Creating few columns of the Vandermonde matrix without using vander

Question

Aleksandar 2011-8-8

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回答： Steven Lord 2022-4-6

Hi,

I want to create the first n columns of the Vandermonde matrix (I need this matrix in order to fit a curve through given points with the lsqlin function). Of course the commands:

A = vander(x);
A = A(1:n);

would do the work, but since the vector x might have over million values and n could be very small, 10 for example, it would be very inefficient.

I already did it with a for loop:

A = ones(length(x), n);
for i = 2 : n
    A(:, i) = (x.^(i - 1));
end

but I think it can be done more elegant without a loop. Is there a function that does this? Or can someone give me an idea how to do it without a loop? I am thinking that repmat or reshape might be useful.

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Answer 1

Jan 2011-8-8

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编辑：Jan 2017-8-16

在 MATLAB Online 中打开

You calculate the Vandermonde matrix from right to left. Is this intented? Do you want to get the n leftmost columns (smallest values)?

The power operation consumes a lot of time.

function A = ShortVander(x, n)
x = x(:);  % Column vector
A = ones(length(x), n);
for i = 2:n
  A(:, i) = x .* A(:, i-1);  % [EDITED, Thanks Steven]
end

Look into the source code of VANDER to find a very equal approach...

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Steven Morrow 2017-8-15

编辑：Steven Morrow 2017-8-15

The multiplication should be compnenet-wise, i.e. the line in the for loop should be A(:,i) = x.*A(:,i-1)

Are Mjaavatten 2022-4-6

This is nice, but why not let the built-in vander function take the number of columns as an optional second argument?

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Answer 2

Aleksandar 2011-8-8

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I just found a function that does this without any loops, just as I was looking for, and it is using repmat and cumprod:

http://www.mathworks.com/matlabcentral/fileexchange/20896-vandermonde-matrix/content/vanderm.m

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Jan 2011-8-8

编辑：Jan 2017-8-16

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Please compare the times with my posted FOR loop. Under Matlab 2009a and for x = rand(1, 1000) the CUMPROD method needs >50% more time. I've tried this version before I've posted the loop:

function A = ShortVander(x, n)
v = v(:);
A = v(:, ones(1, n));
A(:, 1) = 1;
A = cumprod(A);

But this is also about 50% slower than the loop. If you work with large arrays it has the additional problem, that it needs a lot of temporary memory.

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Answer 3

Steven Lord 2022-4-6

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You could do this pretty easily if you know the powers to which you want to raise the data used to create the Vandermonde matrix.

data = 1:6;
powers = [5 3 1];
V = vander(data)
V = 6×6
           1           1           1           1           1           1
          32          16           8           4           2           1
         243          81          27           9           3           1
        1024         256          64          16           4           1
        3125         625         125          25           5           1
        7776        1296         216          36           6           1
V2 = reshape(data, [], 1).^powers
V2 = 6×3
           1           1           1
          32           8           2
         243          27           3
        1024          64           4
        3125         125           5
        7776         216           6