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- Techniques for Solving Ax = b.
- Elimination is a perfect algorithm, except when the particular problem has special properties-as almost every problem has.
- Section 7.4 will concentrate on the property of sparseness, when most of the entries in A are zero.
- We develop iterative rather than direct methods for solving Ax = b.
- An iterative method is "self-correcting," and never reaches the exact answer.
- The object is to get close more quickly than elimination.
- In some problems, that can be done; in many others, elimination is safer and faster if it takes advantage of the zeros.
- The competition is far from over, and we will identify the spectral radius that controls the speed of convergence to x = A⁻¹b.
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- Techniques for Solving Ax = λx.
- The eigenvalue problem is one of the outstanding successes of numerical analysis.
- We have chosen two or three ideas:
- the QR algorithm
- the family of "power methods,"
- the preprocessing of a symmetric matrix to make it tridiagonal.
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- The Condition Number of a Matrix.
- Section 7.2 attempts to measure the "sensitivity" of a problem: If A and b are slightly changed, how great is the effect on x = A⁻¹b?
- Before starting on that question, we need a way to measure A and the change ΔA.
- The length of a vector is already defined, and now we need the norm of a matrix.
- Then the condition number, and the sensitivity of A will follow from multiplying the norms of A and A⁻¹.
The matrices in this chapter are square.
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