I recently posted an article about self-similar structures that arise in Pascal's triangle. Did you know that the Kronecker product (or direct product) can be used to create matrices that have self-similar structure? The basic idea is to start with a 0/1 matrix and compute a sequence of direct products
There are many ways to multiply scalars, vectors, and matrices, but the Kronecker product (also called the direct product) is multiplication on steroids. The Kronecker product looks scary, but it is actually simple. The Kronecker product is merely a way to pack multiples of a matrix B into a block
A colleague asked me a question regarding my recent post about the Pascal triangle matrix. While responding to his question, I discovered a program that I had written in 1999 that computed with a Pascal triangle matrix. Wow, I've been computing with Pascal's triangle for 15 years! I don't know
I was recently asked about how to use the SAS/IML language to efficiently add a constant to every element of a matrix diagonal. Mathematically, the task is to form the matrix sum A + kI, where A is an n x n matrix, k is a scalar value, and I is the
A few years ago I wrote an article that shows how to compute the log-determinant of a covariance matrix in SAS. This computation is often required to evaluate a log-likelihood function. My algorithm used the ROOT function in SAS/IML to compute a Cholesky decomposition of the covariance matrix. The Cholesky
In a previous post, I stated that you should avoid matrix multiplication that involves a huge diagonal matrix because that operation can be carried out more efficiently. Here's another tip that sometimes improves the efficiency of matrix multiplication: use parentheses to prevent the creation of large matrices. Matrix multiplication is
I love working with SAS Technical Support because I get to see real problems that SAS customers face as they use SAS/IML software. The other day I advised a customer how to improve the efficiency of a computation that involved multiplying large matrices. In this article I describe an important
Dear Rick, I am trying to create a numerical matrix with 100,000 rows and columns in PROC IML. I get the following error: (execution) Unable to allocate sufficient memory. Can IML allocate a matrix of this size? What is wrong? Several times a month I see a variation of this
Just one last short article about properties of the Hilbert matrix. I've already blogged about how to construct a Hilbert matrix in the SAS/IML language and how to compute a formula for the determinant. One reason that the Hilbert matrix is a famous (some would say infamous!) example in numerical
Did you know that SAS/IML 12.1 provides built-in functions that compute the norm of a vector or matrix? A vector norm enables you to compute the length of a vector or the distance between two vectors in SAS. Matrix norms are used in numerical linear algebra to estimate the condition
