Java AppLib: Approximation Library for Java

Under the leadership of Alexander I. Rozhenko

Contents

1  History

The first and main aim of the development of this Library was an implementation of various splines to be constructed by function measurements on 1D or multi-dimensional mesh of scattered points. This job was originated on the spline software library LIDA-3 developed in 1985 at the Novosibirsk Computing Center under the leadership of Prof. Vladimir A. Vasilenko. The current job began at the middle of 1999.

2  AppLib's Features

Package ru.sscc.util.data

is the conceptual kernel of the Library. The RealContainer, RealVector, RealPointer, and RealVectors classes are introduced to provide unified access to user's real data represented as sparse vectors in the float[] or double[] type arrays. Many BLAS operations are implemented for such vectors in these classes and also in the collection of static methods of the RealMath class.

Package ru.sscc.matrix

contains matrix classes. The real rectangular dense matrix, the real symmetric banded matrix, and the real rectangular banded matrix are implemented now. The matrix-by-vector and transposed matrix-by-vector operations are implemented. The collection of common matrix-by-matrix multiplication methods is prepared.

Package ru.sscc.matrix.solve

contains a number of solvers that solve Systems of Linear Algebraic Equations (SLAE). 7 direct solvers are prepared now:

• solver for positive definite symmetric banded matrices;
• 3 solvers for square dense matrices (Cholesky, Gauss, and Craut methods are used);
• 3 solvers for square rectangular matrices (based on Givens rotations, Hausholder reflections, and Gauss eliminations). They find the normal pseudo-solution of SLAE, calculate the matrix range, and construct the matrix null space.

The common iterative refinement algorithm for the solution of SLAE is prepared. The construction of inverse matrix is also possible for any direct solver. For a rectangular matrix, the pseudo-inverse matrix may be constructed. The solution of SLAE with lower triangular and upper triangular matrices is supported.

Package ru.sscc.spline.polynomial

provides the construction of 1D polynomial splines of any odd degree (linear, cubic, quintic, and more). The construction of interpolating and smoothing splines is supported. The automatic choice of the smoothing parameter to satisfy the residual criterion is implemented. The weighted residual criterion is also possible. Uniform and nonuniform meshes of measurement nodes are used.

3  Documentation and Downloads

Current Java AppLib version is 1.22.

Java AppLib API specification   (no frames)

Java AppLib API (zipped version)

Java AppLib classes

We will be very appreciative for your comments, bug reports, and feedbacks to be sent to rozhenko@oapmg.sscc.ru.

4  Future of the Library

We plan to extend the Library in the following directions:

• More matrices (asymmetric banded matrices, cyclic 3-diagonal matrices, sparse matrices).
• More solvers (both direct and iterative).
• Solving of partial and full eigenvalue problems. Constructing of SVD.
• 1D polynomial splines of even degree. Support of B-spline representation for polynomial splines.
• Multi-D spline approximation on scattered meshes based on Duchon's approach.
• Multi-D splines with interval restrictions in mesh nodes.
• Special 1D splines: cubic splines with boundary conditions, periodic cubic splines, trigonometric splines, exponential splines, shape preserving splines, and cubic splines with piecewise continuous constraints.
• Tensor approach joining arbitrary per-component splines into one multi-D tensor spline.

5  Project Team

Alexander I. Rozhenko,

senior researcher, ICMMG SB RAS, Laboratory of Numerical Analysis and Computer Graphics

Nina F. Fursova,

student, Novosibirsk State University, Chair of Numerical Mathematics

Aleksei V. Galkov,

student, Novosibirsk State University, Chair of Numerical Mathematics

Oleg A. Likhachev,

student, Novosibirsk State University, Chair of Numerical Mathematics

Anna E. Nikishkina,

student, Novosibirsk State University, Chair of Numerical Mathematics

Denis V. Petrakov,

post-graduate student, ICMMG SB RAS, Chair of Numerical Mathematics