JAVA AND MATRIX COMPUTATION

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JAVA AND MATRIX COMPUTATION

• Geir Gundersen
• Department of Informatics
• University of Bergen
• Norway
• Joint work with Trond Steihaug

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Has JAVA something to offer in the field of
Sparse Matrix Computations?

• Java Grande Several extension to the Java
  language has been proposed but NOT integrated or
  considered by Sun Microsystems.

• Our vision What has the Java language to offer
  in the field of numerical computation as is?

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Objectives

• Java and Scientific Computing
• Java will be used for (limited) numerical
  computations.
• Jagged Arrays
• Static and Dynamic operations.
• Challenges
• Dense Matrix Computations.
• C.
• Future Topics.

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Benchmarking Java against C and FORTRAN for
Scientific Applications

• Java will loose in general for most kernels, a
  factor of 2-4 times, but some benchmarking shows
  that Java will compete and even win for other
  kernels (on some platforms).
• There is still progress in JVM and compiler
  optimizing. The gap between FORTRAN/C and Java
  will in the future get smaller(?)
• Benchmarking results are important but not in the
  scope of this work.

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Java Arrays

• Java arrays are objects.
• Thus creating an array is object creation.
• The objects of an array of objects are not
  necessarily stored contiguously.
• An array of objects stores references to the
  actual objects.
• The primitive elements of an array are most
  likely stored contiguously.
• An array of primitive elements holds the actual
  values for those elements.
• We utilize that Java Arrays need not to be
  rectangular and each inner array can have its own
  size, and that Array Aliasing is allowed.

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Java Arrays Matrix Examples
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Sparse Matrices

• A sparse matrix is usually defined as a matrix
  where "many" of its elements are equal to zero.
• We benefit both in time and space by working only
  on the nonzero data structure.
• Sparse Matrices have a wide variety of structures
  that defines several different data structures.
• Figures shows Sherman banded, Simplex
  unsymmetrical, symmetric and pent diagonal.

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Sparse Matrices Examples
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Compressed Row Storage

• The most commonly used storage schemes for large
  sparse matrices
• Compressed Row/Column Storage
• These storage schemes have enjoyed several
  decades of research
• The compressed storage schemes have minimal
  memory requirements for storing a general sparse
  matrices.

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Java Sparse Array

• Java Sparse Array (JSA) is a new concept for
  storing sparse matrices made possible with Java.
• One array for storing the references to the value
  arrays and one for storing the references to the
  index arrays.
• There is no need for an enclosing object of the
  arrays.

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Matrix Vector and Vector Matrix

• Static operations.
• Traverse only the data structures involved (only
  the vector c(Ab) is created).
• Numerical results indicates no significant loss
  in efficiency when traversing a 2D jagged array
  compared to a 1D array.

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Sparse Matrix Multiplication

• Dynamic operations.
• Creates each row of the resulting matrix C(AB).
• Numerical results indicates no significant loss
  in efficiency using jagged arrays in dynamic
  operations.
• Symbolic Phase and Numerical Phase.
• Jagged Arrays One phase leads to locality.
• CRS Two separate phases.

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The Update Algorithm
Sparse Matrix Update Sparse Matrix Update Sparse Matrix Update Sparse Matrix Update Sparse Matrix Update Sparse Matrix Update Sparse Matrix Update
Type m n nnz(A) nnz(B) nnz(new(A)) CRS JSA
GRE 115 115 421 7 426 11 0
NOS 5 468 2820 148 2963 13 1
ORSREG 1 2205 14133 449 14557 44 8
BCSSTK 16 4884 147631 2365 149942 183 8
BCSSTK 17 10974 219512 1350 2201041 753 8
E4R0100 17282 553956 324 554138 1806 11
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Jagged Variable Band Storage

• In this data structure, all matrix elements from
  the first nonzero in each row to the last nonzero
  in the row are explicitly stored.
• No loss with Matrix Vector and Vector Matrix in
  efficiency comparing JVBS with traditionally data
  structures.
• No loss with Sparse Matrix Multiplication in
  efficiency comparing JVBS with traditionally data
  structures.
• Traversing only non-zero elements with JSA and
  JVBS. JVBS can compete with JSA in performance.
• Density comparing JSA to JVBS when is JVBS more
  efficient. This since there are no indirect index
  addressing using JVBS.

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VBS Storing tridiagonal matrices

• For tri-diagonal (ui li), matrices, the rows i
  1,2...,m-2 have the same upper and lower
  bandwidth and only one bandwidth array is stored,
  this is accomplished by using array aliasing to
  store the references to this array from their
  respective row positions.
• No modification to algorithms that work static on
  such a structure compared to the original JVBS.
• For more dynamic operations the algorithms need
  to be modified.

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Dense Matrices Row versus Column Traversing
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C Timings

• Jagged arrays (Java like) for example
  doublemn. Not necessarily stored contiguously
  in memory.
• Multidimensional arrays (doublem,n).
  Contiguously stored block in the memory.
• Row-oriented.
• Row versus Column traversing on a
  Multidimensional array (mn) is an average of
  3.54 times.
• The difference between row and column traversing
  on a Jagged array (mn) is an average of 5.71
  times.

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C Timings

• Row traversing numerical results shows that
  Jagged Array are more efficient than
  Multidimensional Array with an average of 1.65
  times.
• Column traversing numerical results shows that
  Jagged Array are slightly more efficient than
  Multidimensional Array.
• High Performance Computing view Jagged Arrays
  rather than Multidimensional Arrays seems
  appropriate.

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C Timings
Row versus Column Traversing Row versus Column Traversing Row versus Column Traversing Row versus Column Traversing Row versus Column Traversing
mn Jagged Array Jagged Array Multidimensional Array Multidimensional Array
mn Row Column Row Column
1000 15 78 31 47
1500 47 109 62 109
2000 78 156 125 172
4500 406 3359 656 3453
5500 593 4125 984 5546
6000 716 6812 1172 6828
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The Impact of Java and C

• The future will be Java and C for commercial and
  educational use.
• Commercial applications written in Java and C
  will include scientific applications.
• Java Portability is especially important for
  high performance application, where the hardware
  architecture has a shorter lifespan than the
  application software.
• C/.NET Wide range of features are promised, but
  unfortunately still a one platform show.
• C versus Java Is C a better alternative then
  Java?
• Java Grande Forum C might be the answer for
  some operations like parallel programming.

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Future Topics

• Solving Large Sparse Linear System of Equations
• Sparse Gaussian Elimination with partial pivoting
• Multidimensional Matrices for Tensor Methods
• Tensor methods gives 3D structures where sparsity
  is an important issue.
• Parallel Java Threads
• Threads allow multiple activities to proceed
  concurrently in the same program.
• Parallel programming can only be achieved on a
  multiple processor platform.
• Suggested extensions to the Java language are
  OpenMP and MPI.

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Concluding Remarks

• We have shown for basic data structures there is
  a lot to gain in utilizing the flexibility
  (independently row updating) .
• Challenges as row versus column traversing for a
  2D square matrices.
• This is just the beginning
• Java Threads leads to parallel computing but only
  on the premises of Java.
• Other data structures must be investigated.
• Graphs.
• Applications
• Optimization and Numerical solution of PDEs.
• People will use Java (and C) for numerical
  computations, therefore it may be useful to
  invest time and resources finding how to use Java
  for numerical computation.