Solving Scalar Linear Systems A Little Theory For Jacobi Iteration

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Solving Scalar Linear Systems A Little Theory
For Jacobi Iteration

• Lecture 15
• MA/CS 471
• Fall 2003

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Review

• Using Kirchoffs second law we build the loop
  current circuit matrix.

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1W

4W
4W
6W
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3W
V4
7W
2W
1W
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V1
Note we have boosted the center cell to ensure
diagonal dominance (hack)
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System
Jacobi iterative approach
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Jacobi v. Gauss-Seidel
Jacobi
Gauss-Seidel
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Convergence Proof
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Definition of Spectral Radius

• We define the spectral radius of a matrix A as

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Stage 1 Unique Limit

• The equation we wish to solve is
• We consider the following iterator
• For some easily invertible matrix Q
• Suppose this iteration does indeed converge as
  , i.e.
• Then xtilde will satisfy

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cont

• And yields
• So if the iteration converges, then the limit
  vector will be a solution to the original system.

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Second Stage of Convergence Proof

• We first prove that the Jacobi (and Gauss-Seidel)
  methods converge if and only if the spectral
  radius of

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Theorem

• Suppose and
  that both Q and A are non-singular. If the
  spectral radius of is
  strictly bounded above by 1 then the iterates
  defined by
• converge to for any starting vector

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Proof

• Let denote the error in the nth
  iterate. We combine the following
  relationships
• to obtain
• simplifying

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Proof cont

• The iteration converges with
  increasing n if and only if (proof
  omitted).

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Third Stage of Convergence

• Now we are left with the task of proving that for
  the choice of Q the matrix has
• i.e. we have to prove that the absolute value of
  all of the eigenvalues of the matrix is less
  than one.
• So we use Gershgorins theorem to find the range
  of the eigenvalues of

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Recall Gershgorins Circle Theorem

• Let A be a square NxN matrix. Around every
  element
• aii on the diagonal of the matrix draw a circle
  with
• radius
• Such circles are known as Gershgorins disks.
• Theorem every eigenvalue of A lies in one of
  these Gershgorins disks.

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Jacobi Iteration

• Recall the generic iterative scheme required a Q
  matrix which is easily invertible.
• Lets take Qdiag(A) (i.e. a matrix with zeros
  everywhere, apart from the diagonal entries which
  are the same as those of A)
• Lets write ALDU

D
U
L
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Cont.

• Then the scheme becomes
• i.e.
• We can determine conditions under which this
  scheme will converge.
• Recall the necessary and sufficient condition that

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Cont.

• We can use Gershgorins theorem after we note
  that
• has zero entries on the diagonal so all the
  Gershgorin disks will be centered at zero and
  have maximum radius

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cont

• So if
• Then we are done.
• Note, a matrix which satisfies
• Is called diagonally dominant.

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Summary

• The first stage of the convergence proof showed
  that the unique possible convergent limit of the
  scheme is the actual solution to the linear
  system.
• Secondly, we showed that the scheme converges if
  and only if
• Thirdly, we showed that for the choice of Q
  diagonal of A, that (2) was satisfied.
• i.e. Jacobi iteration converges for any initial
  guess for x if A is diagonally dominant.