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**Unified form language**: A domain-specific language for weak formulations of partial differential equations

Martin S. Alnæs, Anders Logg, Kristian B. Ølgaard, Marie E. Rognes, Garth N. Wells

Article No.: 9

DOI: 10.1145/2566630

We present the Unified Form Language (UFL), which is a domain-specific language for representing weak formulations of partial differential equations with a view to numerical approximation. Features of UFL include support for variational forms and...

**Computing the sparsity pattern of Hessians using automatic differentiation**

Robert Mansel Gower, Margarida Pinheiro Mello

Article No.: 10

DOI: 10.1145/2490254

We compare two methods that calculate the sparsity pattern of Hessian matrices using the computational framework of automatic differentiation. The first method is a forward-mode algorithm by Andrea Walther in 2008 which has been implemented as the...

**How do you compute the midpoint of an interval?**

Frédéric Goualard

Article No.: 11

DOI: 10.1145/2493882

The algorithm that computes the midpoint of an interval with floating-point bounds requires some careful devising to handle all possible inputs correctly. We review several implementations from prominent C/C++ interval arithmetic...

**Optimally packed chains of bulges in multishift QR algorithms**

Lars Karlsson, Daniel Kressner, Bruno Lang

Article No.: 12

DOI: 10.1145/2559986

The QR algorithm is the method of choice for computing all eigenvalues of a dense nonsymmetric matrix *A*. After an initial reduction to Hessenberg form, a QR iteration can be viewed as chasing a small bulge from the top left to the bottom...

**A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc**

Eloy Romero, Jose E. Roman

Article No.: 13

DOI: 10.1145/2543696

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior...

**Algorithm 935**: `IIPBF`, a `MATLAB` toolbox for infinite integral of products of two Bessel functions

J. Tilak Ratnanather, Jung H. Kim, Sirong Zhang, Anthony M. J. Davis, Stephen K. Lucas

Article No.: 14

DOI: 10.1145/2508435

A `MATLAB` toolbox, `IIPBF`, for calculating infinite integrals involving a product of two Bessel functions *J*_{a}(ρ*x*)*J*_{b}(τ *x*), *J*_{a}(ρ...

**Algorithm 936**: A Fortran message processor

Fred T. Krogh

Article No.: 15

DOI: 10.1145/2559993

A code is presented which offers a simple clean way to get output that is very easy to read. Special support is given for the output of error messages which are a part of an application package or subprogram library. The code uses many of the...

**Algorithm 937**: MINRES-QLP for symmetric and Hermitian linear equations and least-squares problems

Sou-Cheng T. Choi, Michael A. Saunders

Article No.: 16

DOI: 10.1145/2527267

We describe algorithm MINRES-QLP and its FORTRAN 90 implementation for solving symmetric or Hermitian linear systems or least-squares problems. If the system is singular, MINRES-QLP computes the unique minimum-length solution (also known as the...

**Algorithm 938**: Compressing circular buffers

John C. Gunther

Article No.: 17

DOI: 10.1145/2559995

Data sequences generated by on-line sensors can become arbitrarily large and must, therefore, be pared down to fit into available memory. For situations where only the most recent data is of interest, this problem can be solved with optimal...