (1982), computes c, s and d as
c={ a/d, d0, 1, d=0,     s={ b/d, d0, 0, d=0, (4)
where σ={ signa, |a|>|b| signb, |a||b| .
The value z defined as
z={ s, |s|<c  or  c=0 1/c, 0<|c|s (5)
is also computed and this enables c and s to be reconstructed from the single value z as
c={ 0, z=1 (1-z2)1/2, |z|<1 1/z, |z|>1     s={ 1, z=1 z, |z|<1 (1-c2)1/2, |z|>1 .  
The other Chapter F06 routines for constructing Givens rotations are based on the computation of the tangent, t=tanθ. t is computed as
t={ 0, b=0 b/a, |b||a|.flmax,b0 sign(b/a).flmax, |b|>|a|.flmax sign(b).flmax, b0,a=0 (6)
where flmax=1/flmin and flmin is the small positive value returned by x02amf. The values of c and s are then computed or reconstructed via t as
c={ 1/(1+t2)1/2, eps|t|1/eps 1, |t|<eps 1/|t|, |t|>1/eps     s={ c.t, eps|t|1/eps t, |t|<eps signt, |t|>1/eps (7)
where eps is the machine precision. Note that c is always non-negative in this scheme and that the same expressions are used in the initial computation of c and s from a and b as in any subsequent recovery of c and s via t. This is the approach used by many of the NAG Library routines that require plane rotations. d is computed simply as
You need not be too concerned with the above detail, since routines are provided for setting up, recovering and applying such rotations.
Another use of plane rotations is to choose c and s so that for given x, y and z
( -c s -s c ) ( x y y z ) ( c -s s -c )=( a 0 0 b ) . (8)
In such an application the matrix R is often termed a Jacobi rotation matrix. The routine that generates a Jacobi rotation (f06bef) first computes the tangent t and then computes c and s via t as described above for the Givens rotation.

2.2.2 Complex plane rotations

In the complex case a plane rotation matrix for the (i,j) plane, Rij is a unitary matrix and, analogously to the real case, it is usual to choose Rij so that
R=( -c¯ s¯ -s c ) ,   |c|2+|s|2=1, (9)
where a¯ denotes the complex conjugate of a.
The BLAS (see Lawson et al. (1979)) do not contain a routine for the generation of complex rotations, and so the routines in Chapter F06 are all based upon computing c and s via t=b/a in an analogous manner to the real case. R can be chosen to have either c real, or s real and there are routines for both cases.
When c is real then it is non-negative and the transformation
( -c s¯ -s c ) ( a b )=( d 0 ) (10)
is such that if a is real then d is also real.
When s is real then the transformation
( -c¯ s -s c ) ( a b )=( d 0 ) (11)
is such that if b is real then d is also real.

2.2.3 Elementary real (Householder) reflections

There are a number of routines in the chapter concerned with setting up and applying Householder transformations. This section discusses the real case and the next section looks at the complex case. For further background information see Golub and Van Loan (1996).
A real elementary reflector, P, is a matrix of the form
P=I-μuuT,   μuTu=2, (12)
where μ is a scalar and u is a vector, and P is both symmetric and orthogonal. In the routines in Chapter F06, u is expressed in the form
u=( ζ z ) ,   ζ​ a scalar (13)
because in many applications ζ and z are not contiguous elements. The usual use of elementary reflectors is to choose μ and u so that for given α and x
P ( α x )=( β 0 ) ,   α​ and ​β​ scalars. (14)
Such a transformation is often termed a Householder transformation. There are two choices of μ and u available in Chapter F06.
The first form of the Householder transformation is compatible with that used by LINPACK (see Dongarra et al. (1979)) and has
μ=1/ζ. (15)
This choice makes ζ satisfy
The second form, and the form used by many of the NAG Library routines, has
μ=1 (16)
which makes
In both cases the special setting
ζ=0 (17)
is used by the routines to flag the case where P=I.
Note that while there are routines to apply an elementary reflector to a vector, there are no routines available in Chapter F06 to apply an elementary reflector to a matrix. This is because such transformations can readily and efficiently be achieved by calls to the matrix-vector Level 2 BLAS routines. For example, to form PA for a given matrix
PA = (I-μuuT)A=A-μuuTA = A-μubT,  b=ATu, (18)
and so we can call a matrix-vector product routine to form b=ATu and then call a rank-one update routine to form (A-μubT). Of course, we must skip the transformation when ζ has been set to zero.

2.2.4 Elementary complex (Householder) reflections

A complex elementary reflector, P, is a matrix of the form
P=I-μuuH,   μuHu=2,   μ​ real,  
where uH denotes the complex conjugate of uT, and P is both Hermitian and unitary. For convenience in a number of applications this definition can be generalized slightly by allowing μ to be complex and so defining the generalized elementary reflector as
P=I-μuuH,   |μ|2uHu=μ+μ¯ (19)
for which P is still unitary, but is no longer Hermitian.
The Chapter F06 routines choose μ and ζ so that
Re(μ)=1,   Im(ζ)=0 (20)
and this reduces to (12) with the choice (16) when μ and u are real. This choice is used because μ and u can now be chosen so that in the Householder transformation (14) we can make
and, as in the real case,
Rather than returning μ and ζ as separate arguments the Chapter F06 routines return the single complex value θ defined as
θ=ζ+i.Im(μ),  i=-1.  
Obviously ζ and μ can be recovered as
ζ=Re(θ),  μ=1+i.Im(θ).  
The special setting
is used to flag the case where P=I, and
Re(θ)0,  Im(θ)0  
is used to flag the case where
P=( γ 0 0 I ) ,   γ​ a scalar (21)
and in this case θ actually contains the value of γ. Notice that with both (18) and (21) we merely have to supply θ¯ rather than θ in order to represent PH.

3 Recommendations on Choice and Use of Available Routines

3.1 Naming Scheme

3.1.1 NAG names

Table 1 shows the naming scheme for the routines in this chapter.
Table 1
Level-0 Level-1 Level-2 Level-3
integer Chapter F06 routine F06D_F
‘real’ BLAS routine F06A_F F06E_F F06P_F F06Y_F
‘real’ Chapter F06 routine F06B_F F06F_F F06Q_F
‘complex’ BLAS routine F06G_F F06S_F F06Z_F
‘complex’ Chapter F06 routine F06C_F F06H_F F06T_F
‘mixed type’ BLAS routine F06J_F
‘mixed type’ Chapter F06 routine F06K_F F06V_F
‘real’ and ‘complex’ LAPACK routines F06W_F F06W_F
The heading ‘mixed type’ is for routines where a mixture of data types is involved, such as a routine that returns the real Euclidean length of a complex vector. In future marks of the Library, routines may be included in categories that are currently empty and further categories may be introduced.

3.1.2 BLAS names

Those routines which conform to the specifications of the BLAS may be called either by their NAG names or by their BLAS names.
In many implementations of the NAG Library, references to BLAS names may be linked 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. Use of BLAS names is recommended for efficiency.
References to NAG routine names (beginning F06-) are always linked to the code provided in the NAG Library and may be significantly slower than the equivalent BLAS routine.
The names of the Level-2 and Level-3 BLAS follow a simple scheme (which is similar to that used for LAPACK routines in Chapters F07 and F08). Each name has the structure XYYZZZ, where the components have the following meanings:
Thus the routine dgemv performs a matrix-vector product involving a real general matrix in double precision; the corresponding routine for a complex general matrix is zgemv.
The names of the Level-1 BLAS mostly follow the same convention for the initial letter (S-, C-, D- or Z-), except for a few involving data of mixed type, where the first two characters are precision-dependent.

3.1.3 LAPACK names

There are some LAPACK routines in this chapter that have BLAS-like functionalty. Four are equivalent to BLAS routines but for matrices stored in Rectangular Full Packed (RFP) format. The naming convention for these is as above with the addition of the matrix types:
HF (complex) Hermitian (RFP storage)
TF triangular (RFP storage)
SF symmetric (RFP storage)
There are an additonal two that compute norms of RFP matrices. These have second and third letters LA (signifying LAPACK), fourth letter N (signifying norm), and fifth and sixth letter signifying matrix type as above. For example zlanhf computes the norm of a Hermitian matrix in RFP format.

3.2 The Level-0 Scalar Routines

The Level-0 routines perform operations on scalars or on vectors or matrices of order 2.

3.3 The Level-1 Vector Routines

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

3.4 The Level-2 Matrix-vector and Matrix Routines

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

3.5 The Level-3 Matrix-matrix Routines

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

3.6 Vector Arguments

Vector arguments (except in the Level-1 Sparse BLAS) 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 incx=2, then the elements of x are in locations x(1),x(3),,x(2n-1) of the array x and the intermediate locations x(2),x(4),,x(2n-2) are not referenced.
When incx>0, the vector element xi is in the array element x(1+(i-1)×incx). When incx0, the elements are stored in the reverse order so that the vector element xi is in the array element x(1-(n-i)×incx) and hence, in particular, the element xn is in x(1). The declared length of the array x in the calling subroutine must be at least (1+(n-1)×|incx|).
Negative increments are permitted only for:
Zero increments are formally permitted for Level-1 routines with more than one argument (in which case the element x(1) is accessed repeatedly), but their use is strongly discouraged since the effect may be implementation-dependent. There is usually an alternative routine in this chapter, with a simplified argument list, to achieve the required purpose. Zero increments are not permitted in the Level-2 BLAS.
In the Level-1 Sparse BLAS, each routine operates on two vectors x and y. The vector x is stored as a compressed sparse vector, and is represented by the three arguments nz, x and indx; nz is the number of ‘interesting’ (usually nonzero) elements of x, and indx is a one-dimensional index array such that
x(indx(k))=x(k),  k=1,2,,nz.  
The (mathematical) length of the vector, n say, does not need to be supplied; it is assumed that 1indx(k)n. For example, the vector
could be represented with nz=3, x=(4,1,6), indx=(2,5,9). The second vector y is stored conventionally, and is represented simply by the one-dimensional array y, with yi in Y(i); the increment is assumed to be 1. Only the elements Y(indx(k)) are referenced.
Non-positive values of nz are permitted, in which case the routines return immediately — except that functions set their value to zero before returning. For those routines where y is an output argument the values in the array indx must be distinct; violating this condition may yield incorrect results.

3.7 Matrix Arguments and Storage Schemes

In this chapter the following different storage schemes are used for 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 routines in Chapters F01, F02, F03 and F04.)
Chapter F01 provides some utility routines for conversion between storage schemes.
In the examples, * indicates an array element which need not be set and is not referenced by the routines. The examples illustrate only the relevant leading rows and columns of the arrays; array arguments may of course have additional rows or columns, according to the usual rules for passing array arguments in Fortran.

3.7.1 Conventional storage

Please see Section 3.3.1 in the F07 Chapter Introduction for full details.

3.7.2 Packed storage

Please see Section 3.3.2 in the F07 Chapter Introduction for full details.

3.7.3 Rectangular Full Packed (RFP) storage

Please see Section 3.3.3 in the F07 Chapter Introduction for full details.

3.7.4 Band storage

Please see Section 3.3.4 in the F07 Chapter Introduction for full details.

3.7.5 Unit triangular matrices

Please see Section 3.3.5 in the F07 Chapter Introduction for full details.

3.7.6 Real diagonal elements of complex Hermitian matrices

Please see Section 3.3.6 in the F07 Chapter Introduction for full details.

3.7.7 Spiked matrices

A few routines in this chapter (f06qsf, f06qwf, f06tsf and f06twf) deal with upper spiked matrices. These are upper triangular matrices with an additional nonzero row or column below the diagonal.
The position of the spike is defined by indices k1 and k2; it is assumed that k1<k2. A row spike has nonzero elements in the k2th row, ak2,k for k=k1,k1+1,,k2-1; a column spike has nonzero elements in the k1th column, ak+1,k1 for k=k1,k1+1,,k2-1. For example, when n=6, k1=2 and k2=5:
Row spike Column spike
( a11 a12 a13 a14 a15 a16 a22 a23 a24 a25 a26 a33 a34 a35 a36 a44 a45 a46 a52 a53 a54 a55 a56 a66 ) ( a11 a12 a13 a14 a15 a16 a22 a23 a24 a25 a26 a32 a33 a34 a35 a36 a42 a44 a45 a46 a52 a55 a56 a66 )
The storage scheme adopted by the routines in this chapter is for the upper triangular part of the spiked matrix to be stored conventionally in a two-dimensional array a, with the subdiagonal elements of the spike stored in a separate vector.

3.8 Option Parameters

Many of the routines in this chapter have one or more option arguments, of type CHARACTER. The descriptions in the routine documents refer only to upper-case values (for example uplo='U' or uplo='L'); however, in every case, the corresponding lower-case characters may be supplied (with the same meaning). Any other value is illegal.
A longer character string can be passed as the actual argument, making the calling program more readable, but only the first character is significant. (This is a feature of Fortran.) For example:
Call dtrsv('Upper','Transpose','Non-unit',...)
The following option arguments are used in this chapter:

3.8.1 Matrix norms

The option argument norm specifies different matrix norms whose definitions are given here for reference (for a general m×n matrix A):
If A is symmetric or Hermitian, A1=A.
The argument norm can also be used to specify the maximum absolute value maxi,j|aij| (if norm='M'), but this is not a norm in the strict mathematical sense.

3.9 Error Handling

Routines in this chapter do not use the usual NAG Library error-handling mechanism, involving the argument IFAIL.
If one of the Level-2 or Level-3 BLAS routines is called with an invalid value of one of its arguments, then an error message is output on the error message unit (see x04aaf), giving the name of the routine and the number of the first invalid argument, and execution of the program is terminated. The following values of arguments are invalid:
Zero values for the matrix dimensions m, n or k are considered valid.
The other routines in this chapter do not report any errors in their arguments. Normally, if called, for example, with an unspecified value for one of the option arguments, or with a negative value of one of the problem dimensions m or n, they simply do nothing and return immediately.

4 Functionality Index

Level 0 (Scalar) operations,  
complex numbers,  
apply similarity rotation to 2×2 Hermitian matrix   f06chf
generate a plane rotation, storing the tangent, real cosine   f06caf
generate a plane rotation, storing the tangent, real sine   f06cbf
quotient of two numbers, with overflow flag   f06clf
recover cosine and sine from given tangent, real cosine   f06ccf
recover cosine and sine from given tangent, real sine   f06cdf
real numbers,  
apply similarity rotation to 2×2 symmetric matrix   f06bhf
compute (a2+b2)   f06bnf
compute Euclidean norm from scaled form   f06bmf
eigenvalue of 2×2 symmetric matrix   f06bpf
generate a Jacobi plane rotation   f06bef
generate a plane rotation   f06aaf
generate a plane rotation storing the tangent   f06baf
quotient of two numbers, with overflow flag   f06blf
recover cosine and sine from given tangent   f06bcf
Level 1 (Vector) operations,  
complex vector(s),  
add scalar times a vector to another vector   f06gcf
apply a complex plane rotation   f06hpf
apply an elementary reflection to a vector   f06htf
apply a real plane rotation   f06kpf
apply plane rotation,  
real cosine, complex sine   f06hmf
broadcast a scalar into a vector   f06hbf
copy a real vector to a complex vector   f06kff
copy a vector   f06gff
dot product of two vectors, conjugated   f06gbf
dot product of two vectors, unconjugated   f06gaf
Euclidean norm of a vector   f06jjf
generate an elementary reflection   f06hrf
generate a sequence of plane rotations   f06hqf
index of element of largest absolute value   f06jmf
multiply vector by a complex scalar   f06gdf
multiply vector by a complex scalar, preserving input vector   f06hdf
multiply vector by a real scalar   f06jdf
multiply vector by a real scalar, preserving input vector   f06kdf
multiply vector by complex diagonal matrix   f06hcf
multiply vector by real diagonal matrix   f06kcf
multiply vector by reciprocal of a real scalar   f06kef
negate a vector   f06hgf
sum of absolute values of vector-elements   f06jkf
swap two vectors   f06ggf
update Euclidean norm in scaled form   f06kjf
integer vector(s),  
broadcast a scalar into a vector   f06dbf
copy a vector   f06dff
real vector(s),  
add scalar times a vector to another vector   f06ecf
apply an elementary reflection to a vector (Linpack style)   f06fuf
apply an elementary reflection to a vector (NAG style)   f06ftf
apply a symmetric plane rotation to two vectors   f06fpf
apply plane rotation   f06epf
broadcast a scalar into a vector   f06fbf
copy a vector   f06eff
cosine of angle between two vectors   f06faf
dot product of two vectors   f06eaf
elements of largest and smallest absolute value   f06flf
Euclidean norm of a vector   f06ejf
generate an elementary reflection (Linpack style)   f06fsf
generate an elementary reflection (NAG style)   f06frf
generate a sequence of plane rotations   f06fqf
index of element of largest absolute value   f06jlf
index of last non-negligible element   f06klf
multiply vector by a scalar   f06edf
multiply vector by a scalar, preserving input vector   f06fdf
multiply vector by diagonal matrix   f06fcf
multiply vector by reciprocal of a scalar   f06fef
negate a vector   f06fgf
sum of absolute values of vector-elements   f06ekf
swap two vectors   f06egf
update Euclidean norm in scaled form   f06fjf
weighted Euclidean norm of a vector   f06fkf
Level 2 (Matrix-vector and matrix) operations,  
complex matrix and vector(s),  
apply sequence of plane rotations to a rectangular matrix,  
complex cosine, real sine   f06tyf
real cosine, complex sine   f06txf
real cosine and sine   f06vxf
compute a norm or the element of largest absolute value,  
band matrix   f06ubf
general matrix   f06uaf
Hermitian band matrix   f06uef
Hermitian matrix   f06ucf
Hermitian matrix, packed form   f06udf
Hermitian matrix, RFP format   f06wnf
Hermitian tridiagonal matrix   f06upf
Hessenberg matrix   f06umf
symmetric band matrix   f06uhf
symmetric matrix   f06uff
symmetric matrix, packed form   f06ugf
trapezoidal matrix   f06ujf
triangular band matrix   f06ulf
triangular matrix, packed form   f06ukf
tridiagonal matrix   f06unf
compute upper Hessenberg matrix by applying sequence of plane rotations to an upper triangular matrix   f06tvf
compute upper spiked matrix by applying sequence of plane rotations to an upper triangular matrix   f06twf
matrix initialization   f06thf
matrix-vector product,  
Hermitian band matrix   f06sdf
Hermitian matrix   f06scf
Hermitian packed matrix   f06sef
rectangular band matrix   f06sbf
rectangular matrix   f06saf
symmetric matrix   f06taf
symmetric packed matrix   f06tcf
triangular band matrix   f06sgf
triangular matrix   f06sff
triangular packed matrix   f06shf
permute rows or columns of a matrix,  
permutations represented by an integer array   f06vjf
permutations represented by a real array   f06vkf
QR factorization by sequence of plane rotations,  
of rank-1 update of upper triangular matrix   f06tpf
of upper triangular matrix augmented by a full row   f06tqf
QR factorization of UZ or RQ factorization of ZU, where U is upper triangular and Z is a sequence of plane rotations   f06ttf
QR or RQ factorization by sequence of plane rotations,  
of upper Hessenberg matrix   f06trf
of upper spiked matrix   f06tsf
rank-1 update,  
Hermitian matrix   f06spf
Hermitian packed matrix   f06sqf
rectangular matrix, conjugated vector   f06snf
rectangular matrix, unconjugated vector   f06smf
symmetric matrix   f06tbf
symmetric packed matrix   f06tdf
rank-2 update,  
Hermitian matrix   f06srf
Hermitian packed matrix   f06ssf
matrix copy, rectangular or trapezoidal   f06tff
solution of a system of equations,  
triangular band matrix   f06skf
triangular matrix   f06sjf
triangular packed matrix   f06slf
unitary similarity transformation of a Hermitian matrix,  
as sequence of plane rotations   f06tmf
real matrix and vector(s),  
apply sequence of plane rotations to a rectangular matrix   f06qxf
compute a norm or the element of largest absolute value,  
band matrix   f06rbf
general matrix   f06raf
Hessenberg matrix   f06rmf
matrix initialization   f06qhf
symmetric band matrix   f06ref
symmetric matrix   f06rcf
symmetric matrix, packed form   f06rdf
symmetric matrix, RFP format   f06waf
symmetric tridiagonal matrix   f06rpf
trapezoidal matrix   f06rjf
triangular band matrix   f06rlf
triangular matrix, packed form   f06rkf
tridiagonal matrix   f06rnf
compute upper Hessenberg matrix by applying sequence of plane rotations to an upper triangular matrix   f06qvf
compute upper spiked matrix by applying sequence of plane rotations to an upper triangular matrix   f06qwf
matrix-vector product,  
rectangular band matrix   f06pbf
rectangular matrix   f06paf
symmetric band matrix   f06pdf
symmetric matrix   f06pcf
symmetric packed matrix   f06pef
triangular band matrix   f06pgf
triangular matrix   f06pff
triangular packed matrix   f06phf
orthogonal similarity transformation of a symmetric matrix,  
as sequence of plane rotations   f06qmf
permute rows or columns of a matrix,  
permutations represented by an integer array   f06qjf
permutations represented by a real array   f06qkf
QR factorization by sequence of plane rotations,  
of rank-1 update of upper triangular matrix   f06qpf
of upper triangular matrix augmented by a full row   f06qqf
QR factorization of UZ or RQ factorization of ZU, where U is upper triangular and Z is a sequence of plane rotations   f06qtf
QR or RQ factorization by sequence of plane rotations,  
of upper Hessenberg matrix   f06qrf
of upper spiked matrix   f06qsf
rank-1 update,  
rectangular matrix   f06pmf
symmetric matrix   f06ppf
symmetric packed matrix   f06pqf
rank-2 update,  
matrix copy, rectangular or trapezoidal   f06qff
symmetric matrix   f06prf
symmetric packed matrix   f06psf
solution of a system of equations,  
triangular band matrix   f06pkf
triangular matrix   f06pjf
triangular packed matrix   f06plf
Level 3 (Matrix-matrix) operations,  
complex matrices,  
matrix-matrix product,  
one matrix Hermitian   f06zcf
one matrix symmetric   f06ztf
one matrix triangular   f06zff
two rectangular matrices   f06zaf
rank-2k update,  
of a Hermitian matrix   f06zrf
of a symmetric matrix   f06zwf
rank-k update,  
of a Hermitian matrix   f06zpf
of a Hermitian matrix, RFP format   f06wqf
of a symmetric matrix   f06zuf
solution of triangular systems of equations   f06zjf
solution of triangular systems of equations, RFP format   f06wpf
real matrices,  
matrix-matrix product,  
one matrix symmetric   f06ycf
one matrix triangular   f06yff
rectangular matrices   f06yaf
rank-2k update of a symmetric matrix   f06yrf
rank-k update,  
of a symmetric matrix   f06ypf
of a symmetric matrix, RFP format   f06wcf
solution of triangular systems of equations   f06yjf
solution of triangular systems of equations, RFP format   f06wbf
Sparse level 1 (vector) operations,  
complex vector(s),  
add scalar times sparse vector to a full vector   f06gtf
dot product of a sparse and a full vector (conjugated)   f06gsf
dot product of a sparse and a full vector (unconjugated)   f06grf
gather and set to zero a sparse vector   f06gvf
gather sparse vector   f06guf
scatter sparse vector   f06gwf
real vector(s),  
add scalar times sparse vector to a full vector   f06etf
apply plane rotation to a sparse and a full vector   f06exf
dot product of a sparse and a full vector   f06erf
gather and set to zero a sparse vector   f06evf
gather sparse vector   f06euf
scatter sparse vector   f06ewf
BLAS Routines

5 Auxiliary Routines Associated with Library Routine Arguments


6 Withdrawn or Deprecated Routines


7 References

Dodson D S and Grimes R G (1982) Remark on Algorithm 539 ACM Trans. Math. Software 8 403–404
Dodson D S, Grimes R G and Lewis J G (1991) Sparse extensions to the Fortran basic linear algebra subprograms ACM Trans. Math. Software 17 253–263
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