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82 lines
2.4 KiB
Matlab
82 lines
2.4 KiB
Matlab
% ENUME MICHAŁ SZOPIŃSKI
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% PROJECT B NUMBER 60
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% TASK 1
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% https://github.com/Lachcim/szopinski-enume
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% define available algorithms
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algorithms = {
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'bisection', @bisect;
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'Newton''s algorithm', @newton
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};
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% find all root brackets
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interval = [2, 11];
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brackets = rootbrac(@taskfunc, interval(1), interval(2));
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% find and graph zeros using both algorithms
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printroots(@taskfunc, algorithms, interval, brackets, ...
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'Approximate zeros of function', 'zeros');
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% use Newton's algorithm to find roots of polynomial
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algorithms = {'Newton''s algorithm', @newton};
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interval = [1, 7];
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brackets = rootbrac(@polynomial, interval(1), interval(2));
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printroots(@polynomial, algorithms, interval, brackets, ...
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'Approximate real roots of polynomial', 'realroots');
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% the function as given in the task
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function y = taskfunc(x)
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y = 0.7 * x * cos(x) - log(x + 1);
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end
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% uses the bisection algorithm to find the root of a function within the given bracket
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function [zero, steps] = bisect(func, a, b, tolerance)
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% initialize empty array of steps
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steps = double.empty(2, 0);
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% iterate algorithm until the error is within tolerance
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while 1
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% calculate midpoint
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zero = (a + b) / 2;
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steps(:, size(steps, 2) + 1) = [zero, func(zero)];
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% stop test
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if abs(a - b) < tolerance; break; end
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% choose next sub-interval based on sign mismatch
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if sign(func(a)) ~= sign(func(zero))
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b = zero;
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else
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a = zero;
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end
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end
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end
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% uses Newton's algorithm to find the root of a function
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function [zero, steps] = newton(func, a, b, tolerance)
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% initialize step array and calculate derivative step
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steps = double.empty(2, 0);
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step = sqrt(eps);
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% calculate first approximation of zero - midpoint of the bracket
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zero = (a + b) / 2;
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steps(:, size(steps, 2) + 1) = [zero, func(zero)];
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% iterate algorithm until the error is within tolerance
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while 1
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% calculate next approximation of zero
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prevzero = zero;
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derivative = (func(zero + step) - func(zero - step)) / (2 * step);
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zero = zero - func(zero) / derivative;
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steps(:, size(steps, 2) + 1) = [zero, func(zero)];
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% prevent divergence during approximation
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if zero < a || zero > b
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error('Divergent iteration');
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end
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% stop test
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if abs(zero - prevzero) <= tolerance; break; end
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end
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end
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