What’s New in MATLAB R2018a?

MATLAB R2018a was released in March 2018. With each biannual release I try to give a brief overview of the changes in MATLAB (not the toolboxes) that are of most interest to me. These are not comprehensive summaries of what’s new and you should check the release notes for full details.

From the MATLAB “R2018a at a Glance” page.

Complex empty arrays, such as that generated with complex([]) now have an (empty) imaginary part instead of being real.

% R2017b
>> complex([])
ans =
     []

% R2018a
>> complex([])
ans =
  0×0 empty complex double matrix

This is a consequence of a change in the way MATLAB stores complex arrays. Prior to R2018a it stored the real parts together and the imaginary parts together. In R2081 it uses an interleaved format in which the real and imaginary parts of a number are stored together; this is the storage format used in the C and C++ languages. For more details see MATLAB Support for Interleaved Complex API in C MEX Functions. Unless you write MEX files or create complex empty arrays, this change should have no effect on you.

The Live Editor continues to gain improved functionality, including embedded sliders and drop-down menus.

Legends can now have multiple columns, specified with the NumColumns property for Legend objects. I must admit that I had not realized that is was already possible to arrange legends in a horizontal (1-by-n) rather than vertical orientation (n-by-1) using the Orientation property.

Tall arrays have several new capabilities, including the ability to compute a least squares solution to an overdetermined linear system Ax = b with a tall A, which is done by QR factorization.

MATLAB starts up faster and has further optimizations in the execution engine.

The GraphPlot Object has some additional options for the 'force', 'force3', and 'circle' layouts.

I noticed that the recent Windows 10 Spring Creators Update broke the Symbolic Toolbox. An update to fix this (and various other issues) is available at this page (MathWorks login required).

For a concise but wide-ranging introduction and reference to MATLAB, see MATLAB Guide (third edition, 2017)

How to Program log z

While Fortran was the first high-level programming language used for scientific computing, Algol 60 was the vehicle for publishing mathematical software in the early 1960s. Algol 60 had real arithmetic, but complex arithmetic had to be programmed by working with real and imaginary parts. Functions of a complex variable were not built-in, and had to be written by the programmer.

I’ve written a number of papers on algorithms to compute the (principal) logarithm of a matrix. The problem of computing the logarithm of a complex scalar—given a library routine that handles real arguments—might appear trivial, by comparison. That it is not can be seen by looking at early attempts to provide such a function in Algol 60.

The paper

J. R. Herndon (1961). Algorithm 48: Logarithm of a complex number. Comm. ACM, 4(4), 179.

presents an Algol 60 code for computing $\log z$ , for a complex number $z$ . It uses the arctan function to obtain the argument of a complex number.

The paper

A. P. Relph (1962). Certification of Algorithm 48: Logarithm of a complex number. Comm. ACM, 5(6), 347.

notes three problems with Herndon’s code: it fails for $z$ with zero real part, the imaginary part of the logarithm is on the wrong range (it should be $(-\pi,\pi]$ for the principal value), and the code uses log (log to the base 10) instead of ln (log to the base $e$ ). The latter error suggests to me that the code had never actually been run, as for almost any argument it would produce an incorrect value. This is perhaps not surprising since Algol 60 compilers must have only just started to become available in 1961.

The paper

M. L. Johnson and W. Sangren, W. (1962). Remark on Algorithm 48: Logarithm of a complex number. Comm. CACM, 5(7), 391.

contains more discussion about avoiding division by zero and getting signs correct. In

D. S. Collens (1964). Remark on remarks on Algorithm 48: Logarithm of a complex number. Comm. ACM, 7(8), 485.

Collens notes that Johnson and Sangren’s code wrongly gives $\log 0 = 0$ and has a missing minus sign in one statement. Finally, Collens gives in

D. S. Collens (1964). Algorithm 243: Logarithm of a complex number: Rewrite of Algorithm 48. Comm. CACM, 7(11), 660.

a rewritten algorithm that fixes all the earlier errors.

So it took five papers over a three year period to produce a correct Algol 60 code for the complex logarithm! Had those authors had the benefit of today’s interactive computing environments that period could no doubt have been shortened, but working with multivalued complex functions is necessarily a tricky business, as I have explained in earlier posts here and here.

Lectures on Multiprecision Algorithms in Kácov

At the end of May, I was one of four lecturers at the ESSAM school on Mathematical Modelling, Numerical Analysis and Scientific Computing, held in Kácov, about a hour’s drive south-east of Prague in the Czech Republic.

The event was superbly organized by Josef Malek, Miroslav Rozlozník, Zdenek Strakos and Miroslav Tuma. This was a relaxed and friendly event, and the excellent weather enabled most meals to be taken on the terrace of the family-run Sporthotel Kácov in which we were staying.

Françoise Tisseur lecturing on the nonlinear eigenvalue problem.

I gave three lectures of about one hour each on Multiprecision Algorithms. The slides are available from this link. Here is an abstract for the lectures:

Today’s computing environments offer multiple precisions of floating-point arithmetic, ranging from quarter precision (8 bits) and half precision (16 bits) to double precision (64 bits) and even quadruple precision (128 bits, available only in software), as well as arbitrary precision arithmetic (again in software). Exploiting the available precisions is essential in order to reduce the time to solution, minimize energy consumption, and (when necessary) solve ill-conditioned problems accurately.

In this course we will describe the precision landscape, explain how we can exploit different precisions in numerical linear algebra, and discuss how to analyze the accuracy and stability of multiprecision algorithms.

Lecture 1. IEEE standard arithmetic and availability in hardware and software. Motivation for low precision from applications, including machine learning. Exploiting reduced communication cost of low precision. Issued relating to rounding error analyses in low precision. Simulating low precision for testing purposes. Challenges of implementing algorithms in low precision.

Lecture 2. Basics of rounding error analysis, illustrated with summation. Why increasing precision is not a panacea. Software for high precision and its cost. Case study: the matrix logarithm in high precision.

Lecture 3. Solving very linear systems (possibly very ill conditioned and/or sparse) using mixed precision: iterative refinement in three precisions. A hybrid direct-iterative method: GMRES-IR.

I gave an earlier version of these lectures in March 2018 at the EU Regional School held at the Aachen Institute for Advanced Study in Computational Engineering Science (AICES), Germany. This single two and a half hour lecture was recorded and can be viewed on YouTube. The slides are available here.

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