On the determinant of the Hilbert matrix

Last week I described the Hilbert matrix of size n, which is a famous square matrix in numerical linear algebra. It is famous partially because its inverse and its determinant have explicit formulas (that is, we know them exactly), but mainly because the matrix is ill-conditioned for moderate values of […]
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Vector and matrix norms in SAS

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 […]
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Optimizing a function of an integral

Last week I showed how to find parameters that maximize the integral of a certain probability density function (PDF). Because the function was a PDF, I could evaluate the integral by calling the CDF function in SAS. (Recall that the cumulative distribution function (CDF) is the integral of a PDF.) […]
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Optimizing a function that evaluates an integral

SAS programmers use the SAS/IML language for many different tasks. One important task is computing an integral. Another is optimizing functions, such as maximizing a likelihood function to find parameters that best fit a set of data. Last week I saw an interesting problem that combines these two important tasks. […]
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Define functions with default parameter values in SAS/IML

One of my favorite new features of SAS/IML 12.1 enables you to define functions that contain default values for parameters. This is extremely useful when you want to write a function that has optional arguments. Example: Centering a data vector It is simple to specify a SAS/IML module with a […]
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A simple way to find the root of a function of one variable

Finding the root (or zero) of a function is an important computational task because it enables you to solve nonlinear equations. I have previously blogged about using Newton's method to find a root for a function of several variables. I have also blogged about how to use the bisection method […]
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How to compute the incomplete beta function in SAS

While sorting through an old pile of papers, I discovered notes from a 2012 SAS conference that I had attended. Next to the abstract for one presentation, I had scrawled a note to myself that read "BLOG about the incomplete beta function!" Okay, Rick, whatever you say! In statistics, the […]
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Compute contours of the bivariate normal CDF

This is the last post in my recent series of articles on computing contours in SAS. Last month a SAS customer asked how to compute the contours of the bivariate normal cumulative distribution function (CDF). Answering that question in a single blog post would have resulted in a long article, […]
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The gradient of the bivariate normal cumulative distribution

I'm spoiled by the internet. I've grown so accustomed to being able to instantly find an answer to any query—no matter how obscure—that I am surprised when I don't find what I am looking for. The other day I was trying to find a mathematical result: a formula for the […]
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Compute a contour (level curve) in SAS

Like many other computer packages, SAS can produce a contour plot that shows the level sets of a function of two variables. For example, I've previously written blogs that use contour plots to visualize the bivariate normal density function and to visualize the cumulative normal distribution function. However, sometimes you […]
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