Numerical Linear Algebra - Computational Sciences - Lecture Slides, Slides of Computational Techniques

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Numerical Linear Algebra

What is Numerical Linear Algebra?• Ax = b• The same algebra learned in high school.

Common terms used in NLA (cont.)• Matrix – n

x

m 2-dimensional array of

values

Types of Matrices

• Dense Matrices – most values are filled

Common terms used in NLA (cont.)• Eigenvalues – a scalar c such that, given

an n x n matrix A and a vector x, Ax = cx,the vector x is known as the Eigenvector

• Least Squares Solution – a vector x in

Ax = b that minimizes the length of thevector Ax – b. A is an m x n matrix and ban m x 1 vector.

Numerical Linear Algebra

History

Hermann Grassman

H. Grassman (cont.)

• “Father of Linear Algebra”•^

Wrote Die lineale Ausdehnungslehre, einneuer Zweig der Mathematik^ – Introduced exterior algebra– Symbols representing points, lines, etc. can

be manipulated using certain rules.

Johann Carl Friedrich Gauss

JCF Gauss (cont.)

• Best known for Gaussian Elimination
  • Used to simplify Ax = b if A is a dense matrix
    • Math prodigy
      • Claimed to know the existence of non-

Euclidean geometry at 15 Contributed to astronomy

Arthur Caley (cont.)

• Theory of Algebraic Invariances• Work with matrices
  • Developed foundation for quantum mechanics
    • Divided geometry into types
      • Euclidean, non-Euclidean, n-dimensional

Cholesky Decomposition

• Used to simplify matrices that are

symmetric and positive definite

• Used in parallel computing

Sparse Matrices: Fill

• New non-zero elements introduced by

column modification

• Preserving sparsity requires that we limit

fill

• Finding optimal row, column ordering to

minimize fill is NP-complete combinatorialproblem