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**The science of deriving dense linear algebra algorithms**

Paolo Bientinesi, John A. Gunnels, Margaret E. Myers, Enrique S. Quintana-Ortí, Robert A. van de Geijn

Pages: 1-26

DOI: 10.1145/1055531.1055532

In this article we present a systematic approach to the derivation of families of high-performance algorithms for a large set of frequently encountered dense linear algebra operations. As part of the derivation a constructive proof of the correctness...

**Representing linear algebra algorithms in code: the FLAME application program interfaces**

Paolo Bientinesi, Enrique S. Quintana-Ortí, Robert A. van de Geijn

Pages: 27-59

DOI: 10.1145/1055531.1055533

In this article, we present a number of Application Program Interfaces (APIs) for coding linear algebra algorithms. On the surface, these APIs for the MATLAB M-script and C programming languages appear to be simple, almost trivial, extensions of...

**Parallel out-of-core computation and updating of the QR factorization**

Brian C. Gunter, Robert A. Van De Geijn

Pages: 60-78

DOI: 10.1145/1055531.1055534

This article discusses the high-performance parallel implementation of the computation and updating of QR factorizations of dense matrices, including problems large enough to require out-of-core computation, where the matrix is stored on disk. The...

**Using AD to solve BVPs in MATLAB**

L. F. Shampine, Robert Ketzscher, Shaun A. Forth

Pages: 79-94

DOI: 10.1145/1055531.1055535

The MATLAB program bvp4c solves two--point boundary value problems (BVPs) of considerable generality. The numerical method requires partial derivatives of several kinds. To make solving BVPs as easy as possible, the default in bvp4c is to approximate...

**SlideCont: An Auto97 driver for bifurcation analysis of Filippov systems**

Fabio Dercole, Yuri A. Kuznetsov

Pages: 95-119

DOI: 10.1145/1055531.1055536

SLIDECONT, an AUTO97 driver for sliding bifurcation analysis of discontinuous piecewise-smooth autonomous systems, known as Filippov systems, is described in detail. Sliding bifurcations are those in which some sliding on the discontinuity boundary...

**SFCGen: A framework for efficient generation of multi-dimensional space-filling curves by recursion**

Guohua Jin, John Mellor-Crummey

Pages: 120-148

DOI: 10.1145/1055531.1055537

Because they are continuous and self-similar, space-filling curves have been widely used in mathematics to transform multi-dimensional problems into one-dimensional forms. For scientific applications, reordering computation by certain space-filling...

**Algorithm 840: computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions---prolate elements**

John P. Boyd

Pages: 149-165

DOI: 10.1145/1055531.1055538

High order domain decomposition methods using a basis of Legendre polynomials, known variously as “spectral elements” or “*p*-type finite elements,” have become very popular. Recent studies suggest that accuracy and...

**Algorithm 841: BHESS: Gaussian reduction to a similar banded Hessenberg form**

Gary W. Howell, Nadia Diaa

Pages: 166-185

DOI: 10.1145/1055531.1055539

BHESS uses Gaussian similarity transformations to reduce a general real square matrix to similar upper Hessenberg form. Multipliers are bounded in root mean square by a user-supplied parameter. If the input matrix is not highly nonnormal and the...