Algorithm for Fractional power calculation

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Life is Wonderful 2022-2-28

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https://ww2.mathworks.cn/matlabcentral/answers/1660285-algorithm-for-fractional-power-calculation

编辑： Life is Wonderful 2022-3-13

1. fractional base is no problem with current implementation.

2. To support fractional exponents, get the n-th root for any given number b. How to implement algorithm to get a numerical approximation ?

3. Current approach is inefficient, because it loops e times

b = [-32:32]; % example input values
e = [-3:3];   % example input values but doesn't support fraction's
power_function(b,e)
p = 1;
if e < 0
    e = abs(e);
    multiplier = 1/b;
else
    multiplier = b;
end
for k = 1:e
    p(:) = p * multiplier; %  n-th root for any given number 
end

回答（1 个）

Alan Stevens 2022-2-28

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在 MATLAB Online 中打开

How about using the Newton-Raphson algorithm. Here's the basic idea:

% x^n = b
% Let f(x) = x^n - b
% dfdx(x) = n*x^(n-1)
%
% Use Newton-Raphson iteration
% x1 = initial guess
% err = 1;
% while err > tol
%       x = x1 - f(x1)/dfdx(x1);
%       err = abs(x - x1);
%       x1 = x
% end

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Life is Wonderful 2022-3-12

编辑：Life is Wonderful 2022-3-13

在 MATLAB Online 中打开

Hi Walter,

Please see below code which is computing exp() and taylor series. At the 1 iteration final exp() value is calculated/converge. Please find the attachment

I could see with one multiplication and one addition, I could get the exp() THDN = -108dBc with x = +23 & -145 with x = -23 dBc which is very much efficient. The same works for fixed point as well . One need to be careful with division operator ( this takes lots time ).

I have added Excel spread sheet ( to replicate my below implementation ) for offline calculation. It has upto 10 fraction place of accuracy ( which pretty good for any standard as acceptance criteria). Please find the attachment.

Next step - With below code, do you have a proposal where number iteration is reduced [ I think one can optimize 90 iteration ] and If one do a loop unroll [smaller than 5 iteration ] , it can be done one max 5 steps. Any proposal/ suggestion ?

Thanks !!

n = 1e2;
x = 23.0; % for  taylor series [ -22:22] for 32 bit DSP
inpval = 23.0;  % for built-in exp()
% create a function and pass the args
expval = 1.0; % initialize as double
fprintf('%20s|%30s|%26s|%25s|%25s|\n--------------------+------------------------------+--------------------------+-------------------------+-------------------------+\n',"indx","expval","exp(x)",'diff','Nbits');
for i = n-1:-1:1
    expval = 1 + x * expval / i;
    fprintf('%20d|%30.10f|%26.15f|%25.10f|%25.10f|\n',i,expval,exp(inpval),abs(exp(inpval) - expval), log2(exp(inpval)));
end