# SIMSTAT: Numerical Analysis

Zeros of a Polynomial
Simply enter the coefficients from the terminal, clipboard or file and the real and imaginary parts of the zeros will be calculated.

Determinants, Eigenvalues and Inverses
Use this option to test if a matrix is singular.

Singular Value Decomposition
Use this option to explore singular design matrices. If you intend to use designed experiments where the probability of singularity is high due to unforseen row or column dependencies, then this option should be used with your design matrix. Typical errors are those leading to design matrices where one column is a linear combination of the other columns, which can happen with integer design matrices or when using proportions or percentages as variables. What this option does is to take a matrix A and calculate the left singular vectors (U), the singular values (S), and the right singular vectors (V), as in
A = U*S*(V-transpose).
You can view the matrices U, S or V-transpose, or write them to file.

LU factorisation of a matrix
This computes the factorisation A = PLU where P is a pivot matrix, L is lower triangular/trapezoidal, and U is upper triangular/trapezoidal. The matrix 1-norm and matrix infinity-norm are also calculated and, for square matrices, the condition numbers for the 1-norm and infinity-norm are estimated.

QR factorisation of a matrix
This computes the factorisation A = QR where Q is an orthogonal matrix and R is upper triangular or trapezoidal.

Cholesky factorisation of a matrix
This computes the factorisation A = R*R-transpose, where A is a positive-definite symmetric matrix, like a covariance matrix, and R is lower triangular.

Matrix multiplication
Given two matrices A and B, this routine calculates the various products C, as in
C = A*B,
C = A-transpose*B,
C = A*B-transpose, or
C = A-transpose*B-transpose,
as long as the matrices have appropriate dimensions. The product C can be viewed or saved to file.

Evaluation of quadratic forms
Given a square n by n matrix A and a n by 1 vector x this procedure evaluates the quadratic form x-transpose*A*x. If A is positive-definite then the quadratic form x-Transpose*A-inverse*x can also be calculated, which is needed when A is a covariance matrix to evaluate Mahalanobis distances.

Ax = b
You can solve this matrix equation for A nonsingular and square, or you can use the L1, L2 or L-infinity norms if the system is overdetermined.

(A - lambda*B)x = 0
The standard eigenvalue problem for symmetrix A and B and B positive definite can be solved to give eigenvalues and eigenvectors.

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