Lecture2 Pss 2
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Transcript of Lecture2 Pss 2
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Iterative Methods (Conjugate Gradient)
=
When A is symmetric and positive definite, i.e.,
=; > 0Some properties of inner product
< , > = =
= < , > = < , >
=
< + , > = < , > + < , >< , > = Lemma: If A is symmetric and positive definite, then the problem of solving =
is equivalent to minimising the quadratic form
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= < , > 2 < , >See pg. 205 for proof of this Lemma.
Offshoot of this lemma
the iteration process
(+) = () + ()
where
= < , >< , >
With
=
(Steepest descent)
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Conjugate Gradient (pg 211 pseudocode)
,
,
,
,
,
= < , > = ,
+
< +
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If we perturb the equation we have
+ = + A) = = +
= +
=
+
= +
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= +
If is very small A is very poorly conditioned. i.e A is very close to singular.If we can precondition A, say such that
= So thatis better conditioned. How does one choose the preconditioner?Simplest case; choose the diagonal elements of A. (See pg. 217) for pseudo code.
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Example: Heat conduction equation solved with PCG.
Nice to write formal matrix equations but when we code we apply smart
approaches.
= =
=
= + Let us discretise the problem with finite difference = ; = + ; = , .
+
+
=
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=
Residual
=
where
+ + + = Need to multiply only non-zero elements.
Show code for PCG.
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Eigenvalue problems
Occur in vibrations, molecular modelling and many more areas
= xwhere is a scalar. = 0
Which implies is singular or
= 0
= Characteristic Polynomial
=
1 2 1
0 1 3
2 1 1
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1 2 10 1 32 1 1
= 0
1 1 1 3 26 21 = 0whose roots are
4, 12 + 72 , 12 72 Complex eigenvalues occur in conjugate pairs.
Power Method
Will compute the largest eigenvalue in magnitude provided(a) There is a single eigenvalues with the largest modulus
(b) There is a linearly independent set of n eigen vectors.
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1 > 2 3 . If
(
)are the eigen vectors, i.e
() = ()If (0) = (1) + (2) + + ()then
(1) =0 ; (2) =1 ; () =1 =0
=1 +2 + . . + = 11 + 22 + . . +
= 1 1 + 21 2 + . . + 1
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Let be a linear functional (which can be just a component of x) , then = (+1)(()) 1 Pseudo code (page 230; Exercise.. code this).
,
,
,
= 1,2
This is a very restrictive algorithm. Not much applicability. In practice we useQR factorization. Many routines can be found in EISPACK, LAPACK
