Tag: Numerical Analysis

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Rick Wicklin 0
Symbolic derivatives in SAS

Did you know that you can get SAS to compute symbolic (analytical) derivatives of simple functions, including applying the product rule, quotient rule, and chain rule? SAS can form the symbolic derivatives of single-variable functions and partial derivatives of multivariable functions. Furthermore, the derivatives are output in a form that

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Rick Wicklin 0
The singular value decomposition: A fundamental technique in multivariate data analysis

The singular value decomposition (SVD) could be called the "billion-dollar algorithm" since it provides the mathematical basis for many modern algorithms in data science, including text mining, recommender systems (think Netflix and Amazon), image processing, and classification problems. Although the SVD was mathematically discovered in the late 1800s, computers have

Programming Tips
Rick Wicklin 0
The arithmetic-geometric mean

All statisticians are familiar with the classical arithmetic mean. Some statisticians are also familiar with the geometric mean. Whereas the arithmetic mean of n numbers is the sum divided by n, the geometric mean of n nonnegative numbers is the n_th root of the product of the numbers. The geometric

Rick Wicklin 0
The Lambert W function in SAS

This article describes how you can evaluate the Lambert W function in SAS/IML software. The Lambert W function is defined implicitly: given a real value x, the function's value w = W(x) is the value of w that satisfies the equation w exp(w) = x. Thus W is the inverse

Rick Wicklin 0
Halley's method for finding roots

Edmond Halley (1656-1742) is best known for computing the orbit and predicting the return of the short-period comet that bears his name. However, like many scientists of his era, he was involved in a variety of mathematical and scientific activities. One of his mathematical contributions is a numerical method for

Rick Wicklin 0
Compute the rank of a matrix in SAS

A common question from statistical programmers is how to compute the rank of a matrix in SAS. Recall that the rank of a matrix is defined as the number of linearly independent columns in the matrix. (Equivalently, the number of linearly independent rows.) This article describes how to compute the

Rick Wicklin 0
Create a custom PDF and CDF in SAS

In my previous post, I showed how to approximate a cumulative density function (CDF) by evaluating only the probability density function. The technique uses the trapezoidal rule of integration to approximate the CDF from the PDF. For common probability distributions, you can use the CDF function in Base SAS to

Rick Wicklin 0
Reversing the limits of integration in SAS

In SAS software, you can use the QUAD subroutine in the SAS/IML language to evaluate definite integrals on an interval [a, b]. The integral is properly defined only for a < b, but mathematicians define the following convention, which enables you to make sense of reversing the limits of integration:

Rick Wicklin 0
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

Rick Wicklin 0
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

Rick Wicklin 0
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.)

Rick Wicklin 0
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.

Rick Wicklin 0
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

Rick Wicklin 0
Create phase portraits in SAS

In a previous post, I showed how to solve differential equations in SAS by using the ODE subroutine in the SAS/IML language, which solves initial value problems. This article describes how to draw phase portraits for two classic differential equations: the equations of motion for the simple harmonic oscillator and

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