Julia – 71 -algebra lineare – 2

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Matrici speciali
Matrices with special symmetries and structures arise often in linear algebra and are frequently associated with various matrix factorizations. Julia features a rich collection of special matrix types, which allow for fast computation with specialized routines that are specially developed for particular matrix types.

The following tables summarize the types of special matrices that have been implemented in Julia, as well as whether hooks to various optimized methods for them in LAPACK are available.

Type            Description
Hermitian       Hermitian matrix
UpperTriangular Upper triangular matrix
LowerTriangular Lower triangular matrix
Tridiagonal     Tridiagonal matrix
SymTridiagonal  Symmetric tridiagonal matrix
Bidiagonal      Upper/lower bidiagonal matrix
Diagonal        Diagonal matrix
UniformScaling  Uniform scaling operator

operazioni elementari

Matrix type       +   -   *   \   Other functions with optimized methods
Hermitian                     MV  inv(), sqrtm(), expm()
UpperTriangular           MV  MV  inv(), det()
LowerTriangular           MV  MV  inv(), det()
SymTridiagonal    M   M   MS  MV  eigmax(), eigmin()
Tridiagonal       M   M   MS  MV     
Bidiagonal        M   M   MS  MV     
Diagonal          M   M   MV  M   inv(), det(), logdet(), /()
UniformScaling    M   M   MVS MVS /()


Key         Description
M (matrix)  An optimized method for matrix-matrix operations is available
V (vector)  An optimized method for matrix-vector operations is available
S (scalar)  An optimized method for matrix-scalar operations is available

l’operatore UniformScaling
A UniformScaling operator represents a scalar times the identity operator, λ*I. The identity operator I is defined as a constant and is an instance of UniformScaling. The size of these operators are generic and match the other matrix in the binary operations +, -, * and \. For A+I and A-I this means that A must be square. Multiplication with the identity operator I is a noop (except for checking that the scaling factor is one) and therefore almost without overhead.

Fattorializzazione di matrici
Matrix factorizations (a.k.a. matrix decompositions) compute the factorization of a matrix into a product of matrices, and are one of the central concepts in linear algebra.

The following table summarizes the types of matrix factorizations that have been implemented in Julia. Details of their associated methods can be found in the Linear Algebra section of the standard library documentation.

Type            Description
Cholesky        Cholesky factorization
CholeskyPivoted Pivoted Cholesky factorization
LU              LU factorization
LUTridiagonal   LU factorization for Tridiagonal matrices
UmfpackLU       LU factorization for sparse matrices (computed by UMFPack)
QR              QR factorization
QRCompactWY     Compact WY form of the QR factorization
QRPivoted       Pivoted QR factorization
Hessenberg      Hessenberg decomposition
Eigen           Spectral decomposition
SVD             Singular value decomposition
GeneralizedSVD  Generalized SVD

Argomento che andrebbe ampliato, secondo me 😊


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