private double GetdF(Double[] V, Int32 Index, double accuracy)
{
double[] temp = new double[V.Length];
for (Int32 i = 0; i < V.Length; i++)
{
if (i == Index)
{
temp.SetValue(V[i] + accuracy, i);
}
else
{
temp.SetValue(V[i], i);
}
}
return((FunctionClass.Function3(temp) - FunctionClass.Function3(V)) / accuracy);
}
private void HookJeevesButton_Click(object sender, EventArgs e)
{
OutBox.Text = string.Empty;
double eps = Convert.ToDouble(EpsilonBox.Text); // Точность вычислений
double step = Convert.ToDouble(StepBox.Text); // Начальный шаг
// Начальное приближение
double x = Convert.ToDouble(x0Box.Text);
double y = Convert.ToDouble(y0Box.Text);
double y0 = FunctionClass.Function2(x, y);
int IterCount = 0;
double fx;
double lenght, y1, y2, y3, x1n, x2n, x1nn, x2nn;
int fl = 0;
lenght = 1;
do
{
fx = FunctionClass.Function2(x, y);
OutBox.Text += "x = " + Math.Round(x, 6).ToString() + " y = " + Math.Round(y, 6).ToString() + " f(x, y) = " + Math.Round(fx, 6).ToString() + "\r\n";
fl = 0;
y1 = FunctionClass.Function2(x + step, y);
if (y1 < y0)
{
x1n = x + step;
}
else
{
y1 = FunctionClass.Function2(x - step, y);
if (y1 < y0)
{
x1n = x - step;
}
else
{
x1n = x;
y1 = y0;
}
}
y2 = FunctionClass.Function2(x1n, y + step);
if (y2 < y1)
{
x2n = y + step;
}
else
{
y2 = FunctionClass.Function2(x1n, y - step);
if (y2 < y1)
{
x2n = y - step;
}
else
{
x2n = y;
y2 = y1;
}
}
if ((x1n == x) && (x2n == y))
{
if (step > eps)
{
fl = 1;
step = step / 2;
}
}
else
{
fl = 1;
do
{
x1nn = x1n + lenght * (x1n - x);
x2nn = x2n + lenght * (x2n - y);
y3 = FunctionClass.Function2(x1nn, x2nn);
x = x1nn;
y = x2nn;
lenght = lenght * 0.5;
} while (y2 < y3);
IterCount++;
y0 = y3;
}
} while (fl == 1);
fx = FunctionClass.Function2(x, y);
MinXBox.Text = x.ToString();
MinYBox.Text = y.ToString();
MinFBox.Text = fx.ToString();
IterCountBox.Text = IterCount.ToString();
}
private void BuildButton_Click(object sender, EventArgs e)
{
// Границы вычисления функции по осям X и Y
double minX = Convert.ToDouble(MinXBox.Text);
double maxX = Convert.ToDouble(MaxXBox.Text);
double minY = Convert.ToDouble(MinYBox.Text);
double maxY = Convert.ToDouble(MaxYBox.Text);
// Масштаб
double intervalX = maxX - minX;
double intervalY = maxY - minY;
double zoomX = intervalX / bitmap.Width; // 1 пиксель : 1 ед.измерения оси X
double zoomY = intervalY / bitmap.Height; // 1 пиксель : 1 ед.измерения оси Y
double eps = Convert.ToDouble(EpsilonBox.Text); // Точность
int levelCount = Convert.ToInt32(LevelBox.Text); // Количество линий уровня
// Уровни
int countX = Convert.ToInt32(bitmap.Width);
int countY = Convert.ToInt32(bitmap.Height);
double[,] values = new double[countX, countY]; // Сетка функции
double[] levels = new double[levelCount]; // Уровни функции
for (int i = 0; i < bitmap.Width; i++)
{
for (int j = 0; j < bitmap.Height; j++)
{
double x = minX + i * zoomX;
double y = minY + j * zoomY;
double z = FunctionClass.Function2(x, y);
values[i, j] = z;
}
}
double minValue = values[0, 0]; // Локальный минимум функции
double maxValue = values[0, 0]; // Локальный максимум функции
for (int i = 0; i < values.GetLength(0); i++)
{
for (int j = 0; j < values.GetLength(1); j++)
{
if (values[i, j] < minValue)
{
minValue = values[i, j];
}
if (values[i, j] > maxValue)
{
maxValue = values[i, j];
}
}
}
int levelStepX = bitmap.Width / levelCount / 2;
int levelStepY = bitmap.Height / levelCount / 2;
for (int i = 0; i < levelCount; i++)
{
levels[i] = values[i * levelStepX, i *levelStepY];
}
// Рисуем линии уровней
ProgressBar.Value = 0;
ProgressBar.Maximum = bitmap.Width * bitmap.Height * levels.Length;
for (int i = 0; i < bitmap.Width; i++)
{
for (int j = 0; j < bitmap.Height; j++)
{
for (int k = 0; k < levels.Length; k++)
{
double x = minX + i * zoomX;
double y = minY + j * zoomY;
double z = FunctionClass.Function2(x, y);
if (z >= levels[k] - eps && z <= levels[k] + eps)
{
int c = (int)(255 * Math.Abs(z - levels[k]) / eps);
DrawPoint(i, bitmap.Height - j - 1, Color.FromArgb(c, c, c));
}
ProgressBar.Value++;
}
}
}
}
private void GaussButton_Click(object sender, EventArgs e)
{
OutBox.Text = string.Empty;
double eps = Convert.ToDouble(EpsilonBox.Text); // Точность вычислений
double step = Convert.ToDouble(StepBox.Text); // Начальный шаг
// Начальное приближение
double x = Convert.ToDouble(x0Box.Text);
double y = Convert.ToDouble(y0Box.Text);
int IterCount = 0;
double fx, prevX, prevY, curX = x, curY = y;
double leftBorder, rightBorder;
double x1, xm, x2;
double y1, ym, y2;
double z1, zm, z2;
do
{
IterCount++;
prevX = curX;
prevY = curY;
// По 1 координате
leftBorder = prevX - step;
rightBorder = prevX + step;
x1 = leftBorder + (rightBorder - leftBorder) / 4;
xm = (leftBorder + rightBorder) / 2;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
z1 = FunctionClass.Function2(x1, prevY);
zm = FunctionClass.Function2(xm, prevY);
z2 = FunctionClass.Function2(x2, prevY);
while (Math.Abs(rightBorder - leftBorder) > eps)
{
if ((z1 > zm) && (zm < z2))
{
leftBorder = x1;
rightBorder = x2;
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
}
else if ((z1 > zm) && (zm > z2))
{
leftBorder = xm;
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
xm = (leftBorder + rightBorder) / 2;
zm = FunctionClass.Function2(xm, prevY);
}
else if ((z1 < zm) && (zm < z2))
{
rightBorder = xm;
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
xm = (leftBorder + rightBorder) / 2;
zm = FunctionClass.Function2(xm, prevY);
}
z1 = FunctionClass.Function2(x1, prevY);
z2 = FunctionClass.Function2(x2, prevY);
}
curX = (leftBorder + rightBorder) / 2;
// По 2 координате
leftBorder = prevY - step;
rightBorder = prevY + step;
y1 = leftBorder + (rightBorder - leftBorder) / 4;
ym = (leftBorder + rightBorder) / 2;
y2 = rightBorder - (rightBorder - leftBorder) / 4;
z1 = FunctionClass.Function2(curX, y1);
zm = FunctionClass.Function2(curX, ym);
z2 = FunctionClass.Function2(curX, y2);
while (Math.Abs(rightBorder - leftBorder) > eps)
{
if ((z1 > zm) && (zm < z2))
{
leftBorder = y1;
rightBorder = y2;
y1 = leftBorder + (rightBorder - leftBorder) / 4;
y2 = rightBorder - (rightBorder - leftBorder) / 4;
}
else if ((z1 > zm) && (zm > z2))
{
leftBorder = ym;
y1 = leftBorder + (rightBorder - leftBorder) / 4;
y2 = rightBorder - (rightBorder - leftBorder) / 4;
ym = (leftBorder + rightBorder) / 2;
zm = FunctionClass.Function2(curX, ym);
}
else if ((z1 < zm) && (zm < z2))
{
rightBorder = ym;
y1 = leftBorder + (rightBorder - leftBorder) / 4;
y2 = rightBorder - (rightBorder - leftBorder) / 4;
ym = (leftBorder + rightBorder) / 2;
zm = FunctionClass.Function2(curX, ym);
}
z1 = FunctionClass.Function2(curX, y1);
z2 = FunctionClass.Function2(curX, y2);
}
curY = (leftBorder + rightBorder) / 2;
}while (Math.Abs(FunctionClass.Function2(curX, curY) - FunctionClass.Function2(prevX, prevY)) > eps);
fx = FunctionClass.Function2(curX, curY);
MinXBox.Text = curX.ToString();
MinYBox.Text = curY.ToString();
MinFBox.Text = fx.ToString();
IterCountBox.Text = IterCount.ToString();
}
private void NewtonButton_Click(object sender, EventArgs e)
{
OutBox.Text = string.Empty;
// Шаг 1
int k = 0; // Счетчик итераций
double x0, y0; // Начальное приближение
double a = 1; // Шаг для модифицированного метода
x0 = Convert.ToDouble(x0Box.Text);
y0 = Convert.ToDouble(y0Box.Text);
eps = Convert.ToDouble(EpsilonBox.Text);
//a = Convert.ToDouble(StepBox.Text);
double x = x0, y = y0; // Текущая точка
double[] grad = new double[2]; // Градиент
double [,] matrix = new double[2, 2]; // Матрица Гессе
double[,] invertMatrix = new double[2, 2]; // Обратная матрица Гессе
double[] p = new double[2]; // Направление спуска
double[] curPoint = new double[2]; // Текущая точка
double[] nextPoint = new double[2]; // Следущая точка
calculate:
OutBox.Text += "Итерация № " + k.ToString() + "\r\n" +
"Текущая точка (" + x + ", " + y + ")\r\n" +
"Значение функции в точке " + FunctionClass.Function2(x, y) + "\r\n";
// Вычисляем градиент
grad = GetGrad(x, y);
OutBox.Text += "Градиент\r\n (" + grad[0] + ", " + grad[1] + ")\r\n";
// Вычисляем матрицу Гессе
matrix = GetHesse(x, y);
OutBox.Text += "Матрица Гессе\r\n" +
"| " + matrix[0, 0] + " | " + matrix[0, 1] + " |\r\n" +
"| " + matrix[1, 0] + " | " + matrix[1, 1] + " |\r\n";
// Шаг 2
// Определяем обратную матрицу
invertMatrix = InvertMatrix(matrix);
if (invertMatrix == null)
{
return;
}
OutBox.Text += "Обратная матрица Гессе\r\n" +
"| " + invertMatrix[0, 0] + " | " + invertMatrix[0, 1] + " |\r\n" +
"| " + invertMatrix[1, 0] + " | " + invertMatrix[1, 1] + " |\r\n";
// Шаг 3
// Определяем направление спуска
p = GetDirect(grad, invertMatrix);
OutBox.Text += "Направление спуска\r\n (" + p[0] + ", " + p[1] + ")\r\n";
// Шаг 4
// Определяем следующую точку
curPoint[0] = x;
curPoint[1] = y;
//a = GoldenSection(curPoint, eps, -10, 10, p);
nextPoint = GetNextPoint(curPoint, a, p);
OutBox.Text += "Новая точка (" + nextPoint[0] + ", " + nextPoint[1] + ")\r\n";
// Шаг 5
// Вычисление в новой точке градиента
grad = GetGrad(nextPoint[0], nextPoint[1]);
// Шаг 6
// Если точность не достигнута, то продолжаем итерации
x = nextPoint[0];
y = nextPoint[1];
if ((Math.Sqrt(grad[0] * grad[0] + grad[1] * grad[1]) > eps && Math.Abs(FunctionClass.Function2(curPoint[0], curPoint[1]) - FunctionClass.Function2(nextPoint[0], nextPoint[1])) > eps))
{
k++;
OutBox.Text += "________________________________\r\n";
goto calculate;
}
double f = FunctionClass.Function2(x, y);
OutBox.Text += "*************************************\r\n" +
"Минимум функции равен " + f + "\r\n" +
"в точке (" + x + ", " + y + ")\r\n";
MinXBox.Text = x.ToString();
MinYBox.Text = y.ToString();
MinFBox.Text = f.ToString();
IterCountBox.Text = (k + 1).ToString();
}
public double GetDfDy(double _x, double _y)
{
return((FunctionClass.Function2(_x, _y + Math.Pow(eps, 2)) - FunctionClass.Function2(_x, _y)) / Math.Pow(eps, 2));
}
// Метод Коши (Наискорейший градиентный спуск)
private string Grad(double[] x0, double eps, double h)
{
// 1 шаг
string result = "";
double[] x = new double[x0.Length], grad = new double[x0.Length];
double deltaX = 0;
Int32 index = 0;
x0.CopyTo(x, 0);
result += "Начальная точка:\r\n";
for (Int32 i = 0; i < x0.Length; i++)
{
result += "x[" + i + "]=" + x0[i] + "\r\n";
}
result += "F(x[0], x[1]) = " + FunctionClass.Function3(x0) + "\r\n\r\n";
do
{
index++;
// 2 шаг
result += "Вычисление градиенты: \r\n";
for (Int32 i = 0; i < x.Length; i++)
{
grad[i] = GetdF(x0, i, eps);
result += grad[i] + "\r\n";
}
// 3 шаг отпадает, т.к. нет ограничений на кол-во итераций
// 4 шаг
h = GoldenSection(x, eps, leftBorder, rightBorder);
result += "Шаг: " + h + "\r\n";
// 5 шаг
for (Int32 i = 0; i < x.Length; i++)
{
x[i] = x0[i] - h * GetdF(x0, i, eps);
}
result += "\r\nНовая точка: \r\n";
for (Int32 i = 0; i < x.Length; i++)
{
result += "x[" + i + "] = " + x[i] + "\r\n";
deltaX = Math.Pow(x[i] - x0[i], 2);
}
result += "F(x[0], x[1]) = " + FunctionClass.Function3(x) + "\r\n\r\n";
x.CopyTo(x0, 0);
}
// 6 шаг
while (Math.Sqrt(Math.Abs(deltaX)) > eps);
result += "Минимум:\r\n";
for (Int32 i = 0; i < x.Length; i++)
{
result += "x[" + i + "] = " + x[i] + "\r\n";
}
result += "Значение: " + FunctionClass.Function3(x);
MinXBox.Text = x[0].ToString();
MinYBox.Text = x[1].ToString();
MinFBox.Text = FunctionClass.Function3(x).ToString();
IterCountBox.Text = index.ToString();
return(result);
}
private double GoldenSection(out string log, double leftBorder, double rightBorder, double accuracy)
{
log = "Метод золотого сечения:\r\n\r\n" +
"Входные данные:\r\n" +
" Левая граница a = " + leftBorder + "\r\n" +
" Правая граница b = " + rightBorder + "\r\n" +
" Точность eps = " + accuracy + "\r\n\r\n" +
"Итерации:\r\n\r\n";
double tau = (-1 + Math.Sqrt(5)) / 2.0;
double x1 = leftBorder + (rightBorder - leftBorder) * tau;
double x2 = rightBorder - (rightBorder - leftBorder) * tau;
double xm = (leftBorder + rightBorder) / 2;
double y1 = FunctionClass.Function1(x1);
double y2 = FunctionClass.Function1(x2);
double ym = FunctionClass.Function1(xm);
log += "a = " + leftBorder + " " + "b = " + rightBorder + " " + "xm = " + xm + " " + "x1 = " + x1 + " " + "x2 = " + x2 + "\r\n";
log += "F(a) = " + FunctionClass.Function1(leftBorder) + " " + "F(b) = " + FunctionClass.Function1(rightBorder) + " " + "F(xm) = " + ym + " " + "F(x1) = " + y1 + " " + "F(x2) = " + y2 + "\r\n\r\n";
int i = 0;
while (Math.Abs(rightBorder - leftBorder) > accuracy)
{
i++;
if ((y1 < y2))
{
log += "# " + i + " y1 < y2\r\n";
leftBorder = x2;
x2 = x1;
x1 = leftBorder + (rightBorder - leftBorder) * tau;
log += "x1 = " + x1 + "\r\n";
log += "x2 = " + x2 + "\r\n";
}
else if ((y1 > y2))
{
log += "# " + i + " y1 > y2\r\n";
rightBorder = x1;
x1 = x2;
x2 = rightBorder - (rightBorder - leftBorder) * tau;
log += "x1 = " + x1 + "\r\n";
log += "x2 = " + x2 + "\r\n";
}
y1 = FunctionClass.Function1(x1);
y2 = FunctionClass.Function1(x2);
xm = (leftBorder + rightBorder) / 2;
ym = FunctionClass.Function1(xm);
log += "# " + i + " " + xm + "\r\n\r\n";
}
return(xm);
}
private double Bisection(out string log, double leftBorder, double rightBorder, double accuracy)
{
log = "Метод половинного деления:\r\n\r\n" +
"Входные данные:\r\n" +
" Левая граница a = " + leftBorder + "\r\n" +
" Правая граница b = " + rightBorder + "\r\n" +
" Точность eps = " + accuracy + "\r\n\r\n" +
"Итерации:\r\n\r\n";
double x1 = leftBorder + (rightBorder - leftBorder) / 4;
double xm = (leftBorder + rightBorder) / 2;
double x2 = rightBorder - (rightBorder - leftBorder) / 4;
double y1 = FunctionClass.Function1(x1);
double ym = FunctionClass.Function1(xm);
double y2 = FunctionClass.Function1(x2);
log += "a = " + leftBorder + " " + "b = " + rightBorder + " " + "xm = " + xm + " " + "x1 = " + x1 + " " + "x2 = " + x2 + "\r\n";
log += "F(a) = " + FunctionClass.Function1(leftBorder) + " " + "F(b) = " + FunctionClass.Function1(rightBorder) + " " + "F(xm) = " + ym + " " + "F(x1) = " + y1 + " " + "F(x2) = " + y2 + "\r\n\r\n";
int i = 0;
while (Math.Abs(rightBorder - leftBorder) > accuracy)
{
i++;
if ((y1 > ym) && (ym < y2))
{
log += "# " + i + " y1 > ym && ym < y2\r\n";
leftBorder = x1;
rightBorder = x2;
log += "a = " + leftBorder + " " + "b = " + rightBorder + "\r\n";
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
log += "x1 = " + x1 + " " + "x2 = " + x2 + "\r\n";
}
else if ((y1 > ym) && (ym > y2))
{
log += "# " + i + " y1 > ym && ym > y2\r\n";
leftBorder = xm;
log += "a = " + leftBorder + "\r\n";
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
xm = (leftBorder + rightBorder) / 2;
log += "x1 = " + x1 + " " + "xm = " + xm + " " + "x2 = " + x2 + "\r\n";
ym = FunctionClass.Function1(xm);
}
else if ((y1 < ym) && (ym < y2))
{
log += "# " + i + " y1 < ym && ym < y2\r\n";
rightBorder = xm;
log += "b = " + rightBorder + "\r\n";
x1 = leftBorder + (rightBorder - leftBorder) / 4;
x2 = rightBorder - (rightBorder - leftBorder) / 4;
xm = (leftBorder + rightBorder) / 2;
log += "x1 = " + x1 + " " + "xm = " + xm + " " + "x2 = " + x2 + "\r\n";
ym = FunctionClass.Function1(xm);
}
y1 = FunctionClass.Function1(x1);
y2 = FunctionClass.Function1(x2);
log += "y1 = " + y1 + " " + "ym = " + ym + " " + "y2 = " + y2 + "\r\n";
log += "xm = " + xm + "\r\n\r\n";
}
return(xm);
}