public static MathObject AddToBothSides(this MathObject obj, MathObject item)
{
if (obj is Equation)
return (obj as Equation).a + item == (obj as Equation).b + item;
throw new Exception();
}
static MathObject GenerateLagrangePolynomial(double[] xs, double[] ys)
{
var xsSet = new HashSet<double>(xs);
if (xsSet.Count < xs.Length)
{
throw new ArgumentException("x values not unique");
}
var ysSet = new HashSet<double>(ys);
if (ysSet.Count < ys.Length)
{
throw new ArgumentException("y values not unique");
}
var x = new Symbol("x");
var members = new MathObject[ys.Length];
for (int i = 0; i < ys.Length; i++)
{
members[i] = ys[i] * GenerateBasisPolynomial(x, xs, i);
}
var sum = new Sum(members);
return sum.AlgebraicExpand();
}
static void AssertEqual(MathObject a, Double b, double tolerance = 0.00000001)
{
var x = (DoubleFloat)a;
var y = new DoubleFloat(b);
if (Math.Abs(x.val - y.val) > tolerance)
Console.WriteLine("{0} and {1} are not equal", x.val, y.val);
}
public static MathObject AssertEqToDouble(this MathObject a, MathObject b, double tolerance = 0.000001)
{
if (
Math.Abs(
((a as Equation).b as DoubleFloat).val
-
((b as Equation).b as DoubleFloat).val)
> tolerance)
{
Console.WriteLine("{0} and {1} are not equal", a, b);
}
return a;
}
public MathObject Substitute(MathObject a, double b)
{
return Substitute(a, new DoubleFloat(b));
}
public static MathObject Sin(MathObject arg)
{
return new Sin(arg).Simplify();
}
public static MathObject Atan2(MathObject a, MathObject b)
{
return new Symbolism.Atan2(a, b).Simplify();
}
public Point(MathObject x_val, int y_val)
{
x = x_val; y = new Integer(y_val);
}
public MathObject Substitute(MathObject a, MathObject b)
{
return bas.Substitute(a, b) ^ exp.Substitute(a, b);
}
public Power(MathObject a, MathObject b)
{
bas = a; exp = b;
}
public static MathObject Term(MathObject u)
{
if (u is Product && ((Product)u).elts[0] is Number)
return new Product() { elts = ((Product)u).elts.Cdr() };
if (u is Product) return u;
return new Product() { elts = new List<MathObject>() { u } };
}
public static MathObject Exponent(MathObject u)
{
if (u is Power) return ((Power)u).exp;
return new Integer(1);
}
public static MathObject Const(MathObject u)
{
if (u is Product)
{
var res = (Product)u;
if (res.elts[0] is Integer) return res.elts[0];
if (res.elts[0] is Fraction) return res.elts[0];
if (res.elts[0] is DoubleFloat) return res.elts[0];
return new Integer(1);
}
return new Integer(1);
}
public static bool Compare(MathObject u, MathObject v)
{
if (u is DoubleFloat && v is DoubleFloat) return ((DoubleFloat)u).val < ((DoubleFloat)v).val;
if (u is DoubleFloat && v is Integer) return ((DoubleFloat)u).val < ((Integer)v).val;
if (u is DoubleFloat && v is Fraction) return
((DoubleFloat)u).val < ((double)((Fraction)v).numerator.val) / ((double)((Fraction)v).denominator.val);
if (u is Integer && v is DoubleFloat) return ((Integer)u).val < ((DoubleFloat)v).val;
if (u is Fraction && v is DoubleFloat) return
((double)((Fraction)u).numerator.val) / ((double)((Fraction)u).denominator.val) < ((DoubleFloat)v).val;
if (u is Integer)
return Compare(new Fraction((Integer)u, new Integer(1)), v);
if (v is Integer)
return Compare(u, new Fraction((Integer)v, new Integer(1)));
if (u is Fraction && v is Fraction)
{
var u_ = (Fraction)u;
var v_ = (Fraction)v;
// a / b < c / d
//
// (a d) / (b d) < (c b) / (b d)
return
(u_.numerator.val * v_.denominator.val)
<
(v_.numerator.val * u_.denominator.val);
}
if (u is Symbol && v is Symbol)
return
String.Compare(
((Symbol)u).name,
((Symbol)v).name) < 0;
if (u is Product && v is Product)
{
// var a = new List<MathObject>(((Product)u).elts.Cdr());
var a = new List<MathObject>(((Product)u).elts);
a.Reverse();
// var b = new List<MathObject>(((Product)v).elts.Cdr());
var b = new List<MathObject>(((Product)v).elts);
b.Reverse();
return O3(a, b);
}
if (u is Sum && v is Sum)
{
// var a = new List<MathObject>(((Sum)u).elts.Cdr());
var a = new List<MathObject>(((Sum)u).elts);
a.Reverse();
// var b = new List<MathObject>(((Sum)v).elts.Cdr());
var b = new List<MathObject>(((Sum)v).elts);
b.Reverse();
return O3(a, b);
}
if (u is Power && v is Power)
{
var u_ = (Power)u;
var v_ = (Power)v;
return (u_.bas == v_.bas) ?
Compare(u_.exp, v_.exp) :
Compare(u_.bas, v_.bas);
}
if (u is Function && v is Function)
{
var u_ = (Function)u;
var v_ = (Function)v;
return u_.name == v_.name ?
O3(u_.args, v_.args) :
String.Compare(u_.name, v_.name) < 0;
}
if ((u is Integer || u is Fraction || u is DoubleFloat)
&&
!(v is Integer || v is Fraction || v is DoubleFloat))
return true;
if (u is Product &&
(v is Power || v is Sum || v is Function || v is Symbol))
return Compare(u, new Product(v));
if (u is Power && (v is Sum || v is Function || v is Symbol))
return Compare(u, new Power(v, new Integer(1)));
if (u is Sum && (v is Function || v is Symbol))
return Compare(u, new Sum(v));
if (u is Function && v is Symbol)
{
var u_ = (Function)u;
var v_ = (Symbol)v;
return u_.name == v_.name ?
false :
Compare(new Symbol(u_.name), v);
}
return !Compare(v, u);
}
public static MathObject Base(MathObject u)
{
if (u is Power) return ((Power)u).bas;
return u;
}
//////////////////////////////////////////////////////////////////////
// overloads for 'int'
public Point(int x_val, int y_val)
{
x = new Integer(x_val); y = new Integer(y_val);
}
public Point(int x_val, MathObject y_val)
{
x = new Integer(x_val); y = y_val;
}
public static MathObject QuadraticEquation(MathObject a, MathObject b, MathObject c, int solution=0)
{
if (a == new Integer(0) || a == new DoubleFloat(0.0))
throw new Exception("a is zero. Equation is not quadratic.");
var discriminant = b * b - 4 * a * c;
var half = new Integer(1) / 2;
if (solution == 0)
return (-b + (discriminant ^ half)) / (2 * a);
if (solution == 1)
return (-b - (discriminant ^ half)) / (2 * a);
throw new Exception();
}
//////////////////////////////////////////////////////////////////////
public static Point FromAngle(MathObject angle, MathObject mag)
{
return new Point(Trig.Cos(angle) * mag, Trig.Sin(angle) * mag);
}
public static MathObject Sq(MathObject a)
{
return (a ^ 2);
}
public static MathObject Cos(MathObject arg)
{
return new Cos(arg).Simplify();
}
public static MathObject Sqrt(MathObject a)
{
return (a ^ (new Integer(1) / 2));
}
public static MathObject MultiplyBothSidesBy(this MathObject obj, MathObject item)
{
//if (obj is Equation)
// return (obj as Equation).a * item == (obj as Equation).b * item;
if (obj is Equation)
return new Equation(
(obj as Equation).a * item,
(obj as Equation).b * item,
(obj as Equation).Operator);
if (obj is And) return (obj as And).Map(elt => elt.MultiplyBothSidesBy(item));
throw new Exception();
}
public Obj AtTime(MathObject t)
{
var dt = t - time;
return
new Obj()
{
time = t,
acceleration = acceleration,
velocity = velocity + acceleration * dt,
position = position + velocity * dt + acceleration * dt * dt / 2
};
}
public Point ProjectileInclineIntersection(MathObject theta)
{
if (theta != null &&
velocity.x != null &&
velocity.y != null &&
acceleration.y != null &&
acceleration.y != 0 &&
acceleration.y != 0.0)
{
var d =
2 * (Trig.Sin(theta) - velocity.y / velocity.x * Trig.Cos(theta))
* ((velocity.x / Trig.Cos(theta)) ^ 2)
/ acceleration.y;
return
new Point(
position.x + d * Trig.Cos(theta),
position.y + d * Trig.Sin(theta));
}
throw new Exception();
}
public Point VelocityAtTime(MathObject t)
{
var dt = t - time;
if (Misc.NotNull(velocity.x, velocity.y, acceleration.x, acceleration.y))
return velocity + acceleration * dt;
throw new Exception();
}
public static MathObject AssertEqTo(this MathObject a, MathObject b)
{
if (!(a == b)) Console.WriteLine((a == b).ToString());
return a;
}
public Point(MathObject x_val, MathObject y_val)
{
x = x_val; y = y_val;
}
public static MathObject MultiplyBothSidesBy(this MathObject obj, MathObject item)
{
if (obj is Equation)
return (obj as Equation).a * item == (obj as Equation).b * item;
throw new Exception();
}
public MathObject Substitute(MathObject a, int b)
{
return Substitute(a, new Integer(b));
}