MECH2700 Engineering Analysis I

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MECH2700 Engineering Analysis I

• Condition number
• Jacobi Method

(Section 2.5 2.6 of Schilling and Harris)
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Ax b Learning objectives

• Know how to formulate linear system equations
  using vector and matrix notations.
• Know how to compute and interpret the 1, 2, and
  infinity norms of matrices and vectors.
• Know how to compute and interpret the condition
  number of a matrix.
• Be able write computer programs that can
  iteratively solve Axb problems using
• The Jacobi method
• The Gauss-Seidel method
• Know the types of problems for which these
  iterative methods are applicable and why
• Understand the conditions under which these
  iterative methods converge and why.

Today
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Ideas from previous lecture

• Vector norms - measures of the size or magnitude
  of a vector
• 1-norm sum of abs values of elements
• 2-norm
• infinity-norm abs value of largest element

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Recap Matrix norms

• Norm of a matrix represents maximum scaling
  effect of that matrix.
• Like vectors there are various norms we can use
• 1 norm
• 2 norm
• infinity norm

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Ideas from previous lecture
1-norm
infinity-norm
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Recap- Beam support
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Condition number
Rather that working with absolute errors, its
preferable to work with relative errors
Condition number gives the magnification factor
for relative errors in the solution of x given
relative errors in b.
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Example
Consider
Note scaling factor measured in the specified
norm. Here we are using the 1-norm
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Example - beam supports
When a is small relative to l, errors scaled
significantly
a/l
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Effects of ill-conditioning

• A simple rule of thumb
• For system with condition number of k, one can
  expect a loss of log10(k) decimal places in the
  accuracy of solution
• If k 5000, log10(k) 3.6 So expect to lose
  three to four decimal places of accuracy
• If k50,000, log10(k) 4.7 so expect to lose 4
  to 5 decimal places of accuracy

See page 68 of Schilling and Harris
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Hilbert matrix
Hilbert matrix arises in the solution of
polynomial curve fitting
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Hilbert matrix - a poorly conditioned matrix

• function k hilbertcond(n)
• for i 1n
• H hilb(i) matlab command
• k(i)cond(H)
• fprintf('kH(d) f\n', i, k(i))
• end
• kH(1) 1.000000
• kH(2) 19.281470
• kH(3) 524.056778
• kH(4) 15513.738739
• kH(5) 476607.250243
• kH(6) 14951058.641724
• kH(7) 475367356.277700
• kH(8) 15257575253.665625

k
size
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Weighted chain
Problem determine shape of chain under gravity
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Jacobi solution of weighted chain
Simplifying assumptions - links of unit
length - Unit tension in link - Net horizontal
forces small
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Jacobi solution of weighted chain
Resolve vertical forces
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Weighted chain
Consider 5 link chain
Let bj 1.0 x0x50
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Weighted chain
Consider 5 link chain
Let bj 1.0 x0x50
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Weighted chain
At each node
For a large number of links number of equations
is large
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Iterative solutions of Axb

• Weve already seem some examples of iterative
  solutions, e.g. Newtons method for root solving.
• Iterative solutions have the characteristic that
  they converge to a solution from an initial
  guess.
• Iterative solutions are efficient
• Iterative solutions of Axb are virtually the
  only way of solving large problems, e.g. A of
  order 105x105.

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Karl Jacob Jacobi (1804-1851)
Jacobi made important contributions to partial
differential equations of the first order and
applied them to the differentialequations of
dynamic systems.
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Jacobi Algorithm - derivation
Diagonal elements
Off diagonal elements
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Jacobi algorithm derivation
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Jacobi algorithm derivation
Form iterative scheme
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Jacobi algorithm
Solve
Step 5 Choose xo and iterate using scheme above
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5 link chain
Diagonal component of A
Off diagonal
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5 link chain
Iterative scheme.
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Weighted chain
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5 link chain
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5 link chain
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Weighted chain
Converges after about 50 iterations
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5 link chain
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Approximating a continuum - 50 link chain
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Next lecture

• Using matrix norms to determine rates of
  convergence.
• Gauss Seidel algorthims - faster convergence
• Successive over relaxation (SOR) - optimally fast
  convergence.