/// <summary>
/// Subtracts one complex polynomial from another.
/// </summary>
/// <param name="poly1">The first complex polynomial.</param>
/// <param name="poly2">The second complex polynomial.</param>
/// <returns>The result of subtracting poly2 from poly1.</returns>
public static CPolynomial Subtract(CPolynomial poly1, CPolynomial poly2)
{
int len1 = poly1.Degree + 1;
int len2 = poly2.Degree + 1;
int maxLen = Math.Max(len1, len2);
int minLen = Math.Min(len1, len2);
CPolynomial result = new CPolynomial(maxLen);
for (int i = 0; i < minLen; i++)
{
result[i] = poly1[i] - poly2[i];
}
if (len1 == maxLen)
{
for (int i = minLen; i < maxLen; i++)
{
result[i] = poly1[i];
}
}
else
{
for (int i = minLen; i < maxLen; i++)
{
result[i] = -poly2[i];
}
}
return(result);
}
void IXmlSerializable.ReadXml(XmlReader reader)
{
string value = reader.ReadInnerXml();
CPolynomial poly = Parse(value, CultureInfo.InvariantCulture);
_coeffs = poly._coeffs;
}
/// <summary>
/// Divides two complex polynomials.
/// </summary>
/// <param name="poly1">The first complex polynomial.</param>
/// <param name="poly2">The second complex polynomial.</param>
/// <param name="remainder">Remainder of division of the first polynomial in the second.</param>
/// <returns>Quotient of division of poly1 on poly2.</returns>
/// <exception cref="System.DivideByZeroException">The polynomial poly2 is zero.</exception>
public static CPolynomial Divide(CPolynomial poly1, CPolynomial poly2, out CPolynomial remainder)
{
int len1 = poly1.Degree + 1;
int len2 = poly2.Degree + 1;
CPolynomial q = new CPolynomial(len1 - len2 + 1);
CPolynomial r = Clean(poly1);
Complex a = poly2[len2 - 1];
for (int i = q.Length - 1; i >= 0; i--)
{
q[i] = r[r.Length - 1] / a;
CPolynomial rt = new CPolynomial(r.Length - 1);
for (int j = rt.Length - 1, k = len2 - 2; j > rt.Length - len2; j--, k--)
{
rt[j] = r[j] - q[i] * poly2[k];
}
for (int j = rt.Length - len2; j >= 0; j--)
{
rt[j] = r[j];
}
r = rt;
}
remainder = r;
return(q);
}
/// <summary>
/// Returns the remainder of dividing the first polynomial in the second.
/// </summary>
/// <param name="poly1">The first complex polynomial.</param>
/// <param name="poly2">The second complex polynomial.</param>
/// <returns>The remainder of the division poly1 to poly2.</returns>
public static CPolynomial Modulus(CPolynomial poly1, CPolynomial poly2)
{
CPolynomial remainder;
Divide(poly1, poly2, out remainder);
return(remainder);
}
/// <summary>
/// Divides a complex polynomial by a scalar.
/// </summary>
/// <param name="poly">A complex polynomial.</param>
/// <param name="number">A complex number.</param>
/// <returns>The result of dividing poly by number.</returns>
/// <exception cref="System.DivideByZeroException">The number is zero.</exception>
public static CPolynomial Divide(CPolynomial poly, Complex number)
{
CPolynomial result = new CPolynomial(poly.Degree + 1);
for (int i = 0; i < result.Length; i++)
{
result[i] = poly[i] / number;
}
return(result);
}
/// <summary>
/// Returns the complex polynomial without leading zeros.
/// </summary>
/// <param name="poly">A complex polynomial.</param>
/// <returns>The poly without leading zeros.</returns>
public static CPolynomial Clean(CPolynomial poly)
{
CPolynomial result = new CPolynomial(poly.Degree + 1);
for (int i = 0; i < result.Length; i++)
{
result[i] = poly[i];
}
return(result);
}
/// <summary>
/// Returns the normalized complex polynomial.
/// </summary>
/// <returns>The normalized complex polynomial.</returns>
public CPolynomial Normalize()
{
CPolynomial result = new CPolynomial(Degree + 1);
Complex divisor = _coeffs[result.Length - 1];
for (int i = 0; i < result.Length; i++)
{
result[i] = _coeffs[i] / divisor;
}
return(result);
}
/// <summary>
/// Returns the antiderivative of the complex polynomial.
/// </summary>
/// <returns>The antiderivative of this instance.</returns>
public CPolynomial Antiderivative()
{
int len = Degree + 1;
CPolynomial result = new CPolynomial(len + 1);
for (int i = 1; i < result.Length; i++)
{
result[i] = _coeffs[i - 1] / i;
}
return(result);
}
/// <summary>
/// Divides a complex polynomial by a binomial of the form (x - c).
/// </summary>
/// <param name="poly">A complex polynomial.</param>
/// <param name="c">A binomial coefficient in the constant term.</param>
/// <param name="remainder">Remainder of division of the polynomial in the binomial.</param>
/// <returns>Quotient of division of poly on the binomial (x - c).</returns>
public static CPolynomial DivBinom(CPolynomial poly, Complex c, out Complex remainder)
{
int len = poly.Degree + 1;
CPolynomial result = new CPolynomial(len - 1);
remainder = poly[len - 1];
for (int i = result.Length - 1; i >= 0; i--)
{
result[i] = remainder;
remainder = remainder * c + poly[i];
}
return(result);
}
/// <summary>
/// Returns the second derivative of the complex polynomial.
/// </summary>
/// <returns>The second derivative of this instance.</returns>
public CPolynomial SecondDerivative()
{
int len = Degree + 1;
if (len <= 2)
{
return(new CPolynomial(1));
}
CPolynomial result = new CPolynomial(len - 2);
for (int i = 2; i < len; i++)
{
result[i - 2] = this[i] * i * (i - 1);
}
return(result);
}
/// <summary>
/// Returns the first derivative of the complex polynomial.
/// </summary>
/// <returns>The first derivative of this instance.</returns>
public CPolynomial FirstDerivative()
{
int len = Degree + 1;
if (len <= 1)
{
return(new CPolynomial(1));
}
CPolynomial result = new CPolynomial(len - 1);
for (int i = 1; i < len; i++)
{
result[i - 1] = this[i] * i;
}
return(result);
}
/// <summary>
/// Returns a complex polynomial that has the specified roots.
/// </summary>
/// <param name="roots">An array of complex roots.</param>
/// <returns>The complex polynomial with the specified roots.</returns>
public static CPolynomial FromRoots(IList <Complex> roots)
{
int n = roots.Count;
CPolynomial result = new CPolynomial(n + 1);
result[n] = 1.0;
for (int i = 0; i < n; i++)
{
for (int j = n - i - 1; j < n; j++)
{
result[j] = result[j] - roots[i] * result[j + 1];
}
}
return(result);
}
/// <summary>
/// Multiplies two complex polynomials.
/// </summary>
/// <param name="poly1">The first complex polynomial.</param>
/// <param name="poly2">The second complex polynomial.</param>
/// <returns>The product of poly1 and poly2.</returns>
public static CPolynomial Multiply(CPolynomial poly1, CPolynomial poly2)
{
int len1 = poly1.Degree + 1;
int len2 = poly2.Degree + 1;
CPolynomial result = new CPolynomial(len1 + len2 - 1);
for (int i = 0; i < len1; i++)
{
for (int j = 0; j < len2; j++)
{
result[i + j] += poly1[i] * poly2[j];
}
}
return(result);
}
/// <summary>
/// Returns the value indicating whether two instances of complex polynomial are equal.
/// </summary>
/// <param name="poly1">The first complex polynomial to compare.</param>
/// <param name="poly2">The second complex polynomial to compare.</param>
/// <returns>True if the poly1 and poly2 parameters have the same value; otherwise, false.</returns>
public static bool Equals(CPolynomial poly1, CPolynomial poly2)
{
if (poly1.Degree != poly2.Degree)
{
return(false);
}
int len = poly1.Degree + 1;
for (int i = 0; i < len; i++)
{
if (poly1[i] != poly2[i])
{
return(false);
}
}
return(true);
}
/// <summary>
/// Returns the interpolating polynomial through a set of nodes using the Lagrange method.
/// </summary>
/// <param name="xValues">
/// A real array containing the x-coordinates of interpolation nodes.
/// The elements of this array must be distinct.
/// </param>
/// <param name="yValues">
/// A real array containing the y-coordinates of interpolation nodes.
/// </param>
/// <returns>The interpolating polynomial for the specified nodes.</returns>
/// <exception cref="System.ArgumentException">The arrays xValues and yValues have different lengths.</exception>
public static CPolynomial InterpolatingPolynomial(IList <double> xValues, IList <double> yValues)
{
if (xValues.Count != yValues.Count)
{
throw new ArgumentException("The arrays of abscissas and ordinates have different lengths.");
}
int n = xValues.Count;
CPolynomial L = new CPolynomial(n);
for (int i = 0; i < n; i++)
{
CPolynomial l = new CPolynomial(n);
l[n - 1] = 1;
Complex denom = 1;
for (int j = 0; j < n; j++)
{
if (j != i)
{
int k = (j > i) ? n - j - 1 : n - j - 2;
for (; k < n - 1; k++)
{
l[k] = l[k] - xValues[j] * l[k + 1];
}
denom *= (xValues[i] - xValues[j]);
}
}
Complex factor = yValues[i] / denom;
for (int j = 0; j < n; j++)
{
L[j] += factor * l[j];
}
}
return(L);
}
/// <summary>
/// Returns the n-th derivative of the complex polynomial.
/// </summary>
/// <param name="order">An order of derivative.</param>
/// <returns>The n-th derivative of this instance.</returns>
public CPolynomial NthDerivative(int order)
{
int len = Degree + 1;
if (len <= order)
{
return(new CPolynomial(1));
}
CPolynomial result = new CPolynomial(len - order);
for (int i = order; i < len; i++)
{
int exp = 1, k = i;
for (int j = 0; j < order; j++)
{
exp *= k--;
}
result[i - order] = this[i] * exp;
}
return(result);
}
/// <summary>
/// Initializes a complex polynomial from another instance of a complex polynomial.
/// </summary>
/// <param name="poly">A complex polynomial.</param>
public CPolynomial(CPolynomial poly)
{
_coeffs = (Complex[])poly._coeffs.Clone();
}
/// <summary>
/// Converts string representation of a polynomial
/// in a specified culture-specific format to its complex polynomial equivalent.
/// </summary>
/// <param name="s">A string containing a complex polynomial to convert.</param>
/// <param name="provider">An System.IFormatProvider that supplies culture-specific formatting information about s.</param>
/// <returns>A complex polynomial equivalent to the value specified in s.</returns>
/// <exception cref="System.ArgumentNullException">The string s is null.</exception>
/// <exception cref="System.ArgumentException">The string s is empty string.</exception>
/// <exception cref="System.FormatException">The string s is not a polynomial in a valid format.</exception>
public static CPolynomial Parse(string s, IFormatProvider provider)
{
if (s == null)
{
throw new ArgumentNullException(Properties.Resources.EXC_STRING_NOT_NULL);
}
string str = s.Trim();
if (String.IsNullOrEmpty(str))
{
throw new ArgumentException(Properties.Resources.EXC_STRING_NOT_EMPTY);
}
string varName = String.Empty;
Match matchAll = _polyRegex.Match(str);
if (!matchAll.Success)
{
throw new FormatException(Properties.Resources.EXC_POLY_INCCORECT_FORMAT);
}
Dictionary <int, Complex> p = new Dictionary <int, Complex>();
int maxExp = 0;
foreach (Capture capture in matchAll.Groups["term"].Captures)
{
Match match = _polyPartRegex.Match(capture.Value.Trim());
if (!String.IsNullOrEmpty(match.Value))
{
string sign_s = match.Groups["sign"].Value;
string coef_s = match.Groups["num"].Value;
string var_s = match.Groups["var"].Value;
string exp_s = match.Groups["exp"].Value;
if (!String.IsNullOrEmpty(var_s))
{
if (String.IsNullOrEmpty(varName))
{
varName = var_s;
}
if (varName != var_s)
{
throw new FormatException(Properties.Resources.EXC_POLY_MISMATCH_VAR_NAMES);
}
}
int sign = int.Parse(sign_s + 1, provider);
Complex coef = !String.IsNullOrEmpty(coef_s) ? sign * Complex.Parse(coef_s, provider) : sign;
int exp = !String.IsNullOrEmpty(exp_s) ? int.Parse(exp_s, provider) : 0;
if (!String.IsNullOrEmpty(var_s) && String.IsNullOrEmpty(exp_s))
{
exp = 1;
}
if (maxExp < exp)
{
maxExp = exp;
}
if (p.ContainsKey(exp))
{
p[exp] += coef;
}
else
{
p.Add(exp, coef);
}
}
}
// Need to find the maximum degree and put here
CPolynomial poly = new CPolynomial(maxExp + 1);
foreach (KeyValuePair <int, Complex> pair in p.OrderBy(x => x.Key))
{
poly[pair.Key] = pair.Value;
}
return(poly);
}
/// <summary>
/// Returns a vector of approximate values of roots of a complex polynomial by Laguerre's method.
/// </summary>
/// <param name="poly">A complex polynomial.</param>
/// <returns>The approximate values of roots of poly.</returns>
/// <exception cref="System.ArgumentException">
/// Number of elements in poly is less than 2 or more than 99.
/// </exception>
public static Complex[] LaguerreRoots(CPolynomial poly)
{
const int maxIters = 100;
const int maxAttempts = 10;
// Remove zero elements standing at the end.
int lidx = 0;
int ridx = poly.Length - 1;
while (ridx >= 0 && poly[ridx] == Complex.Zero)
{
ridx--;
}
while (lidx < poly.Length && poly[lidx] == Complex.Zero)
{
lidx++;
}
int length = ridx + 1;
if (length < 2 || length > 99)
{
throw new ArgumentException("Number of coefficients must be between 1 and 100.");
}
int rootsCount = length - 1;
Complex[] roots = new Complex[rootsCount];
CPolynomial div = new CPolynomial(ridx - lidx + 1);
for (int i = lidx; i <= ridx; i++)
{
div[i - lidx] = poly[i];
}
Random rand = new Random(0);
// Main loop
for (int i = 0; i < ridx - lidx; i++)
{
Complex a = rand.Next(-1000, 1000);
if (a == 0)
{
a = 1;
}
Complex a_old = a + 1;
int currIter = 0;
int currAttempt = 0;
// Degree of the div polynomial
int n = div.Length - 1;
while (!NumericUtil.FuzzyEquals(a, a_old, Machine.Epsilon))
{
if (currIter > maxIters)
{
a = rand.Next(-1000, 1000);
if (a == 0)
{
a = 1;
}
a_old = a + 1;
currIter = 0;
currAttempt++;
if (currAttempt > maxAttempts)
{
throw new NotConvergenceException();
}
continue;
}
currIter++;
Complex p = div.Evaluate(a);
Complex dp = div.FirstDerivative(a);
Complex d2p = div.SecondDerivative(a);
if (Complex.Abs(p) <= Machine.Epsilon)
{
break;
}
Complex G = dp / p;
Complex H = G * G - d2p / p;
Complex D = Complex.Sqrt((n - 1) * (n * H - G * G));
Complex denom = Complex.Abs(G + D) >= Complex.Abs(G - D) ? G + D : G - D;
if (denom == Complex.Zero)
{
denom = Machine.Epsilon;
}
a_old = a;
a = a - n / denom;
}
roots[i] = a;
Complex rem;
div = DivBinom(div, a, out rem);
}
Array.Sort <Complex>(roots, new ComplexComparer());
return(roots);
}
/// <summary>
/// Returns a value indicating whether this instance is equal to a specified
/// complex polynomial.
/// </summary>
/// <param name="obj">A CPolynomial object to compare to this instance.</param>
/// <returns>True if obj is equal to this instance; otherwise, false.</returns>
public bool Equals(CPolynomial obj)
{
return(Equals(this, obj));
}
/// <summary>
/// Divides two complex polynomials.
/// </summary>
/// <param name="poly1">The first complex polynomial.</param>
/// <param name="poly2">The second complex polynomial.</param>
/// <returns>Quotient of division of poly1 on poly2.</returns>
/// <exception cref="System.DivideByZeroException">The polynomial poly2 is zero.</exception>
public static CPolynomial Divide(CPolynomial poly1, CPolynomial poly2)
{
CPolynomial remainder;
return(Divide(poly1, poly2, out remainder));
}