How to do a rank-1 approximation?

I am able to perform most all this question is asking. I am NEW to coding (let me preface with that), and as such, we have not done anything with parts of what this question is asking (gasp). Here is what I did thus far in MatLab and the Question. I also do not understand why the answers are displayed as fractions when I did not type "format rat" in ... off in the workspace, the answers are displayed rounded to 4 decimal points. But, I am unsure how to get them displayed as such in my main display. THANK YOU!

• Use the svd() function in MATLAB to compute , the rank-1 approximation of A. Clearly state what is, rounded to 4 decimal places. Also, compute the root-mean square error (RMSE) between A and .

Answer 1

[Karl]

You can check the display format that's set, and change if necessary:

% Show the current display format.
display(format)

  DisplayFormatOptions with properties:

    NumericFormat: "short"
      LineSpacing: "loose"

% Set short fixed point format, with 4 digits after the decimal point.
format short

When you know the functions that you need to use, but want to check their details, the help available at the MATLAB prompt can be really useful:

help help

 help - Help for functions in Command Window
    This MATLAB function displays the help text for the functionality
    specified by name, such as a function, method, class, toolbox, or
    variable.

    Syntax
      help name
      help

    Input Arguments
      name - Functionality name
        character vector | string scalar

    See also doc, lookfor, more, what, which, whos

    Introduced in MATLAB before R2006a
    Documentation for help
       doc help

    Other uses of help

       dsp/help    parallel-computing/help    simulink/help
    Folders named help

       toolstrip/help    ppt/help    xml/help              xml_xpath/help
       dom/help          re/help     xml_transform/help

From the help for svd, the formula that you've used for A1 isn't quite right.

From the help for rmse, you need to specify the two arrays that you want to compare.

[Jessica]

Thank you, Karl. I have decimals now. I still just am unsure about the rank-1 approximation of A (and how to go about coding that). Any help or advice there?

Here's what displays now:

A = [1 2 3; 3 3 4; 5 6 7]

A = 3x3

1 2 3 3 3 4 5 6 7

[U, S, V] = svd(A)

U = 3x3

-0.2904 0.9504 -0.1114 -0.4644 -0.2418 -0.8520 -0.8367 -0.1957 0.5115

S = 3x3

12.5318 0 0 0 0.9122 0 0 0 0.3499

V = 3x3

-0.4682 -0.8261 -0.3136 -0.5581 0.0012 0.8298 -0.6851 0.5635 -0.4616

A1 = U*S*V.'

A1 = 3x3

1.0000 2.0000 3.0000 3.0000 3.0000 4.0000 5.0000 6.0000 7.0000

rmse(A, A1)

ans = 1x3

1.0e-14 * 0.1456 0.0690 0.1026

[Torsten]

https://math.stackexchange.com/questions/2964843/finding-the-best-rank-one-approximation-of-the-matrix-bf-a

A = [1 2 3; 3 3 4; 5 6 7]

A = 3×3

1 2 3 3 3 4 5 6 7

[U, S, V] = svd(A)

U = 3×3

-0.2904 0.9504 -0.1114 -0.4644 -0.2418 -0.8520 -0.8367 -0.1957 0.5115

S = 3×3

12.5318 0 0 0 0.9122 0 0 0 0.3499

V = 3×3

-0.4682 -0.8261 -0.3136 -0.5581 0.0012 0.8298 -0.6851 0.5635 -0.4616

A1 = S(1,1)*U(:,1)*V(:,1).'

A1 = 3×3

1.7039 2.0313 2.4935 2.7243 3.2477 3.9867 4.9087 5.8517 7.1832

A2 = A1 + S(2,2)*U(:,2)*V(:,2).'

A2 = 3×3

0.9878 2.0324 2.9820 2.9065 3.2474 3.8624 5.0561 5.8515 7.0826

A3 = A2 + S(3,3)*U(:,3)*V(:,3).'

A3 = 3×3

1.0000 2.0000 3.0000 3.0000 3.0000 4.0000 5.0000 6.0000 7.0000

norm(A-A1,'fro')/3

ans = 0.3257

rmse(A,A1,'All')

ans = 0.3257

norm(A-A2,'fro')/3

ans = 0.1166

rmse(A,A2,'All')

ans = 0.1166

norm(A-A3,'fro')/3

ans = 1.1028e-15

rmse(A,A3,'All')

ans = 1.1028e-15

[Jessica]

Thank you so much! Upon looking up what 'fro' meant, that is what I was missing: Frobenius norm! I can totally do Euclidean norm by hand -- it's this coding that is brand new to me that has me all jumbled.

I appreciate you all for jumping in to my rescue!