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Biomechanics_5

# Biomechanics_5 - Lecture 5 BMEn 3001 Biomechanics 12...

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Lecture 5 BMEn 3001 Biomechanics 12 September 2008

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Last Lecture • Matrix Inverse • The eigenproblem (in brief) • Condition Number
Topics for Today • Special Matrices – Symmetric – Banded – Sparse • Iterative Solution Techniques

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Symmetric Matrices • As we discussed earlier, a square matrix A is symmetric iff A ij = A ji . • It turns out that a symmetric matrix can be decomposed by a special subclass of LU-type factorizations. • The Cholesky decomposition reduces A to a product of a lower-triangular matrix and its transpose: A = Q Q T
Cholesky Decomposition • The Cholesky Decomposition is similar to the (Doolittle) LU decomposition except for some bells and whistles, which may be seen in any standard text. • Cholesky decomposition takes about half of the time and storage space of LU.

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Sparse Matrices • Most of the matrices one encounters in class are small (lending themselves to solution by hand) and have no special structure. • In practice, however, one often encounters large matrices with many zeroes. Such matrices are called sparse .
Banded Matrices • Consider a matrix that has all of its nonzero elements bunched along the diagonal.

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Banded Matrices • Consider a matrix that has all of its nonzero elements bunched along the diagonal.
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Biomechanics_5 - Lecture 5 BMEn 3001 Biomechanics 12...

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