Linear Algebra Bonus Question

FancyFunction.m

% The Question in Question-
%     Find a polynomial p(t) such that p(n) and the derivative p'(n) 
%     are both equal to n for n = 1, 2, 3, 4, and 5.
% 'degree' is the largest n we want this to be true for
% Loses precision around degree = 8
function ans = FancyFunction(degree)

    % Max power we're raising a particular degree to.
    % There are degree * 2 equations for any solution.
    % As 0 based, - 1
    order = degree * 2 - 1;

    % Construct matrix so polynomial of degree is evaluated at each n 
    % and polynomial derivative of degree is evaluated at each n 
    equations = vertcat(...
        (1:degree)'.^(0:order),... % Polynomial equations
        horzcat(zeros(degree,1),(1:degree)'.^(0:order - 1).*(1:order))...  % Polynomial derivative equations
    )

    % build solutions so that f(n) = n and f'(n) = n
    solutions = vertcat((1:degree)',(1:degree)')

    % Find and Flip solution so that matrix is usable in tests
    ans = flipud(inv(equations)*solutions)';
end

TestFancyFunction.m

% For details on polyval, polyder (Shorthand for Polynomial Valuation and Polynominal Derivative) see Octave API
function valid = TestFancyFunction()

    % We only care about degree 5
    % Find solution and derivative
    V  = FancyFunction(5)
    dV = polyder(V)

    % Since precision lost, declare acceptable 
    % error margin epsilon to accept as true
    epsilon = 1e-8;

    % Initialize valid to true. 
    valid   = 1;

    % If not found to be valid, valid = 0
    for i = 1:5
        valid = abs(polyval(V,i)  - i) < epsilon && ...
                abs(polyval(dV,i) - i) < epsilon && ...
                valid
    end

end

zOutput.txt

octave:1> TestFancyFunction
equations =

         1         1         1         1         1         1         1         1         1         1
         1         2         4         8        16        32        64       128       256       512
         1         3         9        27        81       243       729      2187      6561     19683
         1         4        16        64       256      1024      4096     16384     65536    262144
         1         5        25       125       625      3125     15625     78125    390625   1953125
         0         1         2         3         4         5         6         7         8         9
         0         1         4        12        32        80       192       448      1024      2304
         0         1         6        27       108       405      1458      5103     17496     59049
         0         1         8        48       256      1280      6144     28672    131072    589824
         0         1        10        75       500      3125     18750    109375    625000   3515625

solutions =

   1
   2
   3
   4
   5
   1
   2
   3
   4
   5

V =

   2.4306e-01  -6.4931e+00   7.4792e+01  -4.8621e+02   1.9599e+03  -5.0626e+03   8.3494e+03  -8.4447e+03   4.7367e+03  -1.1200e+03

dV =

   2.1875e+00  -5.1944e+01   5.2354e+02  -2.9173e+03   9.7993e+03  -2.0250e+04   2.5048e+04  -1.6889e+04   4.7367e+03

valid =  1
valid =  1
valid =  1
valid =  1
valid =  1
ans =  1