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MQL5 ReferenceMatrix and Vector MethodsOpenBLASFactored Calculations 

Factored Calculations

This help section describes a set of functions for the numerical solution of linear algebra problems based on matrix factorizations and LAPACK library methods, wrapped in the matrix, matrixf, matrixc, and matrixcf classes.

These functions are focused on highly efficient and reliable calculations applicable to various types of matrices: real and complex, single- and double-precision. They cover key problems of applied linear algebra, including:

  • Solving systems of linear equations (SLAEs);
  • Estimating the condition number of a matrix;
  • Inverse transformations and computing inverse matrices;
  • Special SVD-based techniques for obtaining pseudoinverses and polar decompositions.

These functions work in conjunction with matrix prefactorizations such as Cholesky, LDL, PLU, and others, implemented in the corresponding modules. The algorithms used replicate the behavior of LAPACK and allow for efficient processing of both dense and specialized matrix structures (e.g., tridiagonal).

Below are all available functions, grouped by the factorization method used.

Function

Action

PLULinearEquationsSolution

Solves a system of linear equations  A * X = B,  A**T * X = B, or  A**H * X = B with an LU-factored square coefficient matrix AF computed by FactorizationPLURaw, with multiple right-hand sides. LAPACK function GETRS.

PLUInverse

Computes the inverse of an LU-factored general matrix AF computed by FactorizationPLURaw. LAPACK function GETRI.

PLUCondNumReciprocal

Estimates the reciprocal of the condition number of a general matrix A in either the one-norm or infinity-norm, using the LU factorization computed by FactorizationPLURaw. LAPACK function GECON.

PLUQLinearEquationsSolution

Solves a system of linear equations A * X = scale * RHS with a general N-by-N matrix A using the LU-factoization with complete pivoting computed by FactorizationPLUQRaw. LAPACK function GESC2.

PLUGeTridLinearEquationsSolution

Solves a system of linear equations A * X = B,  A**T * X = B, or  A**H * X = B with a tridiagonal matrix A using the LU-factorization computed by FactorizationPLUGeTridRaw, with multiple right-hand sides. LAPACK function GTTRS.

PLUGeTridCondNumReciprocal

Estimates the reciprocal of the condition number of a general tridiagonal matrix A in either the one-norm or infinity-norm, using the LU factorization computed by FactorizationPLUGeTridRaw. LAPACK function GTCON.

LDLLinearEquationsSolution

Solves a system of linear equations  A * X = B  with a real symmetric or complex Hermitian indefinite matrix using the factorization A = U**T * D * U or A = L * D * L**T computed by FactorizationLDLRaw, with multiple right-hand sides. LAPACK functions SYTRS, HETRS.

LDLInverse

Computes the inverse of a real symmetric or complex Hermitian indefinite matrix using the factorization A = U**T * D * U or A = L * D * L**T computed by FactorizationLDLRaw. LAPACK functions SYTRI, HETRI.

LDLCondNumReciprocal

Estimates the reciprocal of the condition number of a real symmetric or complex Hermitian indefinite matrix A, using the LDLT factorization computed by FactorizationLDLRaw. LAPACK functions SYCON, HECON.

LDLComplexSyLinearEquationsSolution

Solves a system of linear equations  A * X = B  with a complex symmetric indefinite matrix using the factorization A = U**T * D * U or A = L * D * L**T computed by FactorizationLDLComplexSyRaw, with multiple right-hand sides. LAPACK function SYTRS.

LDLComplexSyInverse

Computes the inverse of a complex symmetric indefinite matrix using the factorization A = U**T * D * U or A = L * D * L**T computed by FactorizationLDLComplexSyRaw. LAPACK function SYTRI.

LDLComplexSyCondNumReciprocal

Estimates the reciprocal of the condition number of a complex symmetric indefinite matrix A, using the LDLT factorization computed by FactorizationLDLComplexSyRaw. LAPACK function SYCON.

LDLSyTridPDLinearEquationsSolution

Solves a system of linear equations  A * X = B  with a real symmetric or complex Hermitian positive-definite tridiagonal matrix using the factorization A = L * D * L**T computed by FactorizationLDLSyTridPD, with multiple right-hand sides. LAPACK function PTTRS.

LDLSyTridPDCondNumReciprocal

Estimates the reciprocal of the condition number of a real symmetric or complex Hermitian positive-definite tridiagonal matrix A using the LDLT factorization computed by FactorizationLDLSyTridPD. LAPACK function PTCON.

CholeskyLinearEquationsSolution

Solves a system of linear equations  A * X = B  with a real symmetric or complex Hermitian positive-definite matrix using the factorization A = L * L**T computed by FactorizationCholesky, with multiple right-hand sides. LAPACK function POTRS.

CholeskyInverse

Computes the inverse of a real symmetric or complex Hermitian positive-definite matrix using the LLT factorization computed by FactorizationCholesky. LAPACK function POTRI.

CholeskyCondNumReciprocal

Estimates the reciprocal of the condition number of a real symmetric or complex Hermitian positive-definite matrix A using the LDL factorization computed by FactorizationCholesky. LAPACK function POCON.

SylvesterEquationSchur

Solves Sylvester equation for real quasi-triangular or complex triangular matrices: A*X + X*B = C where  A and B are both upper triangular. A is m-by-m and B is n-by-n; the right hand side C and the solution X are m-by-n. LAPACK function TRSYL.

SylvesterEquationSchurBlocked

Solves Sylvester equation for real quasi-triangular or complex triangular matrices: A*X + X*B = C where  A and B are both upper triangular. A is m-by-m and B is n-by-n; the right hand side C and the solution X are m-by-n. LAPACK function TRSYL3. This is the block (BLAS level 3) version of TRSYL. Faster up to 5 times but not so accurate.

Pseudo Inverse

There is no special function in the OpenBLAS library to calculate the pseudo-inverse of a matrix. However, for this purpose can be used singular value decomposition (SVD)

Polar Decomposition

There is no special function in the OpenBLAS library to calculate the polar decomposition of a matrix. However, for this purpose can be used singular value decomposition (SVD)

The functions provide support for various matrix types (real, complex, single-precision, and double-precision) and cover the following key tasks.

Condition Number Estimation

Unlike the true condition number, the estimate is approximate but much faster. Here, CondNumReciprocal denotes the inverse of the condition number (i.e., 1/cond_number).

Functions of the *CondNumReciprocal type are designed to estimate the reciprocal of the condition number of a matrix:

Matrix Inverse

Functions of the *Inverse type calculate the inverse matrix based on the corresponding factorization:

Linear System Solvers

*LinearEquationsSolution functions solve linear equations of the form A * X = B:

  • Both matrix and vector right-hand sides are supported;
  • The transposition (or Hermitian conjugation) of the coefficient matrix is taken into account via ENUM_EQUATIONS_FORM;
  • Separate functions are provided for tridiagonal and generalized matrices.

Поддержка расширенных LU-факторизаций

The PLUQLinearEquationsSolution function solves the system with scaling:

A * X = scale * B

based on full LU factorization with row- and column-wise permutations.

Special Methods (based on SVD)

While the library functions do not directly implement polar decomposition or pseudoinverse matrices, this section contains implementation examples based on singular value decomposition:

  • PolarDecomposition —  decomposition of A = Q * P, where Q  is an orthogonal matrix and P is a symmetric positive-definite matrix;
  • PseudoInverse — calculation of the pseudoinverse of a matrix A+ = V * Σ⁺ * Uᵀ.

 

Notes

  • All functions return true on success and false on error.
  • Prerequisites include previously performed factorizations (FactorizationCholesky, FactorizationLDLRaw, GETRF, etc.).
  • Calculation results may be sensitive to numerical stability, especially for small values of rcond.