Glossary
- AH
- Denotes the conjugate transpose of a general matrixA.See alsoconjugate matrix.
- AT
- Denotes the transpose of a general matrixA.See alsotranspose.
- band matrix
- A generalm-by-nmatrixAsuch thatforaij= 0|, wherei-j| >l1 <. For example, any tridiagonal matrix is a band matrix.l< min(m,n)
- band storage
- A special storage scheme for band matrices. A matrix is stored in a two-dimensional 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, matrix-vector, and matrix-matrix 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 user-designed BRNG into the VSL and use it along with the predefined VSL basic generators.
- Bunch-Kaufman factorization
- Representation of a real symmetric or complex Hermitian matrixAin the form(orA=PUDUHPT) whereA=PLDLHPTPis a permutation matrix,UandLare upper and lower triangular matrices with unit diagonal, andDis a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.UandLhave 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks ofD.
- c
- When found as the first letter of routine names,cindicates the usage of single-precision complex data type.
- CBLAS
- C interface to the BLAS.SeeBLAS.
- CDF
- Cumulative Distribution Function. The function that determines probability distribution for univariate or multivariate random variableX. For univariate distribution the cumulative distribution function is the function of real argumentx, which for everyxtakes a value equal to probability of the event. For multivariate distribution the cumulative distribution function is the function of a real vectorA:X≤x, which, for everyx= (x1,x2, ...,xn)x, takes a value equal to probability of the event.A= (X1≤x1&X2≤x2, & ..., &Xn≤xn)
- Cholesky factorization
- Representation of a symmetric positive-definite or, for complex data, Hermitian positive-definite matrixAin the formorA=UHU, whereA=LLHLis a lower triangular matrix andUis an upper triangular matrix.
- condition number
- The numberdefined for a given square matrixκ(A)Aas follows:.κ(A) = ||A|| ||A||−1
- conjugate matrix
- The matrixAHdefined for a given general matrixAas follows: (.AH)ij= (aji)*
- conjugate number
- The conjugate of a complex numberisz=a+bi.z*=a-bi
- d
- When found as the first letter of routine names,dindicates the usage of double-precision real data type.
- dot product
- The number denotedand defined for given vectorsx·yxandyas follows:.x·y=ΣixiyiHerexiandyistand for thei-th elements ofxandy, respectively.
- double precision
- A floating-point data type. On Intel® processors, this data type allows you to store real numbersxsuch that2.23*10. For this data type, the machine precision< |−308x| < 1.79*10308εis approximately 10, which means that double-precision numbers usually contain no more than 15 significant decimal digits. For more information, refer to−15Intel® 64 and IA-32 Architectures Software Developer's Manual, Volume 1: Basic Architecture.
- eigenvalue
- Seeeigenvalue problem.
- eigenvalue problem
- A problem of finding non-zero vectorsxand numbersλ(for a given square matrixA) such that. Here the numbersAx=λxλare called the eigenvalues of the matrixAand the vectorsxare called the eigenvectors of the matrixA.
- eigenvector
- Seeeigenvalue problem.
- elementary reflector(Householder matrix)
- Matrix of a general form, whereH=I−τvvTvis a column vector andτis a scalar. In LAPACK elementary reflectors are used, for example, to represent the matrixQin theQRfactorization (the matrixQis represented as a product of elementary reflectors).
- factorization
- Representation of a matrix as a product of matrices.See alsoBunch-Kaufman factorization, Cholesky factorization,LUfactorization,LQfactorization,QRfactorization, 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 matrixAis stored in a two-dimensional arraya, with the matrix elementstored in the array elementaij.a(i,j)
- Hermitian matrix
- A square matrixAthat is equal to its conjugate matrixAH. The conjugateAHis defined as follows:(.AH)ij= (aji)*
- I
- Seeidentity matrix.
- identity matrix
- A square matrixIwhose diagonal elements are 1, and off-diagonal elements are 0. For any matrixA,andAI=A.IA=A
- i.i.d.
- Independent Identically Distributed.
- in-place
- Qualifier of an operation. A function that performs its operation in-place takes its input from an array and returns its output to the same array.
- Intel® oneAPI Math Kernel Library
- Abbreviation for.Intel® oneAPIMath Kernel Library
- inverse matrix
- The matrix denoted as Aand defined for a given square matrix−1Aas follows:.AA=−1A−1A=Idoes not exist for singular matricesA−1A.
- LQfactorization
- Representation of anm-by-nmatrixAasorA=LQ. HereA= (L0)QQis ann-by-northogonal (unitary) matrix. For,m≤nLis anm-by-mlower triangular matrix with real diagonal elements; for,m>nwhereL1is ann-by-nlower triangular matrix, andL2is a rectangular matrix.
- LUfactorization
- Representation of a generalm-by-nmatrixAas, whereA=PLUPis a permutation matrix,Lis lower triangular with unit diagonal elements (lower trapezoidal if) andm>nUis upper triangular (upper trapezoidal if).m<n
- machine precision
- The numberεdetermining the precision of the machine representation of real numbers. For Intel® architecture, the machine precision is approximately 10for single-precision data, and approximately 10−7for double-precision data. The precision also determines the number of significant decimal digits in the machine representation of real numbers.−15See alsodouble precision and single precision.
- MPI
- Message Passing Interface. This standard defines the user interface and functionality for a wide range of message-passing capabilities in parallel computing.
- MPICH
- A freely available, portable implementation of MPI standard for message-passing libraries.
- orthogonal matrix
- A real square matrixAwhose transpose and inverse are equal, that is,, and thereforeAT=A-1. All eigenvalues of an orthogonal matrix have the absolute value 1.AAT=ATA=I
- 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 one-dimensional array.
- PDF
- Probability Density Function. The function that determines probability distribution for univariate or multivariate continuous random variableX. The probability density functionf(x) is closely related with the cumulative distribution functionF(x).For univariate distribution the relation isFor multivariate distribution the relation is
- positive-definite matrix
- A square matrixAsuch thatfor any non-zero vectorAx·x> 0x. Here · denotes the dot product.
- pseudorandom number generator
- A completely deterministic algorithm that imitates truly random sequences.
- QRfactorization
- Representation of anm-by-nmatrixAas, whereA = QRQis anm-by-morthogonal (unitary) matrix, andRisn-by-nupper triangular with real diagonal elements (if) or trapezoidal (ifm≥n) matrix.m<n
- 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,sindicates the usage of single-precision real data type.
- ScaLAPACK
- Stands for Scalable Linear Algebra PACKage.
- Schur factorization
- Representation of a square matrixAin the form. Here T is an upper quasi-triangular matrix (for complexA=ZTZHA, triangular matrix) called the Schur form ofA; the matrixZis orthogonal (for complexA, unitary). Columns ofZare called Schur vectors.
- single precision
- A floating-point data type. On Intel® processors, this data type allows you to store real numbersxsuch that1.18*10. For this data type, the machine precision (< |−38x| < 3.40*1038ε) is approximately 10, which means that single-precision numbers usually contain no more than 7 significant decimal digits. For more information, refer to−7Intel® 64 and IA-32 Architectures Software Developer's Manual, Volume 1: Basic Architecture.
- singular matrix
- A matrix whose determinant is zero. IfAis a singular matrix, the inversedoes not exist, and the system of equationsA-1does not have a unique solution (that is, there exist no solutions or an infinite number of solutions).Ax=b
- singular value
- The numbers defined for a given general matrixAas the eigenvalues of the matrixAAH.See alsoSVD.
- SMP
- Abbreviation for Symmetric MultiProcessing.offers performance gains through parallelism provided by the SMP feature.Intel® oneAPI Math Kernel Library
- sparse BLAS
- Routines performing basic vector operations on sparse vectors. Sparse BLAS routines take advantage of vectors' sparsity: they allow you to store only non-zero elements of vectors.SeeBLAS.
- sparse vectors
- Vectors in which most of the components are zeros.
- storage scheme
- The way of storing matrices.Seefull storage, packed storage, and band storage.
- SVD
- Abbreviation for Singular Value Decomposition.See alsoSingular value decomposition section in"LAPACK Auxiliary and Utility Routines".
- symmetric matrix
- A square matrixAsuch that.aij=aji
- transpose
- The transpose of a given matrixAis a matrixsuch that (AT(rows ofAT)ij=ajiAbecome columns ofAT, and columns ofAbecome rows ofAT).
- trapezoidal matrix
- A matrixAsuch that, whereA= (A1A2)A1is an upper triangular matrix,A2is a rectangular matrix.
- triangular matrix
- A matrixAis called an upper (lower) triangular matrix if all its subdiagonal elements (superdiagonal elements) are zeros. Thus, for an upper triangular matrixwhenaij= 0; for a lower triangular matrixi>jwhenaij= 0.i<j
- tridiagonal matrix
- A matrix whose non-zero elements are in three diagonals only: the leading diagonal, the first subdiagonal, and the first super-diagonal.
- unitary matrix
- A complex square matrixAwhose conjugate and inverse are equal, that is, that is,, and thereforeAH=A-1. All eigenvalues of a unitary matrix have the absolute value 1.AAH=AHA = I
- 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,zindicates the usage of double-precision complex data type.