Scope of the PBLAS

The design of the software is as consistent as possible with that of the BLAS; thus, the experienced linear algebra programmer will have the same basic tools available in both the sequential and parallel programming worlds.

In real arithmetic the operations for the PBLAS have the following form:

• Level 1 - Vector-vector operations
  { }*
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  Index and value of the first maximal element in absolute value of a vector.
• Level 2 - Matrix-vector operations
  • Matrix-vector products
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  • Rank-1 update of a general matrix
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  • Rank-1 and rank-2 updates of a symmetric matrix
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  • Multiplication by a triangular matrix
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  • Solving a triangular system of equations
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    • 
• Level 3 - Matrix-matrix operations
  • Matrix-matrix products
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  • Rank- and rank- updates of a symmetric matrix
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    • 
    • 
    • 
  • Multiplication by a triangular matrix
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    • 
    • 
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  • Solving multiple triangular systems of equations
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  • Matrix transposition
    • 

  Here and are scalars, and are vectors, , and are rectangular matrices (in some cases square and symmetric), and is an upper or lower triangular matrix (and nonsingular for the triangular solves).

  Analogous operations are proposed in complex arithmetic. Conjugate transposition is specified as well as simple transposition. Additional operations are provided for scaling a complex vector by a real scalar and updates of a Hermitian matrix as follows:

  { }*
  
  
  
  
  
  
  
  
  with and real for the vector-vector and matrix-matrix operations, and
  { }*
  
  
  
  
  
  
  with real.