Tridiagonal matrix-Condition number

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??

Answer 1

[John D'Errico]

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.