public static Tuple<Function, string> FunctionVariation(Function I, double level = 1, double from = 0, double to = 1, string by = "x")
{
StringBuilder sb = new StringBuilder(
@"<!DOCTYPE html>
<?xml-stylesheet type='text/xsl' href='mathml.xsl'?>
<html
xmlns='http://www.w3.org/1999/xhtml'
xmlns:math='http://www.w3.org/1998/Math/MathML'
>
<head><meta http-equiv='Content-Type' content='text/html; charset=UTF-8' /></head>
<body>
<b>Вариация функции:</b></br>"
);
sb.AppendFormat("<math><mrow><mi>I</mi><mo>=</mo><mrow><msubsup><mo>∫</mo><mn>{1}</mn><mn>{2}</mn></mrow><mrow><mfenced>{0}</mfenced></mrow><mo>d</mo><mi>{3}</mi></mrow></math></br>", I.ToMathML(), from, to, by);
string[] vars = I.Variables;
Dictionary<string, IMatrixElement> varsValues = new Dictionary<string, IMatrixElement>();
Variable Alpha = new Variable("α");
for (int i = 0; i < vars.Length; i++)
{
if (vars[i] != by)
{
varsValues[vars[i]] = new FunctionMatrixElement(new Function(new Variable(vars[i]) + Alpha * new Variable("d" + vars[i])));
}
}
Function I2 = new Function(I.SetVariablesValues(varsValues));
sb.AppendFormat(@"
<math><mrow>
<mi>I</mi><mfenced><mrow><mi>y</mi><mo>+</mo><mi>{4}</mi><mo>d</mo><mi>y</mi></mrow></mfenced><mo>=</mo>
<mrow>
<mrow><msubsup><mo>∫</mo><mn>{1}</mn><mn>{2}</mn></msubsup></mrow>
<mrow>{0}</mrow><mrow><mo>d</mo><mi>{3}</mi></mrow></mrow>
</mrow>
</mrow></math></br>", I2.ToMathML(), from, to, by, Alpha.Name);
var curI = I2;
for (int i = 0; i < level; i++)
{
sb.AppendFormat("</br></br>{0} вариация:</br>", i + 1);
var dI2 = new Function(curI.Derivative(Alpha.Name));
sb.AppendFormat("<math><mrow><msup><mo>d</mo><mn>{4}</mn></msup></mrow><mi>I</mi><mo>=</mo><msubsup><mo>∫</mo><mn>{1}</mn><mn>{2}</mn></msubsup><mfenced><mrow>{0}</mrow></mfenced><mo>d</mo><mi>{3}</mi><math></br></br>", dI2.ToMathML(), from, to, by, i + 1);
sb.AppendFormat("<math><mrow><msup><mo>d</mo><mn>{4}</mn></msup></mrow><mi>I</mi><mo>⁢</mo><mrow><mfenced><mrow>{5}</mrow><mo>=</mo><mi>0</mi></mfenced></mrow><mo>=</mo><msubsup><mo>∫</mo><mn>{1}</mn><mn>{2}</mn></msubsup><mfenced><mrow>{0}<mrow></mfenced><mo>d</mo><mi>{3}</mi><math>", dI2.SetVariableValue(Alpha.Name, 0).ToMathML(), from, to, by, i + 1, Alpha.Name);
curI = dI2;
}
sb.Append("</body></html>");
return new Tuple<Function, string>(new Function(curI.SetVariableValue(Alpha.Name, 0)), sb.ToString());
}
public static Tuple<Matrix, string> NuthonMethod(Function f, Matrix x0, double e1, double e2, int M)
{
string matrixStartLine = "[", matrixElementsSeparator = ", ", matrixEndLine = "]";
StringBuilder sb = new StringBuilder("Метод Ньютона\n");
int k = 0;
double t = 0;
string[] vars = f.Variables;
Matrix varsNames = new Matrix(vars.Length);
int n = vars.Length;
for (int i = 0; i < vars.Length; i++)
{
varsNames[i] = new FunctionMatrixElement(1, vars[i]);
}
Matrix gradF = f.Gradient();
Matrix H = f.GesseMatrix();
sb.AppendFormat("Шаг 1. Задаём: x0 = {0}, e1 = {1}, e2 = {2}, M = {3}\n", x0.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine), e1, e2, M);
sb.AppendFormat(" Градиент функции: ▼f = {0}\n", gradF.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
sb.AppendFormat(" Матрица Гессе: H = {0}\n", H.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
sb.AppendFormat("Шаг 2. Положим: k = {0}\n", k);
List<Matrix> gradFs = new List<Matrix>();
List<Matrix> x = new List<Matrix>();
List<Matrix> d = new List<Matrix>();
x.Add(x0);
Matrix answer = x0;
while (true)
{
sb.AppendFormat("---------------------------{0} иттерация-------------------\n", k);
gradFs.Add(gradF.SetVariablesValuesWithFixedOrder(x[k], varsNames));
sb.AppendFormat("Шаг 3. Градиент: ▼f(x[{0}]) = {1}\n", k, gradFs[k].Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
double normF = gradFs[k].Norm();
sb.AppendFormat("Шаг 4. 1-ый критерий окончания: || ▼f(x[{0}]) || = {1} {2} e1 = {3}\n", k, normF, normF <= e1 ? " <= " : " > ", e1);
if (normF <= e1)
{
answer = x[k];
sb.AppendFormat("1-ый критерий выполнен, следовательно x* = {0}", answer.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
break;
}
sb.AppendFormat("Шаг 5. Проверяем k >= M: {0} {1} {2}\n", k, k < M ? "<" : ">=", M);
if (k >= M)
{
answer = x[k];
sb.AppendFormat("Следовательно x* = {0}", answer.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
break;
}
Matrix Hk = H.SetVariablesValuesWithFixedOrder(x[k], varsNames);
sb.AppendFormat("Шаг 6. H(x[{0}]):\n{1}\n", k, Hk.ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
sb.AppendFormat("Шаг 7. Обратная матрица матрицы H(x[{0}]):\n{1}\n", k, Hk.Reverse().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
bool flag = Hk.Reverse().Silvestr();
sb.AppendFormat("Шаг 8. Проверяем условие H^-1[{0}] > 0: {1}", k, flag);
if (flag)
{
sb.AppendFormat(" => переходим к 9 шагу.\n");
d.Add(new FunctionMatrixElement(-1) * Hk.Reverse() * gradFs[k]);
sb.AppendFormat("Шаг 9. Находим d[{0}]:\n{1}\n", k, d[k].ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
t = 1;
}
else
{
d.Add(new FunctionMatrixElement(-1) * gradFs[k]);
sb.AppendFormat(" => переходим к 10 шагу, положим d[{0}] = -▼f(x[{0}]):{1}\n", k, d[k].ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
Matrix tf = x[k] - new FunctionMatrixElement(1, "t#") * gradFs[k];
Function fi = new Function(f.SetVariablesValues(tf));
sb.AppendFormat("x{0} - t{0}*▼f(x{0}) = {1}\n", k, tf.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
sb.AppendFormat("f(x{0} - t{0}*▼f(x{0})) = {1}\n", k, fi);
Equation et = new Function(fi.Derivative("t#")).EquationSolutionByVariable("t#", new Function(0));
sb.AppendLine(et.ToString());
t = et.RightPart.ToDouble();
}
x.Add(x[k] + new FunctionMatrixElement(t) * d[k]);
sb.AppendFormat("Шаг 10. x{0} = {1}\n", k + 1, x[k + 1].Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
if (k == 0)
{
k++;
sb.AppendFormat("Шаг 11. Т.к. это первая иттерация, то увеличиваем k и возвращаемся к 3 шагу.\n\tk = {0}\n", k);
continue;
}
double normX1 = (x[k + 1] - x[k]).Norm(), normX2 = (x[k] - x[k - 1]).Norm();
double normF1 = Math.Abs((new Function(f.SetVariablesValuesWithFixedOrder(x[k + 1], varsNames)) - f.SetVariablesValuesWithFixedOrder(x[k], varsNames)).ToDouble()),
normF2 = Math.Abs((new Function(f.SetVariablesValuesWithFixedOrder(x[k], varsNames)) - f.SetVariablesValuesWithFixedOrder(x[k - 1], varsNames)).ToDouble());
sb.AppendFormat("Шаг 11. ||x{0} - x{1}|| = {2} {3} {4}\n ||f(x{0}) - f(x{1})|| = {5} {6} {4}\n",
k, k + 1, normX1, normX1 < e2 ? "<" : ">", e2, normF1, normF1 < e2 ? "<" : ">");
sb.AppendFormat(" ||x{0} - x{1}|| = {2} {3} {4}\n ||f(x{0}) - f(x{1})|| = {5} {6} {4}\n",
k - 1, k, normX2, normX2 < e2 ? "<" : ">", e2, normF2, normF2 < e2 ? "<" : ">");
if (normX1 < e2 && normX2 < e2 && normF1 < e2 && normF2 < e2)
{
answer = x[k + 1];
sb.AppendFormat("Следовательно x* = {0}", answer.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
break;
}
else
{
k++;
sb.AppendFormat("Т.к. хотя бы одно условие не выполнено (не <), то увеличиваем k и возвращаемся к 3 шагу.\n\tk = {0}\n", k);
continue;
}
}
return new Tuple<Matrix, string>(answer, sb.ToString());
}
public static Tuple<Matrix, string> BarrierMethod(Function f, Equation[] q, Matrix x0, double e, double r0, double C)
{
List<Function> qm = new List<Function>(), qp = new List<Function>();
StringBuilder sb = new StringBuilder("<h1>Метод барьерных функций</h1></br>");
foreach (var qi in q)
{
if (qi.Sign == "<=")
{
qp.Add(qi.AllToLeft());
}
if (qi.Sign == "=")
{
qm.Add(qi.AllToLeft());
}
}
sb.AppendFormat("<b>Функция</b> f = {0}</br><b>Ограничения:</b></br>", f.ToMathML());
int i = 0;
qm.ForEach(qi => sb.AppendFormat("    q({0}) = {1} = 0</br>", i++, qi.ToMathML()));
i = 0;
qp.ForEach(qi => sb.AppendFormat("    q({0}) = {1} <= 0</br>", i++, qi.ToMathML()));
int k = 0;
string[] vars = f.Variables;
Matrix varsNames = new Matrix(vars.Length);
for (i = 0; i < vars.Length; i++)
{
varsNames[i] = new FunctionMatrixElement(1, vars[i]);
}
sb.AppendFormat("Шаг 1. Задаём: <math><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>=</mo><mrow>{0}</mrow><mo></math>,штраф <math><msub><mi>r</mi><mn>0</mn></msub><mo>=</mo></math>{1}, C = {2}, e = {3}, k = {4}</br>", x0.ToMathML(), r0, C, e, k);
List<Matrix> x = new List<Matrix>();
List<double> r = new List<double>();
Function Sq = new Function();
foreach(var qi in q) {
Sq += qi.AllToLeft().Pow(-1);
}
Function P = new Variable("rk") * Sq;
x.Add(x0);
r.Add(r0);
Matrix answer = x0;
while (true)
{
sb.AppendFormat("---------------------------<b>{0} иттерация</b>-------------------</br>", k);
Function F = f - P.SetVariableValue("rk", r[k]);
sb.AppendFormat("Шаг 2. Вспомогательная функция = {0}</br>", F.ToMathML());
//Function der = F.Derivative();
//Matrix minX = SecondOrderMethods.NuthonMethod(F, x[k], e, e, 5).Item1;
Matrix minX = MinimisationMethods.PenaltyMethod(F, q.ToArray(), x[k], e, r0, C).Item1;
sb.AppendFormat("Шаг 3. Минимум функции F:</br>     <math><msub><mi>x</mi><mn>0</mn></msub><mo>=</mo><msub><mi>x</mi><mn>{0}</mn></msub><mo>=</mo></math>{1}</br>    X* = {2}</br>",
k,
x[k].ToMathML(),
minX.ToMathML()
);
sb.AppendFormat("Шаг 4. Условие окончания: <math><mi>P</mi><mfenced><msup><mi>x</mi><mn>*</mn></msup><mfenced><msub><mi>r</mi><mn>{0}</mn></msub></mfenced><mo>,</mo><msub><mi>r</mi><mn>{0}</mn></msub></mfenced><mo><=?</mo><mi>e</mi><mo>=</mo><mi>{1}</mi></math></br>", k, e);
double p = Math.Abs(new Function(P.SetVariableValue("rk", r[k])).SetVariablesValuesWithFixedOrder(minX, varsNames).ToDouble());
sb.AppendFormat("    {0} {1} {2}</br>", p, p <= e ? "<=" : ">", e);
if (p <= e)
{
answer = minX;
sb.AppendFormat("Следовательно x* = {0}", answer.ToMathML());
break;
}
else
{
r.Add(r[k] / C);
x.Add(minX);
k++;
}
}
return new Tuple<Matrix, string>(answer, sb.ToString());
}
public static Tuple<Matrix, string> PenaltyMethod(Function f, Equation[] q, Matrix x0, double e, double r0, double C)
{
string matrixStartLine = "[", matrixElementsSeparator = ", ", matrixEndLine = "]";
List<Function> qm = new List<Function>(), qp = new List<Function>();
StringBuilder sb = new StringBuilder("<h1>Метод штрафных функций</h1></br>");
foreach (var qi in q)
{
if (qi.Sign == "<=")
{
qp.Add(qi.AllToLeft());
}
if (qi.Sign == "=")
{
qm.Add(qi.AllToLeft());
}
}
sb.AppendFormat("<b>Функция</b> f = {0}</br><b>Ограничения:</b></br>", f.ToMathML());
int i = 0;
qm.ForEach(qi => sb.AppendFormat("    q({0}) = {1} = 0</br>", i++, qi.ToMathML()));
i = 0;
qp.ForEach(qi => sb.AppendFormat("    q({0}) = {1} <= 0</br>", i++, qi.ToMathML()));
int k = 0;
string[] vars = f.Variables;
Matrix varsNames = new Matrix(vars.Length);
for (i = 0; i < vars.Length; i++)
{
varsNames[i] = new FunctionMatrixElement(1, vars[i]);
}
sb.AppendFormat("Шаг 1. Задаём: <math><mrow><msub><mi>x</mi><mn>0</mn></msub></mrow><mo>=</mo><mrow>{0}</mrow><mo></math>,штраф <math><msub><mi>r</mi><mn>0</mn></msub><mo>=</mo></math>{1}, C = {2}, e = {3}, k = {4}</br>", x0.ToMathML(), r0, C, e, k);
List<Matrix> x = new List<Matrix>();
List<double> r = new List<double>();
Function Sm = new Function();
Function Sp = new Function();
qm.ForEach(qi => Sm += qi.Pow(2));
#warning Врятли без этого оно будет работать
//qp.ForEach(qi => Sp += qi.AddFunction(new SliceFunction()).Pow(2));
Function P = new Variable("rk") / 2.0 * (Sm + Sp);
x.Add(x0);
r.Add(r0);
Matrix answer = x0;
while (true)
{
sb.AppendFormat("---------------------------{0} иттерация-------------------\n", k);
sb.AppendFormat("{0}\n", P);
Function F = f + P.SetVariableValue("rk", r[k]);
sb.AppendFormat("Шаг 2. Вспомогательная функция = {0}\n", F);
Matrix minX = SecondOrderMethods.NuthonMethod(F, x[k], e, e, 5).Item1;
sb.AppendFormat("Шаг 3. Минимум функции F:\n\tx[0] = x[{0}] = {1}\n\tX* = {2}\n",
k,
x[k].Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine),
minX.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine)
);
sb.AppendFormat("Шаг 4. Условие окончания: P(x*(r[{0}],r[{0}]) <=? e = {1}\n", k, e);
double p = new Function(P.SetVariableValue("rk", r[k])).SetVariablesValuesWithFixedOrder(minX, varsNames).ToDouble();
sb.AppendFormat(" {0} {1} {2}\n", p, p <= e ? "<=" : ">", e);
if (p <= e)
{
answer = minX;
sb.AppendFormat("Следовательно x* = {0}", answer.Transposition().ToString(matrixStartLine, matrixElementsSeparator, matrixEndLine));
break;
}
else
{
r.Add(C * r[k]);
x.Add(minX);
k++;
}
}
return new Tuple<Matrix, string>(answer, sb.ToString());
}
public Matrix GesseMatrix()
{
string[] vars = this.Variables;
Matrix m = new Matrix(vars.Length, vars.Length);
for (int i = 0; i < vars.Length; i++)
{
for (int j = 0; j < vars.Length; j++)
{
m[i, j] = new FunctionMatrixElement(new Function(this.Derivative(vars[i]).Derivative(vars[j])));
}
}
return m;
}
public Matrix Gradient()
{
string[] vars = this.Variables;
Matrix m = new Matrix(vars.Length, 1);
for (int i = 0; i < vars.Length; i++)
{
m[i, 0] = new FunctionMatrixElement(new Function(this.Derivative(vars[i])));
}
return m;
}