Matrix Numerics 1.1.4

Matrix Numerics is a powerful app for Numerical Linear Algebra that offers a simple and user-friendly interface. It efficiently presents Linear Algebra results involving matrices and vectors with minimal effort. The app harnesses the SwiftLAN engine, a robust framework designed by academics for native Scientific Computation and Modeling in Swift. This state-of-the-art framework utilizes the speed and expressiveness of the Swift programming language to deliver exceptional performance and accuracy. Each algorithm within the framework undergoes extensive research, development, and debugging, aligning closely with the mathematical literature of renowned authors such as G. Golub, D.S Watkins, J.G.F Francis, C. Van Loan, among others.

Matrix Numerics performs a multitude of functions that contribute to its status as a world-class tool for scientists, students, and anyone involved in Linear Algebra regardless of their level of expertise or experience. Some of the key features offered by Matrix Numerics include:

• Addition, subtraction, and multiplication of matrices
• Inversion
• Calculating the condition number
• Determining the rank
• Computing the spectral radius
• Tracing
• Calculating determinants
• Transposition
• Finding the adjoint
• Determining characteristic polynomials
• Calculating eigenvalues and eigenvectors
• Determining singular values
• Performing singular value decomposition
• Computing pseudoinverses
• Performing PLU decomposition
• Conducting Cholesky decomposition
• Utilizing Gram-Schmidt decomposition
• Performing QR decomposition
• Implementing Upper Hessenberg decomposition
• Determining the Real Schur Form
• Calculating various norms: 1-norm, 2-norm, ∞-norm, and Frobenius norm
• Working with Fundamental Subspaces

When dealing with vectors, Matrix Numerics offers the ability to compute:

• Inner products
• Cross products
• Projection vectors
• Correlations between vectors
• Angles between vectors
• Vector lengths
• Vector normalization
• Vector orthonormalization

The app also allows for the approximation of solutions to Linear Systems of Equations and Linear Least Squares using various algorithms tailored to suit the specific system. The available algorithms include:

• Ordinary Least Squares
• Iteratively Reweighted Least Squares
• Pseudoinverse by Singular Value Decomposition (SVD)
• Kaczmarz’s Algorithm
• PLU with Scaled Row Pivoting
• Successive Over Relaxation
• Conjugate Gradient
• Gauss-Seidel Iteration
• Jacobi Iteration

The app also allows for the identification of specific attributes of matrices through a single touch. These attributes include determining whether the matrix is:

• Singular
• Symmetric
• Positive Definite
• Diagonal
• Upper or Lower Diagonal
• Orthogonal
• Convergent
• Row or Column Diagonally Dominant

Matrix Numerics presents numerical results and notations in a visually appealing LaTeX-like style, supporting both decimal and rational modes. All matrix computations are performed directly on the device using the powerful A-Series Apple chips, ensuring that the app can be used offline without requiring an internet connection. Matrix Numerics is compatible with iPhone 6 or newer models and all iPads except mini.

The developers would like to extend their special thanks to J. Dominguez, L. Meza, and L. Smith for their contributions to the app's development.

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