多项式评估的生成方法
-
06-07-2019 - |
题
我试图想出一种优雅的方法来处理一些生成的多项式。对于这个问题,我们将(专门)关注以下情况:
- 命令 是生成的参数 n三阶多项式,其中 n:=order + 1。
- 我 是 0..n 范围内的整数参数
- 多项式在 x_j 处有零点,其中 j = 1..n 且 j ≠ i(此时应该很清楚 StackOverflow 需要一个新功能,或者它已经存在但我不知道)
- 多项式在 x_i 处的计算结果为 1。
由于这个特定的代码示例生成 x_1 ..x_n,我将解释如何在代码中找到它们。点间隔均匀 x_j = j * elementSize / order
分开,在哪里 n = order + 1
.
我生成一个 Func<double, double>
评估这个多项式。
private static Func<double, double> GeneratePsi(double elementSize, int order, int i)
{
if (order < 1)
throw new ArgumentOutOfRangeException("order", "order must be greater than 0.");
if (i < 0)
throw new ArgumentOutOfRangeException("i", "i cannot be less than zero.");
if (i > order)
throw new ArgumentException("i", "i cannot be greater than order");
ParameterExpression xp = Expression.Parameter(typeof(double), "x");
// generate the terms of the factored polynomial in form (x_j - x)
List<Expression> factors = new List<Expression>();
for (int j = 0; j <= order; j++)
{
if (j == i)
continue;
double p = j * elementSize / order;
factors.Add(Expression.Subtract(Expression.Constant(p), xp));
}
// evaluate the result at the point x_i to get scaleInv=1.0/scale.
double xi = i * elementSize / order;
double scaleInv = Enumerable.Range(0, order + 1).Aggregate(0.0, (product, j) => product * (j == i ? 1.0 : (j * elementSize / order - xi)));
/* generate an expression to evaluate
* (x_0 - x) * (x_1 - x) .. (x_n - x) / (x_i - x)
* obviously the term (x_i - x) is cancelled in this result, but included here to make the result clear
*/
Expression expr = factors.Skip(1).Aggregate(factors[0], Expression.Multiply);
// multiplying by scale forces the condition f(x_i)=1
expr = Expression.Multiply(Expression.Constant(1.0 / scaleInv), expr);
Expression<Func<double, double>> lambdaMethod = Expression.Lambda<Func<double, double>>(expr, xp);
return lambdaMethod.Compile();
}
问题: 我还需要评估 ψ′=dψ/dx。为此,我可以将 ψ=scale×(x_0 - x)(x_1 - x)×..×(x_n - x)/(x_i - x) 重写为 ψ=α_n×x^n + α_n×x 的形式^(n-1) + ..+ α_1×x + α_0。这给出 ψ′=n×α_n×x^(n-1) + (n-1)×α_n×x^(n-2) + ..+ 1×α_1。
出于计算原因,我们可以重写最终答案,而无需调用 Math.Pow
写成 ψ′=x×(x×(x×(..) - β_2) - β_1) - β_0。
为了完成所有这些“技巧”(所有非常基本的代数),我需要一种干净的方法来:
- 展开因式分解
Expression
含有ConstantExpression
和ParameterExpression
叶子和基本数学运算(最终要么BinaryExpression
与NodeType
设置为操作) - 这里的结果可以包括InvocationExpression
元素到MethodInfo
为了Math.Pow
我们将在整个过程中以特殊方式处理。 - 然后我对某些指定的导数进行求导
ParameterExpression
. 。结果中的术语,其中调用的右侧参数Math.Pow
常数 2 被替换为ConstantExpression(2)
乘以左边的值(调用Math.Pow(x,1)
已移除)。结果中因相对于 x 为常数而变为零的项将被删除。 - 然后分解出一些特定的实例
ParameterExpression
它们作为调用的左侧参数出现Math.Pow
. 。当调用的右侧变为ConstantExpression
与价值1
, ,我们将调用替换为ParameterExpression
本身。
¹ 将来,我想要一种方法 ParameterExpression
并返回一个 Expression
根据该参数进行评估。这样我就可以聚合生成的函数。我还没到那儿。² 将来,我希望发布一个通用库,用于将 LINQ 表达式用作符号数学。
解决方案
我使用以下方法编写了几个符号数学功能的基础知识 表达访问者 输入.NET 4。它并不完美,但看起来像是可行解决方案的基础。
Symbolic
是一个公共静态类,公开类似的方法Expand
,Simplify
, , 和PartialDerivative
ExpandVisitor
是扩展表达式的内部辅助类型SimplifyVisitor
是简化表达式的内部辅助类型DerivativeVisitor
是一个内部辅助类型,它采用表达式的导数ListPrintVisitor
是一个内部辅助类型,它转换Expression
到 Lisp 语法的前缀表示法
Symbolic
public static class Symbolic
{
public static Expression Expand(Expression expression)
{
return new ExpandVisitor().Visit(expression);
}
public static Expression Simplify(Expression expression)
{
return new SimplifyVisitor().Visit(expression);
}
public static Expression PartialDerivative(Expression expression, ParameterExpression parameter)
{
bool totalDerivative = false;
return new DerivativeVisitor(parameter, totalDerivative).Visit(expression);
}
public static string ToString(Expression expression)
{
ConstantExpression result = (ConstantExpression)new ListPrintVisitor().Visit(expression);
return result.Value.ToString();
}
}
扩展表达式 ExpandVisitor
internal class ExpandVisitor : ExpressionVisitor
{
protected override Expression VisitBinary(BinaryExpression node)
{
var left = Visit(node.Left);
var right = Visit(node.Right);
if (node.NodeType == ExpressionType.Multiply)
{
Expression[] leftNodes = GetAddedNodes(left).ToArray();
Expression[] rightNodes = GetAddedNodes(right).ToArray();
var result =
leftNodes
.SelectMany(x => rightNodes.Select(y => Expression.Multiply(x, y)))
.Aggregate((sum, term) => Expression.Add(sum, term));
return result;
}
if (node.Left == left && node.Right == right)
return node;
return Expression.MakeBinary(node.NodeType, left, right, node.IsLiftedToNull, node.Method, node.Conversion);
}
/// <summary>
/// Treats the <paramref name="node"/> as the sum (or difference) of one or more child nodes and returns the
/// the individual addends in the sum.
/// </summary>
private static IEnumerable<Expression> GetAddedNodes(Expression node)
{
BinaryExpression binary = node as BinaryExpression;
if (binary != null)
{
switch (binary.NodeType)
{
case ExpressionType.Add:
foreach (var n in GetAddedNodes(binary.Left))
yield return n;
foreach (var n in GetAddedNodes(binary.Right))
yield return n;
yield break;
case ExpressionType.Subtract:
foreach (var n in GetAddedNodes(binary.Left))
yield return n;
foreach (var n in GetAddedNodes(binary.Right))
yield return Expression.Negate(n);
yield break;
default:
break;
}
}
yield return node;
}
}
求导数 DerivativeVisitor
internal class DerivativeVisitor : ExpressionVisitor
{
private ParameterExpression _parameter;
private bool _totalDerivative;
public DerivativeVisitor(ParameterExpression parameter, bool totalDerivative)
{
if (_totalDerivative)
throw new NotImplementedException();
_parameter = parameter;
_totalDerivative = totalDerivative;
}
protected override Expression VisitBinary(BinaryExpression node)
{
switch (node.NodeType)
{
case ExpressionType.Add:
case ExpressionType.Subtract:
return Expression.MakeBinary(node.NodeType, Visit(node.Left), Visit(node.Right));
case ExpressionType.Multiply:
return Expression.Add(Expression.Multiply(node.Left, Visit(node.Right)), Expression.Multiply(Visit(node.Left), node.Right));
case ExpressionType.Divide:
return Expression.Divide(Expression.Subtract(Expression.Multiply(Visit(node.Left), node.Right), Expression.Multiply(node.Left, Visit(node.Right))), Expression.Power(node.Right, Expression.Constant(2)));
case ExpressionType.Power:
if (node.Right is ConstantExpression)
{
return Expression.Multiply(node.Right, Expression.Multiply(Visit(node.Left), Expression.Subtract(node.Right, Expression.Constant(1))));
}
else if (node.Left is ConstantExpression)
{
return Expression.Multiply(node, MathExpressions.Log(node.Left));
}
else
{
return Expression.Multiply(node, Expression.Add(
Expression.Multiply(Visit(node.Left), Expression.Divide(node.Right, node.Left)),
Expression.Multiply(Visit(node.Right), MathExpressions.Log(node.Left))
));
}
default:
throw new NotImplementedException();
}
}
protected override Expression VisitConstant(ConstantExpression node)
{
return MathExpressions.Zero;
}
protected override Expression VisitInvocation(InvocationExpression node)
{
MemberExpression memberExpression = node.Expression as MemberExpression;
if (memberExpression != null)
{
var member = memberExpression.Member;
if (member.DeclaringType != typeof(Math))
throw new NotImplementedException();
switch (member.Name)
{
case "Log":
return Expression.Divide(Visit(node.Expression), node.Expression);
case "Log10":
return Expression.Divide(Visit(node.Expression), Expression.Multiply(Expression.Constant(Math.Log(10)), node.Expression));
case "Exp":
case "Sin":
case "Cos":
default:
throw new NotImplementedException();
}
}
throw new NotImplementedException();
}
protected override Expression VisitParameter(ParameterExpression node)
{
if (node == _parameter)
return MathExpressions.One;
return MathExpressions.Zero;
}
}
简化表达式 SimplifyVisitor
internal class SimplifyVisitor : ExpressionVisitor
{
protected override Expression VisitBinary(BinaryExpression node)
{
var left = Visit(node.Left);
var right = Visit(node.Right);
ConstantExpression leftConstant = left as ConstantExpression;
ConstantExpression rightConstant = right as ConstantExpression;
if (leftConstant != null && rightConstant != null
&& (leftConstant.Value is double) && (rightConstant.Value is double))
{
double leftValue = (double)leftConstant.Value;
double rightValue = (double)rightConstant.Value;
switch (node.NodeType)
{
case ExpressionType.Add:
return Expression.Constant(leftValue + rightValue);
case ExpressionType.Subtract:
return Expression.Constant(leftValue - rightValue);
case ExpressionType.Multiply:
return Expression.Constant(leftValue * rightValue);
case ExpressionType.Divide:
return Expression.Constant(leftValue / rightValue);
default:
throw new NotImplementedException();
}
}
switch (node.NodeType)
{
case ExpressionType.Add:
if (IsZero(left))
return right;
if (IsZero(right))
return left;
break;
case ExpressionType.Subtract:
if (IsZero(left))
return Expression.Negate(right);
if (IsZero(right))
return left;
break;
case ExpressionType.Multiply:
if (IsZero(left) || IsZero(right))
return MathExpressions.Zero;
if (IsOne(left))
return right;
if (IsOne(right))
return left;
break;
case ExpressionType.Divide:
if (IsZero(right))
throw new DivideByZeroException();
if (IsZero(left))
return MathExpressions.Zero;
if (IsOne(right))
return left;
break;
default:
throw new NotImplementedException();
}
return Expression.MakeBinary(node.NodeType, left, right);
}
protected override Expression VisitUnary(UnaryExpression node)
{
var operand = Visit(node.Operand);
ConstantExpression operandConstant = operand as ConstantExpression;
if (operandConstant != null && (operandConstant.Value is double))
{
double operandValue = (double)operandConstant.Value;
switch (node.NodeType)
{
case ExpressionType.Negate:
if (operandValue == 0.0)
return MathExpressions.Zero;
return Expression.Constant(-operandValue);
default:
throw new NotImplementedException();
}
}
switch (node.NodeType)
{
case ExpressionType.Negate:
if (operand.NodeType == ExpressionType.Negate)
{
return ((UnaryExpression)operand).Operand;
}
break;
default:
throw new NotImplementedException();
}
return Expression.MakeUnary(node.NodeType, operand, node.Type);
}
private static bool IsZero(Expression expression)
{
ConstantExpression constant = expression as ConstantExpression;
if (constant != null)
{
if (constant.Value.Equals(0.0))
return true;
}
return false;
}
private static bool IsOne(Expression expression)
{
ConstantExpression constant = expression as ConstantExpression;
if (constant != null)
{
if (constant.Value.Equals(1.0))
return true;
}
return false;
}
}
设置表达式的显示格式 ListPrintVisitor
internal class ListPrintVisitor : ExpressionVisitor
{
protected override Expression VisitBinary(BinaryExpression node)
{
string op = null;
switch (node.NodeType)
{
case ExpressionType.Add:
op = "+";
break;
case ExpressionType.Subtract:
op = "-";
break;
case ExpressionType.Multiply:
op = "*";
break;
case ExpressionType.Divide:
op = "/";
break;
default:
throw new NotImplementedException();
}
var left = Visit(node.Left);
var right = Visit(node.Right);
string result = string.Format("({0} {1} {2})", op, ((ConstantExpression)left).Value, ((ConstantExpression)right).Value);
return Expression.Constant(result);
}
protected override Expression VisitConstant(ConstantExpression node)
{
if (node.Value is string)
return node;
return Expression.Constant(node.Value.ToString());
}
protected override Expression VisitParameter(ParameterExpression node)
{
return Expression.Constant(node.Name);
}
}
测试结果
[TestMethod]
public void BasicSymbolicTest()
{
ParameterExpression x = Expression.Parameter(typeof(double), "x");
Expression linear = Expression.Add(Expression.Constant(3.0), x);
Assert.AreEqual("(+ 3 x)", Symbolic.ToString(linear));
Expression quadratic = Expression.Multiply(linear, Expression.Add(Expression.Constant(2.0), x));
Assert.AreEqual("(* (+ 3 x) (+ 2 x))", Symbolic.ToString(quadratic));
Expression expanded = Symbolic.Expand(quadratic);
Assert.AreEqual("(+ (+ (+ (* 3 2) (* 3 x)) (* x 2)) (* x x))", Symbolic.ToString(expanded));
Assert.AreEqual("(+ (+ (+ 6 (* 3 x)) (* x 2)) (* x x))", Symbolic.ToString(Symbolic.Simplify(expanded)));
Expression derivative = Symbolic.PartialDerivative(expanded, x);
Assert.AreEqual("(+ (+ (+ (+ (* 3 0) (* 0 2)) (+ (* 3 1) (* 0 x))) (+ (* x 0) (* 1 2))) (+ (* x 1) (* 1 x)))", Symbolic.ToString(derivative));
Expression simplified = Symbolic.Simplify(derivative);
Assert.AreEqual("(+ 5 (+ x x))", Symbolic.ToString(simplified));
}
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