Glossary
A^{H} 
Denotes the conjugate transpose of a general matrix A. See also conjugate matrix. 
A^{T} 
Denotes the transpose of a general matrix A. See also transpose. 
band matrix 
A general mbyn matrix A such that a_{ij} = 0 for i  j > l, where 1 < l < min(m, n). For example, any tridiagonal matrix is a band matrix. 
band storage 
A special storage scheme for band matrices. A matrix is stored in a twodimensional array: columns of the matrix are stored in the corresponding columns of the array, and diagonals of the matrix are stored in rows of the array. 
BLAS 
Abbreviation for Basic Linear Algebra Subprograms. These subprograms implement vector, matrixvector, and matrixmatrix operations. 
BRNG 
Abbreviation for Basic Random Number Generator. Basic random number generators are pseudorandom number generators imitating i.i.d. random number sequences of uniform distribution. Distributions other than uniform are generated by applying different transformation techniques to the sequences of random numbers of uniform distribution. 
BRNG registration 
Standardized mechanism that allows a user to include a userdesigned BRNG into the VSL and use it along with the predefined VSL basic generators. 
BunchKaufman factorization 
Representation of a real symmetric or complex Hermitian matrix A in the form A = PUDU^{H}P^{T} (or A = PLDL^{H}P^{T}) where P is a permutation matrix, U and L are upper and lower triangular matrices with unit diagonal, and D is a Hermitian blockdiagonal matrix with 1by1 and 2by2 diagonal blocks. U and L have 2by2 unit diagonal blocks corresponding to the 2by2 blocks of D. 
c 
When found as the first letter of routine names, c indicates the usage of singleprecision complex data type. 
CBLAS 
C interface to the BLAS. See BLAS. 
CDF 
Cumulative Distribution Function. The function that determines probability distribution for univariate or multivariate random variable X. For univariate distribution the cumulative distribution function is the function of real argument x, which for every x takes a value equal to probability of the event A: X≤x. For multivariate distribution the cumulative distribution function is the function of a real vector x = (x_{1},x_{2}, ..., x_{n}), which, for every x, takes a value equal to probability of the event A = (X_{1}≤x_{1} & X_{2}≤x_{2}, & ..., & X_{n}≤x_{n}). 
Cholesky factorization 
Representation of a symmetric positivedefinite or, for complex data, Hermitian positivedefinite matrix A in the form A = U^{H}U or A = LL^{H}, where L is a lower triangular matrix and U is an upper triangular matrix. 
condition number 
The number κ(A) defined for a given square matrix A as follows: κ(A) = A A^{−1}. 
conjugate matrix 
The matrix A^{H} defined for a given general matrix A as follows: (A^{H})_{ij} = (a_{ji})^{*}. 
conjugate number 
The conjugate of a complex number z = a + bi is z^{*} = a  bi. 
d 
When found as the first letter of routine names, d indicates the usage of doubleprecision real data type. 
dot product 
The number denoted x · y and defined for given vectors x and y as follows: x · y = Σ_{i}x_{i}y_{i}. Here x_{i} and y_{i} stand for the ith elements of x and y, respectively. 
double precision 
A floatingpoint data type. On Intel® processors, this data type allows you to store real numbers x such that 2.23*10^{−308}<  x  < 1.79*10^{308}. For this data type, the machine precision ε is approximately 10^{−15}, which means that doubleprecision numbers usually contain no more than 15 significant decimal digits. For more information, refer to Intel® 64 and IA32 Architectures Software Developer's Manual, Volume 1: Basic Architecture. 
eigenvalue 
See eigenvalue problem. 
eigenvalue problem 
A problem of finding nonzero vectors x and numbers λ (for a given square matrix A) such that Ax = λx. Here the numbers λ are called the eigenvalues of the matrix A and the vectors x are called the eigenvectors of the matrix A. 
eigenvector 
See eigenvalue problem. 
elementary reflector(Householder matrix) 
Matrix of a general form H = I−τvv^{T}, where v is a column vector and τ is a scalar. In LAPACK elementary reflectors are used, for example, to represent the matrix Q in the QR factorization (the matrix Q is represented as a product of elementary reflectors). 
factorization 
Representation of a matrix as a product of matrices. See also BunchKaufman factorization, Cholesky factorization, LU factorization, LQ factorization, QR factorization, Schur factorization. 
FFTs 
Abbreviation for Fast Fourier Transforms. See"Fourier Transform Functions". 
full storage 
A storage scheme allowing you to store matrices of any kind. A matrix A is stored in a twodimensional array a, with the matrix element a_{ij} stored in the array element a(i,j). 
Hermitian matrix 
A square matrix A that is equal to its conjugate matrix A^{H}. The conjugate A^{H} is defined as follows: (A^{H})_{ij} = (a_{ji})^{*}. 

See identity matrix. 
identity matrix 
A square matrix whose diagonal elements are 1, and offdiagonal elements are 0. For any matrix A, AI = A and IA = A. 
i.i.d. 
Independent Identically Distributed. 
inplace 
Qualifier of an operation. A function that performs its operation inplace takes its input from an array and returns its output to the same array. 
Intel MKL 
Abbreviation for Intel® Math Kernel Library. 
inverse matrix 
The matrix denoted as A^{−1} and defined for a given square matrix A as follows: AA^{−1} = A^{−1}A = I. A^{−1} does not exist for singular matrices A. 
LQ factorization 
Representation of an mbyn matrix A as A = LQ or A = (L 0)Q. Here Q is an nbyn orthogonal (unitary) matrix. For m≤n, L is an mbym lower triangular matrix with real diagonal elements; for m > n,
where L_{1} is an nbyn lower triangular matrix, and L_{2} is a rectangular matrix. 
LU factorization 
Representation of a general mbyn matrix A as A = PLU, where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n) and U is upper triangular (upper trapezoidal if m < n). 
machine precision 
The number ε determining the precision of the machine representation of real numbers. For Intel® architecture, the machine precision is approximately 10^{−7} for singleprecision data, and approximately 10^{−15} for doubleprecision data. The precision also determines the number of significant decimal digits in the machine representation of real numbers. See also double precision and single precision. 
MPI 
Message Passing Interface. This standard defines the user interface and functionality for a wide range of messagepassing capabilities in parallel computing. 
MPICH 
A freely available, portable implementation of MPI standard for messagepassing libraries. 
orthogonal matrix 
A real square matrix A whose transpose and inverse are equal, that is, A^{T} = A^{1}, and therefore AA^{T} = A^{T}A = I. All eigenvalues of an orthogonal matrix have the absolute value 1. 
packed storage 
A storage scheme allowing you to store symmetric, Hermitian, or triangular matrices more compactly. The upper or lower triangle of a matrix is packed by columns in a onedimensional array. 

Probability Density Function. The function that determines probability distribution for univariate or multivariate continuous random variable X. The probability density function f(x) is closely related with the cumulative distribution function F(x). For univariate distribution the relation is
For multivariate distribution the relation is 
positivedefinite matrix 
A square matrix A such that Ax · x > 0 for any nonzero vector x. Here · denotes the dot product. 
pseudorandom number generator 
A completely deterministic algorithm that imitates truly random sequences. 
QR factorization 
Representation of an mbyn matrix A as A = QR, where Q is an mbym orthogonal (unitary) matrix, and R is nbyn upper triangular with real diagonal elements (if m≥n) or trapezoidal (if m < n) matrix. 
random stream 
An abstract source of independent identically distributed random numbers of uniform distribution. In this manual a random stream points to a structure that uniquely defines a random number sequence generated by a basic generator associated with a given random stream. 
RNG 
Abbreviation for Random Number Generator. In this manual the term "random number generators" stands for pseudorandom number generators, that is, generators based on completely deterministic algorithms imitating truly random sequences. 
Rectangular Full Packed (RFP) storage 
A storage scheme combining the full and packed storage schemes for the upper or lower triangle of the matrix. This combination enables using half of the full storage as packed storage while maintaining efficiency by using Level 3 BLAS/LAPACK kernels as the full storage. 
s 
When found as the first letter of routine names, s indicates the usage of singleprecision real data type. 
ScaLAPACK 
Stands for Scalable Linear Algebra PACKage. 
Schur factorization 
Representation of a square matrix A in the form A = ZTZ^{H}. Here T is an upper quasitriangular matrix (for complex A, triangular matrix) called the Schur form of A; the matrix Z is orthogonal (for complex A, unitary). Columns of Z are called Schur vectors. 
single precision 
A floatingpoint data type. On Intel® processors, this data type allows you to store real numbers x such that 1.18*10^{−38} <  x  < 3.40*10^{38}. For this data type, the machine precision (ε) is approximately 10^{−7}, which means that singleprecision numbers usually contain no more than 7 significant decimal digits. For more information, refer to Intel® 64 and IA32 Architectures Software Developer's Manual, Volume 1: Basic Architecture. 
singular matrix 
A matrix whose determinant is zero. If A is a singular matrix, the inverse A^{1} does not exist, and the system of equations Ax = b does not have a unique solution (that is, there exist no solutions or an infinite number of solutions). 
singular value 
The numbers defined for a given general matrix A as the eigenvalues of the matrix AA^{H}. See also SVD. 
SMP 
Abbreviation for Symmetric MultiProcessing. Intel MKL offers performance gains through parallelism provided by the SMP feature. 
sparse BLAS 
Routines performing basic vector operations on sparse vectors. Sparse BLAS routines take advantage of vectors' sparsity: they allow you to store only nonzero elements of vectors. See BLAS. 
sparse vectors 
Vectors in which most of the components are zeros. 
storage scheme 
The way of storing matrices. See full storage, packed storage, and band storage. 
SVD 
Abbreviation for Singular Value Decomposition. See also Singular value decomposition section in "LAPACK Auxiliary and Utility Routines". 
symmetric matrix 
A square matrix A such that a_{ij} = a_{ji}. 
transpose 
The transpose of a given matrix A is a matrix A^{T} such that (A^{T})_{ij} = a_{ji} (rows of A become columns of A^{T}, and columns of A become rows of A^{T}). 
trapezoidal matrix 
A matrix A such that A = (A_{1}A_{2}), where A_{1} is an upper triangular matrix, A_{2} is a rectangular matrix. 
triangular matrix 
A matrix A is called an upper (lower) triangular matrix if all its subdiagonal elements (superdiagonal elements) are zeros. Thus, for an upper triangular matrix a_{ij} = 0 when i > j; for a lower triangular matrix a_{ij} = 0 when i < j. 
tridiagonal matrix 
A matrix whose nonzero elements are in three diagonals only: the leading diagonal, the first subdiagonal, and the first superdiagonal. 
unitary matrix 
A complex square matrix A whose conjugate and inverse are equal, that is, that is, A^{H} = A^{1}, and therefore AA^{H} = A^{H}A = I. All eigenvalues of a unitary matrix have the absolute value 1. 
VML 
Abbreviation for Vector Mathematical Library. See"Vector Mathematical Functions". 
VSL 
Abbreviation for Vector Statistical Library. See"Statistical Functions". 
z 
When found as the first letter of routine names, z indicates the usage of doubleprecision complex data type. 