Tridiagonal matrix-Condition number
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If we have the tridiagonal matrix,that has the number 4 at the main diagonal and the number 1 at the first diagonal below the main diagonal and at the first diagonal above the main diagonal,I get that the condition number,using the infinity norm,is 3,independent from the dimension I give(for n>=20)..Is this right???If yes,why does this happen??Why isn't there any change of the condition number??
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John D'Errico
2013 年 11 月 24 日
編集済み: John D'Errico
2013 年 11 月 25 日
Well, in fact, there IS a change in the condition number. I won't prove that the condition number has an upper limit of 3, as that may be your goal if this is a homework problem. But if we look at the singular values of the matrices, we can get an idea of what is happening.
First though, make sure we are reporting a sufficient number of digits.
format long g
Afun = @(N) diag(repmat(4,1,N)) + diag(repmat(1,1,N-1),1) + diag(repmat(1,1,N-1),-1);
svd(Afun(5))
ans =
5.73205080756888
5
4
3
2.26794919243112
cond(Afun(5),inf)
ans =
2.88461538461538
svd(Afun(20))
ans =
5.97766165245026
5.91114561157228
5.80193773580484
5.65247754863199
5.46610374365965
5.24697960371747
5
4.73068204873279
4.44504186791263
4.14946018717285
3.85053981282715
3.55495813208737
3.26931795126721
3
2.75302039628253
2.53389625634035
2.34752245136801
2.19806226419516
2.08885438842772
2.02233834754974
cond(Afun(20),inf)
ans =
2.99999274315861
As you can see, the condition number seems to approach 3 as a limit. For N = 50 or so, it gets as close to 3 as a double precision number can get.
cond(Afun(50),inf)
ans =
2.99999999999998
So in fact, the condition number DOES vary, but it approaches a limit of 3 quickly enough.
You should be able to show that the singular values are bounded in the range [2,6], so this puts an upper limit on the condition number. That you saw no change in the condition number after a certain point says that you were using a short display format, AND that the number approaches its limit reasonably quickly.
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