Лабы / ЧМ.Лабы.Лисовец / ЧМ.labs.by mice / lab02 / Отчёт по лабораторной работе

Отчёт по лабораторной работе №2

Решение уравнений вида . Методы дихотомии, Ньютона, простых итераций.

Решение данных уравнений подразумевает два этапа:

1. Локализация корней – выделение отрезков, на которых находится не более одного корня.

2. Поиск корня с заданной точностью.

Первый пункт решается следующим способом:

lx = left_bound;

rx = lx + rs_step;

while (rigth_bound >= rx)

ly = slv_func(lx);

ry = slv_func(rx);

if (0 > ry * ly)

disp(sprintf('root found in [%f, %f]', lx, rx));

nx = (rx + lx) / 2; % x0

[...]

end

lx = lx + rs_step;

rx = rx + rs_step;

end

Можно добавить проверку минимумов и максимумов функции.

Второй пункт выполняем тремя различными способами: дихотомия, метод Ньютона, метод простых итераций.

Дихотомия:

disp(sprintf('root found in [%f, %f]', lx, rx));

tlx = lx;

trx = rx;

mx = (trx + tlx) / 2;

sc = 1; % steps count

while (dich_precision <= trx - tlx)

my = slv_func(mx);

if (0 > my * ry)

tlx = mx;

elseif (0 > my * ly)

trx = mx;

end

mx = (trx + tlx) / 2;

ly = slv_func(tlx);

ry = slv_func(trx);

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', mx, sc));

Метод Ньютона:

disp(sprintf('root found in [%f, %f]', lx, rx));

nx = (rx + lx) / 2; % x0 for newton

px = nx + 2*newt_precision;

sc = 1;

while(newt_precision < abs(px-nx))

drv = slv_func(nx+newt_deriv_acc/2) / newt_deriv_acc;

drv = drv - (slv_func(nx-newt_deriv_acc/2)) / newt_deriv_acc;

% drv = slv_func1(nx);

px = nx;

nx = nx - slv_func(nx) / drv;

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', nx, sc));

Метод простых итераций:

disp(sprintf('root found in [%f, %f]', lx, rx));

nx = (rx + lx) / 2; % x0 for simp_iter

px = nx + 2*newt_precision;

sc = 1;

while(newt_precision < abs(px-nx))

px = nx;

nx = slv_funcp(px);

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', nx, sc));

Программа целиком:

clc;

clear;

left_bound = -10;

rigth_bound = 10;

rs_step = 0.001; % roots search step

dich_precision = 0.0000001;

newt_precision = 0.0000001;

newt_deriv_acc = 0.0000001;

sitr_precision = 0.0000001;

if (left_bound > rigth_bound)

disp('Bound check error: (left_bound > rigth_bound)');

return;

end

if (0 >= rs_step)

disp('Invalid roots search step: (0 >= rs_step)');

return;

end

if (0 >= dich_precision)

disp('Invalid precision for [dichotomy]: (0 >= dich_precision)');

return;

end

if (0 >= newt_precision)

disp('Invalid precision for [newton]: (0 >= newt_precision)');

return;

end

if (0 >= newt_deriv_acc)

disp('Invalid derivative accuracy for [newton]: (0 >= newt_deriv_acc)');

return;

end

if (0 >= sitr_precision)

disp('Invalid precision for [simp_iter]: (0 >= sitr_precision)');

return;

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% метод дихотомии

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

disp(sprintf('Searching for roots using [dichotomy]...\n'));

lx = left_bound;

rx = lx + rs_step;

while (rigth_bound >= rx)

ly = slv_func(lx);

ry = slv_func(rx);

if (0 > ry * ly)

disp(sprintf('root found in [%f, %f]', lx, rx));

tlx = lx;

trx = rx;

mx = (trx + tlx) / 2;

sc = 1; % steps count

while (dich_precision <= trx - tlx)

my = slv_func(mx);

if (0 > my * ry)

tlx = mx;

elseif (0 > my * ly)

trx = mx;

end

mx = (trx + tlx) / 2;

ly = slv_func(tlx);

ry = slv_func(trx);

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', mx, sc));

end

lx = lx + rs_step;

rx = rx + rs_step;

end

disp(sprintf('[dichotomy] complete.\n\n'));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% метод Ньютона

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

disp(sprintf('Searching for roots using [newton]...\n'));

lx = left_bound;

rx = lx + rs_step;

while (rigth_bound >= rx)

ly = slv_func(lx);

ry = slv_func(rx);

if (0 > ry * ly)

disp(sprintf('root found in [%f, %f]', lx, rx));

nx = (rx + lx) / 2; % x0 for newton

px = nx + 2*newt_precision;

sc = 1;

while(newt_precision < abs(px-nx))

drv = slv_func(nx+newt_deriv_acc/2) / newt_deriv_acc;

drv = drv - (slv_func(nx-newt_deriv_acc/2)) / newt_deriv_acc;

% drv = slv_func1(nx);

px = nx;

nx = nx - slv_func(nx) / drv;

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', nx, sc));

end

lx = lx + rs_step;

rx = rx + rs_step;

end

disp(sprintf('[newton] complete.\n\n'));

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% метод простых итераций

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

disp(sprintf('Searching for roots using [simp_iter]...\n'));

lx = left_bound;

rx = lx + rs_step;

while (rigth_bound >= rx)

ly = slv_func(lx);

ry = slv_func(rx);

if (0 > ry * ly)

disp(sprintf('root found in [%f, %f]', lx, rx));

nx = (rx + lx) / 2; % x0 for simp_iter

px = nx + 2*sitr_precision;

sc = 1;

while(sitr_precision < abs(px-nx))

px = nx;

nx = slv_funcp(px);

sc = sc + 1;

end

disp(sprintf('root approx value is: %f; %i steps used\n', nx, sc));

end

lx = lx + rs_step;

rx = rx + rs_step;

end

disp(sprintf('[simp_iter] complete.\n\n'));

disp(sprintf('program complete.\n\n'));