This library provides functions operating on matrices. Below, the functions are ordered according to categories and a typical call of the respective function is shown. Most functions are solely an interface to the external LAPACK library.
Note: A' is a short hand notation of transpose(A):
Basic Information
Linear Equations
Matrix Factorizations
Matrix Properties
Matrix Exponentials
Matrix Equations
Matrix Manipulation
| Name | Description |
|---|---|
| Examples | Examples demonstrating the usage of the Math.Matrices functions |
| toString | Convert a matrix into its string representation |
| isEqual | Compare whether two Real matrices are identical |
| solve | Solve real system of linear equations A*x=b with a b vector (Gaussian elimination with partial pivoting) |
| solve2 | Solve real system of linear equations A*X=B with a B matrix (Gaussian elimination with partial pivoting) |
| leastSquares | Solve linear equation A*x = b (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient) |
| leastSquares2 | Solve linear equation A*X = B (exactly if possible, or otherwise in a least square sense; A may be non-square and may be rank deficient) |
| equalityLeastSquares | Solve a linear equality constrained least squares problem |
| LU | LU decomposition of square or rectangular matrix |
| LU_solve | Solve real system of linear equations P*L*U*x=b with a b vector and an LU decomposition (from LU(..)) |
| LU_solve2 | Solve real system of linear equations P*L*U*X=B with a B matrix and an LU decomposition (from LU(..)) |
| eigenValues | Return eigenvalues and eigenvectors for a real, nonsymmetric matrix in a Real representation |
| eigenValueMatrix | Return real valued block diagonal matrix J of eigenvalues of matrix A (A=V*J*Vinv) |
| singularValues | Return singular values and left and right singular vectors |
| QR | Return the QR decomposition of a square matrix with optional column pivoting (A(:,p) = Q*R) |
| hessenberg | Return upper Hessenberg form of a matrix |
| realSchur | Return the real Schur form (rsf) S of a square matrix A, A=QZ*S*QZ' |
| cholesky | Return the Cholesky factorization of a symmetric positive definite matrix |
| balance | Return a balanced form of matrix A to improve the condition of A |
| balanceABC | Return a balanced form of a system [A,B;C,0] to improve its condition by a state transformation |
| trace | Return the trace of matrix A, i.e., the sum of the diagonal elements |
| det | Return determinant of a matrix (computed by LU decomposition; try to avoid det(..)) |
| inv | Return inverse of a matrix (try to avoid inv(..)) |
| rank | Return rank of a rectangular matrix (computed with singular values) |
| conditionNumber | Return the condition number norm(A)*norm(inv(A)) of a matrix A |
| rcond | Return the reciprocal condition number of a matrix |
| norm | Return the p-norm of a matrix |
| frobeniusNorm | Return the Frobenius norm of a matrix |
| nullSpace | Return the orthonormal nullspace of a matrix |
| exp | Return the exponential of a matrix by adaptive Taylor series expansion with scaling and balancing |
| integralExp | Return the exponential and the integral of the exponential of a matrix |
| integralExpT | Return the exponential, the integral of the exponential, and time-weighted integral of the exponential of a matrix |
| continuousLyapunov | Return solution X of the continuous-time Lyapunov equation X*A + A'*X = C |
| continuousSylvester | Return solution X of the continuous-time Sylvester equation A*X + X*B = C |
| continuousRiccati | Return solution X of the continuous-time algebraic Riccati equation A'*X + X*A - X*B*inv(R)*B'*X + Q = 0 (care) |
| discreteLyapunov | Return solution X of the discrete-time Lyapunov equation A'*X*A + sgn*X = C |
| discreteSylvester | Return solution of the discrete-time Sylvester equation A*X*B + sgn*X = C |
| discreteRiccati | Return solution of discrete-time algebraic Riccati equation A'*X*A - X - A'*X*B*inv(R + B'*X*B)*B'*X*A + Q = 0 (dare) |
| sort | Sort the rows or columns of a matrix in ascending or descending order |
| flipLeftRight | Flip the columns of a matrix in left/right direction |
| flipUpDown | Flip the rows of a matrix in up/down direction |
| LAPACK | Interface to LAPACK library (should usually not directly be used but only indirectly via Modelica.Math.Matrices) |
| Utilities | Utility functions that should not be directly utilized by the user |