In addition to the documenting features, conversion ease and performance improvements, Mathcad Prime 3.0 will introduce new numerical improvements. One interesting enhancement is the new Matrix Factorization Functions, which are an essential tool in linear algebra applications. These advances will improve direct solution of linear systems, forming matrix inverse and obtaining least square solution to over-determined systems.
Historically, Mathcad offered four matrix factorization functions:
1. Singular value decomposition (SVD) – A matrix is decomposed into product of unitary matrix, a rectangular diagonal matrix and the conjugate transpose of the unitary matrix. This factorization is used in many signal processing and statistics applications.
2. LU Decomposition (also called LU factorization) – A matrix is factorized into the product of a lower and upper triangular matrices. This used frequently to solve linear systems of equations which is used to solve many discrete and continuous mechanical and electrical systems
3. QR Decomposition (also called QR factorization) – A matrix is decomposed into the product of an orthogonal matrix and an upper triangular matrix. This factorization is often used to solve linear least squares and eigenvalue problems.
4. Cholesky Decomposition – Where a Hermitian positive matrix is decomposed into the product of a lower triangular matrix and its conjugate transpose often used to solve linear systems of equations.
Mathcad Prime 3.0 will introduce the following enhancement to LU, QR and Cholesky decompositions:
- Full pivoting support and enhanced stability
- Significant Performance improvement
- Full complex numbers support
- Full generalization
- Readily accessible results
In addition, the new decomposition functions will support non-square matrices (i.e. n≠m). As well as the option to turn pivoting on and off. Below is a table with a summary of the introduced enhancements:
In summary, these new numerical matrix factorization enhancements will allow engineers more flexibility in solving a wider set of more complex linear algebra problems and solve them at faster speeds with greater ease.