Design of C Matrix Library for FEM Computing Base on Generic Programming

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Design of C Matrix Library for FEM Computing
Base on Generic Programming
Zhou Jianhui
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Existing linear algebra library(1)
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Existing linear algebra library(2)

• Summary
• 1. C code is by no means inherently slower
  than Fortran code. After being optimized, C
  matrix library perform equally or even better
  than that Fortran in term of corresponding codes.
• 2. Almost all high-performance C matrix library
  use generic programming technique to optimize
  performance. Generic programming is proved to be
  of key importance to high-performance numerical
  computation.
• 3. C matrix libraries have more convenient
  interface. The operator overloading of C makes
  it possible to write program as easy as writing
  algebra expressions. For example, if you do
  matrix addition operation, you can write A B
  C, rather than call addition (A,B,C).

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Overview

• As the new C matrix library hereby is
  oriented towards FEM computing, it only
  provides operations of dense, symmetric and
  banded matrix, vector, sub-matrix, etc, rather
  than that of all algebra operations. But it does
  provide optimization specifically for FEM
  computing optimizing.
• The library was developed with references to
  those existing matrix libraries mentioned above,
  which brings some similarities between them
  resulting from the same optimizing technique
  based on generic programming.
• Notably, the library has made some
  significant improvements, including a novel
  architecture, an upgraded optimizing technique
  and a efficient memory management policy. The
  library achieves better performance, user
  interface and capability for further expansion.

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Architecture
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User Interface(1)
Some of the user interface
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User Interface(2)
Some of the user interface
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User Interface(3)
Some of the user interface
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The realization of optimizing by C generic
programming.

• Two of the most widely used optimizing technique
  are
• Expression templates.
• Template metaprograms.

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Expression templates eliminate temporary matrix

• Matrix addition A B C
• Fortran code will act as follows
• while C codes usually generate temporary
  variables implicitly and actually act as follows
• Such temporary matrix occupies too much
  spaces and is time consuming. Such drawbacks can
  be eliminated by expression templates, making C
  code as efficient as Fortran Code.

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Template metaprograms minimize cost of loop

• Inner production of vectors dot V1 V2,
  suppose size of V1,V2 are 3
• General codes can be depicted as
• Float dot 0
  DOT 0
• For(int i 0 i lt 3 i)
  DO 100 I 1, 3
• dot v1i v2i 100
  DOT DOT V1(I) V2(I)
• But with the size of two vectors given in
  advance, the code can be more efficient as
  follows
• dot v10 v20 v11 v21
  v12 v22
• or DOT V1(1) V2(1) V1(2) V2(2) V1(3)
  V2(3)
• Essential difference between them is that the
  latter codes saves the runtime of loop,such as
  increment of looping variant.
• With template metaprograms, C can generate
  codes like the latter form automatically during
  compiling process on condition that size of
  matrix or vector is known beforehand.

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The highlight of the new matrix library

• 1. Improved implementation of expression
  templates, which further simplifies the user
  interface.
• 2. Efficient memory management. The library does
  memory management itself instead of the operating
  system.
• 3. New library architecture, as mentioned before.
• 4. Some specific optimization for FEM computing.
• (1) Special storage strategy and the
  corresponding algorithms for special matrix, such
  as stiffness matrix.
• (2) Use template metaprograms as frequently
  as possible.

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Specific optimizing for FEM computing

• (1) Specific storage strategy and the
  corresponding algorithms for specific matrix
• Example
• In FEM computing, as K is a symmetric matrix,
  thus the multiplication algorithm can be
  optimized so that its calculation can be reduced
  by half

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FEM computing-specific optimizing

• (2) Use template metaprograms as widely as
  possible.
• Example For an eight-node isoparametric element
• The size of all the above matrixes are fixed,
  this is exactly where template metaprograms can
  be used to optimize performance.
• Actually, there are so many matrix operations
  where matrix and vector are size-fixed, that
  template metaprograms can be frequently used to
  achieve high performance.

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Performance Comparison

• The program is compiled under Visual C7.1,
  and run on Intel P?550E machine for 500000 times.
  For instance, the matrix of the isoparametric
  element operations in the program can be depicted
  as follows

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• Thanks for listening!