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**Partitioned Triangular Tridiagonalization**

Miroslav Rozložník, Gil Shklarski, Sivan Toledo

Article No.: 38

DOI: 10.1145/1916461.1916462

We present a partitioned algorithm for reducing a symmetric matrix to a tridiagonal form, with partial pivoting. That is, the algorithm computes a factorization *PAPT* = *LTLT*, where, *P* is a permutation matrix, *L* is lower...

**Solving Very Sparse Rational Systems of Equations**

William Cook, Daniel E. Steffy

Article No.: 39

DOI: 10.1145/1916461.1916463

Efficient methods for solving linear-programming problems in exact precision rely on the solution of sparse systems of linear equations over the rational numbers. We consider a test set of instances arising from exact-precision linear programming...

**SelInv---An Algorithm for Selected Inversion of a Sparse Symmetric Matrix**

Lin Lin, Chao Yang, Juan C. Meza, Jianfeng Lu, Lexing Ying, Weinan E

Article No.: 40

DOI: 10.1145/1916461.1916464

We describe an efficient implementation of an algorithm for computing selected elements of a general sparse symmetric matrix *A* that can be decomposed as *A* = *LDLT*, where *L* is lower triangular and *D* is diagonal....

**Costas Arrays**: Survey, Standardization, and MATLAB Toolbox

Ken Taylor, Scott Rickard, Konstantinos Drakakis

Article No.: 41

DOI: 10.1145/1916461.1916465

A Costas array is an arrangement of *N* dots on an *N*-by-*N* grid, one per row, one per column, such that no two dots share the same displacement vector with any other pair. Costas arrays have applications in SONAR/RADAR systems,...

**An Optimal Iterative Solver for Symmetric Indefinite Systems Stemming from Mixed Approximation**

David J. Silvester, Valeria Simoncini

Article No.: 42

DOI: 10.1145/1916461.1916466

We discuss the design and implementation of a suite of functions for solving symmetric indefinite linear systems associated with mixed approximation of systems of PDEs. The novel feature of our iterative solver is the incorporation of error...

**A Supernodal Approach to Incomplete LU Factorization with Partial Pivoting**

Xiaoye S. Li, Meiyue Shao

Article No.: 43

DOI: 10.1145/1916461.1916467

We present a new supernode-based incomplete LU factorization method to construct a preconditioner for solving sparse linear systems with iterative methods. The new algorithm is primarily based on the ILUTP approach by Saad, and we incorporate a...

**Algorithm 909**: NOMAD: Nonlinear Optimization with the MADS Algorithm

Sébastien Le Digabel

Article No.: 44

DOI: 10.1145/1916461.1916468

NOMAD is software that implements the Mesh Adaptive Direct Search (MADS) algorithm for blackbox optimization under general nonlinear constraints. Blackbox optimization is about optimizing functions that are usually given as costly programs with no...

**Algorithm 910**: A Portable C++ Multiple-Precision System for Special-Function Calculations

Christopher Kormanyos

Article No.: 45

DOI: 10.1145/1916461.1916469

This article presents a portable C++ system for multiple precision calculations of special functions called *e_float*. It has an extendable architecture with a uniform C++ layer which can be used with any suitably prepared MP type. The system...

**Algorithm 911**: Multiple-Precision Exponential Integral and Related Functions

David M. Smith

Article No.: 46

DOI: 10.1145/1916461.1916470

This article describes a collection of Fortran-95 routines for evaluating the exponential integral function, error function, sine and cosine integrals, Fresnel integrals, Bessel functions, and related mathematical special functions using the FM...

**Algorithm 912**: A Module for Calculating Cylindrical Functions of Complex Order and Complex Argument

Masao Kodama

Article No.: 47

DOI: 10.1145/1916461.1916471

The present algorithm provides a module for calculating the cylindrical functions *Jv*(*z*), *Yv*(*z*), *Hv*(1)(*z*), and *Hv*(2)(*z*), where the order *v* is complex and the complex argument *z*...