removed the constraint on low degree polynomials
when building Chebyshev, Hermite, Laguerre or Legendre polynomials git-svn-id: https://svn.apache.org/repos/asf/commons/proper/math/trunk@760901 13f79535-47bb-0310-9956-ffa450edef68
This commit is contained in:
Luc Maisonobe
parent
3664a2cc7f
commit
f5618f7717
|
|
@ -18,7 +18,7 @@ package org.apache.commons.math.analysis.polynomials;
|
|||
|
||||
import java.util.ArrayList;
|
||||
|
||||
import org.apache.commons.math.fraction.Fraction;
|
||||
import org.apache.commons.math.fraction.BigFraction;
|
||||
|
||||
/**
|
||||
* A collection of static methods that operate on or return polynomials.
|
||||
|
|
@ -29,46 +29,46 @@ import org.apache.commons.math.fraction.Fraction;
|
|||
public class PolynomialsUtils {
|
||||
|
||||
/** Coefficients for Chebyshev polynomials. */
|
||||
private static final ArrayList<Fraction> CHEBYSHEV_COEFFICIENTS;
|
||||
private static final ArrayList<BigFraction> CHEBYSHEV_COEFFICIENTS;
|
||||
|
||||
/** Coefficients for Hermite polynomials. */
|
||||
private static final ArrayList<Fraction> HERMITE_COEFFICIENTS;
|
||||
private static final ArrayList<BigFraction> HERMITE_COEFFICIENTS;
|
||||
|
||||
/** Coefficients for Laguerre polynomials. */
|
||||
private static final ArrayList<Fraction> LAGUERRE_COEFFICIENTS;
|
||||
private static final ArrayList<BigFraction> LAGUERRE_COEFFICIENTS;
|
||||
|
||||
/** Coefficients for Legendre polynomials. */
|
||||
private static final ArrayList<Fraction> LEGENDRE_COEFFICIENTS;
|
||||
private static final ArrayList<BigFraction> LEGENDRE_COEFFICIENTS;
|
||||
|
||||
static {
|
||||
|
||||
// initialize recurrence for Chebyshev polynomials
|
||||
// T0(X) = 1, T1(X) = 0 + 1 * X
|
||||
CHEBYSHEV_COEFFICIENTS = new ArrayList<Fraction>();
|
||||
CHEBYSHEV_COEFFICIENTS.add(Fraction.ONE);
|
||||
CHEBYSHEV_COEFFICIENTS.add(Fraction.ZERO);
|
||||
CHEBYSHEV_COEFFICIENTS.add(Fraction.ONE);
|
||||
CHEBYSHEV_COEFFICIENTS = new ArrayList<BigFraction>();
|
||||
CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);
|
||||
CHEBYSHEV_COEFFICIENTS.add(BigFraction.ZERO);
|
||||
CHEBYSHEV_COEFFICIENTS.add(BigFraction.ONE);
|
||||
|
||||
// initialize recurrence for Hermite polynomials
|
||||
// H0(X) = 1, H1(X) = 0 + 2 * X
|
||||
HERMITE_COEFFICIENTS = new ArrayList<Fraction>();
|
||||
HERMITE_COEFFICIENTS.add(Fraction.ONE);
|
||||
HERMITE_COEFFICIENTS.add(Fraction.ZERO);
|
||||
HERMITE_COEFFICIENTS.add(Fraction.TWO);
|
||||
HERMITE_COEFFICIENTS = new ArrayList<BigFraction>();
|
||||
HERMITE_COEFFICIENTS.add(BigFraction.ONE);
|
||||
HERMITE_COEFFICIENTS.add(BigFraction.ZERO);
|
||||
HERMITE_COEFFICIENTS.add(BigFraction.TWO);
|
||||
|
||||
// initialize recurrence for Laguerre polynomials
|
||||
// L0(X) = 1, L1(X) = 1 - 1 * X
|
||||
LAGUERRE_COEFFICIENTS = new ArrayList<Fraction>();
|
||||
LAGUERRE_COEFFICIENTS.add(Fraction.ONE);
|
||||
LAGUERRE_COEFFICIENTS.add(Fraction.ONE);
|
||||
LAGUERRE_COEFFICIENTS.add(Fraction.MINUS_ONE);
|
||||
LAGUERRE_COEFFICIENTS = new ArrayList<BigFraction>();
|
||||
LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);
|
||||
LAGUERRE_COEFFICIENTS.add(BigFraction.ONE);
|
||||
LAGUERRE_COEFFICIENTS.add(BigFraction.MINUS_ONE);
|
||||
|
||||
// initialize recurrence for Legendre polynomials
|
||||
// P0(X) = 1, P1(X) = 0 + 1 * X
|
||||
LEGENDRE_COEFFICIENTS = new ArrayList<Fraction>();
|
||||
LEGENDRE_COEFFICIENTS.add(Fraction.ONE);
|
||||
LEGENDRE_COEFFICIENTS.add(Fraction.ZERO);
|
||||
LEGENDRE_COEFFICIENTS.add(Fraction.ONE);
|
||||
LEGENDRE_COEFFICIENTS = new ArrayList<BigFraction>();
|
||||
LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);
|
||||
LEGENDRE_COEFFICIENTS.add(BigFraction.ZERO);
|
||||
LEGENDRE_COEFFICIENTS.add(BigFraction.ONE);
|
||||
|
||||
}
|
||||
|
||||
|
|
@ -94,9 +94,9 @@ public class PolynomialsUtils {
|
|||
public static PolynomialFunction createChebyshevPolynomial(final int degree) {
|
||||
return buildPolynomial(degree, CHEBYSHEV_COEFFICIENTS,
|
||||
new RecurrenceCoefficientsGenerator() {
|
||||
private final Fraction[] coeffs = { Fraction.ZERO, Fraction.TWO, Fraction.ONE};
|
||||
private final BigFraction[] coeffs = { BigFraction.ZERO, BigFraction.TWO, BigFraction.ONE };
|
||||
/** {@inheritDoc} */
|
||||
public Fraction[] generate(int k) {
|
||||
public BigFraction[] generate(int k) {
|
||||
return coeffs;
|
||||
}
|
||||
});
|
||||
|
|
@ -120,11 +120,11 @@ public class PolynomialsUtils {
|
|||
return buildPolynomial(degree, HERMITE_COEFFICIENTS,
|
||||
new RecurrenceCoefficientsGenerator() {
|
||||
/** {@inheritDoc} */
|
||||
public Fraction[] generate(int k) {
|
||||
return new Fraction[] {
|
||||
Fraction.ZERO,
|
||||
Fraction.TWO,
|
||||
new Fraction(2 * k, 1)};
|
||||
public BigFraction[] generate(int k) {
|
||||
return new BigFraction[] {
|
||||
BigFraction.ZERO,
|
||||
BigFraction.TWO,
|
||||
new BigFraction(2 * k)};
|
||||
}
|
||||
});
|
||||
}
|
||||
|
|
@ -146,12 +146,12 @@ public class PolynomialsUtils {
|
|||
return buildPolynomial(degree, LAGUERRE_COEFFICIENTS,
|
||||
new RecurrenceCoefficientsGenerator() {
|
||||
/** {@inheritDoc} */
|
||||
public Fraction[] generate(int k) {
|
||||
public BigFraction[] generate(int k) {
|
||||
final int kP1 = k + 1;
|
||||
return new Fraction[] {
|
||||
new Fraction(2 * k + 1, kP1),
|
||||
new Fraction(-1, kP1),
|
||||
new Fraction(k, kP1)};
|
||||
return new BigFraction[] {
|
||||
new BigFraction(2 * k + 1, kP1),
|
||||
new BigFraction(-1, kP1),
|
||||
new BigFraction(k, kP1)};
|
||||
}
|
||||
});
|
||||
}
|
||||
|
|
@ -173,12 +173,12 @@ public class PolynomialsUtils {
|
|||
return buildPolynomial(degree, LEGENDRE_COEFFICIENTS,
|
||||
new RecurrenceCoefficientsGenerator() {
|
||||
/** {@inheritDoc} */
|
||||
public Fraction[] generate(int k) {
|
||||
public BigFraction[] generate(int k) {
|
||||
final int kP1 = k + 1;
|
||||
return new Fraction[] {
|
||||
Fraction.ZERO,
|
||||
new Fraction(k + kP1, kP1),
|
||||
new Fraction(k, kP1)};
|
||||
return new BigFraction[] {
|
||||
BigFraction.ZERO,
|
||||
new BigFraction(k + kP1, kP1),
|
||||
new BigFraction(k, kP1)};
|
||||
}
|
||||
});
|
||||
}
|
||||
|
|
@ -190,7 +190,7 @@ public class PolynomialsUtils {
|
|||
* @return coefficients array
|
||||
*/
|
||||
private static PolynomialFunction buildPolynomial(final int degree,
|
||||
final ArrayList<Fraction> coefficients,
|
||||
final ArrayList<BigFraction> coefficients,
|
||||
final RecurrenceCoefficientsGenerator generator) {
|
||||
|
||||
final int maxDegree = (int) Math.floor(Math.sqrt(2 * coefficients.size())) - 1;
|
||||
|
|
@ -228,7 +228,7 @@ public class PolynomialsUtils {
|
|||
*/
|
||||
private static void computeUpToDegree(final int degree, final int maxDegree,
|
||||
final RecurrenceCoefficientsGenerator generator,
|
||||
final ArrayList<Fraction> coefficients) {
|
||||
final ArrayList<BigFraction> coefficients) {
|
||||
|
||||
int startK = (maxDegree - 1) * maxDegree / 2;
|
||||
for (int k = maxDegree; k < degree; ++k) {
|
||||
|
|
@ -238,24 +238,24 @@ public class PolynomialsUtils {
|
|||
startK += k;
|
||||
|
||||
// Pk+1(X) = (a[0] + a[1] X) Pk(X) - a[2] Pk-1(X)
|
||||
Fraction[] ai = generator.generate(k);
|
||||
BigFraction[] ai = generator.generate(k);
|
||||
|
||||
Fraction ck = coefficients.get(startK);
|
||||
Fraction ckm1 = coefficients.get(startKm1);
|
||||
BigFraction ck = coefficients.get(startK);
|
||||
BigFraction ckm1 = coefficients.get(startKm1);
|
||||
|
||||
// degree 0 coefficient
|
||||
coefficients.add(ck.multiply(ai[0]).subtract(ckm1.multiply(ai[2])));
|
||||
|
||||
// degree 1 to degree k-1 coefficients
|
||||
for (int i = 1; i < k; ++i) {
|
||||
final Fraction ckPrev = ck;
|
||||
final BigFraction ckPrev = ck;
|
||||
ck = coefficients.get(startK + i);
|
||||
ckm1 = coefficients.get(startKm1 + i);
|
||||
coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])).subtract(ckm1.multiply(ai[2])));
|
||||
}
|
||||
|
||||
// degree k coefficient
|
||||
final Fraction ckPrev = ck;
|
||||
final BigFraction ckPrev = ck;
|
||||
ck = coefficients.get(startK + k);
|
||||
coefficients.add(ck.multiply(ai[0]).add(ckPrev.multiply(ai[1])));
|
||||
|
||||
|
|
@ -274,7 +274,7 @@ public class PolynomialsUtils {
|
|||
* @return an array of three coefficients such that
|
||||
* P<sub>k+1</sub>(X) = (a[0] + a[1] X) P<sub>k</sub>(X) - a[2] P<sub>k-1</sub>(X)
|
||||
*/
|
||||
Fraction[] generate(int k);
|
||||
BigFraction[] generate(int k);
|
||||
}
|
||||
|
||||
}
|
||||
|
|
|
|||
|
|
@ -324,9 +324,8 @@ System.out println("f(" + interpolationX + ") = " + interpolatedY);</source>
|
|||
one, using traditional coefficients arrays. The <a
|
||||
href="../apidocs/org/apache/commons/math/analysis/polynomials/PolynomialsUtils.html">
|
||||
org.apache.commons.math.analysis.polynomials.PolynomialsUtils</a> utility class provides static
|
||||
factory methods to build Chebyshev, Hermite, Lagrange and Legendre polynomials. Beware that due
|
||||
to overflows in the coefficients computations, these factory methods can only build low degrees
|
||||
polynomials yet.
|
||||
factory methods to build Chebyshev, Hermite, Lagrange and Legendre polynomials. Coefficients
|
||||
are computed using exact fractions so these factory methods can build polynomials up to any degree.
|
||||
</p>
|
||||
</subsection>
|
||||
</section>
|
||||
|
|
|
|||
|
|
@ -177,34 +177,25 @@ public class PolynomialsUtilsTest extends TestCase {
|
|||
}
|
||||
|
||||
public void testHighDegreeLegendre() {
|
||||
try {
|
||||
PolynomialsUtils.createLegendrePolynomial(40);
|
||||
fail("an exception should have been thrown");
|
||||
} catch (ArithmeticException ae) {
|
||||
// expected
|
||||
PolynomialsUtils.createLegendrePolynomial(40);
|
||||
double[] l40 = PolynomialsUtils.createLegendrePolynomial(40).getCoefficients();
|
||||
double denominator = 274877906944.0;
|
||||
double[] numerators = new double[] {
|
||||
+34461632205.0, -28258538408100.0, +3847870979902950.0, -207785032914759300.0,
|
||||
+5929294332103310025.0, -103301483474866556880.0, +1197358103913226000200.0, -9763073770369381232400.0,
|
||||
+58171647881784229843050.0, -260061484647976556945400.0, +888315281771246239250340.0, -2345767627188139419665400.0,
|
||||
+4819022625419112503443050.0, -7710436200670580005508880.0, +9566652323054238154983240.0, -9104813935044723209570256.0,
|
||||
+6516550296251767619752905.0, -3391858621221953912598660.0, +1211378079007840683070950.0, -265365894974690562152100.0,
|
||||
+26876802183334044115405.0
|
||||
};
|
||||
for (int i = 0; i < l40.length; ++i) {
|
||||
if (i % 2 == 0) {
|
||||
double ci = numerators[i / 2] / denominator;
|
||||
assertEquals(ci, l40[i], ci * 1.0e-15);
|
||||
} else {
|
||||
assertEquals(0.0, l40[i], 0.0);
|
||||
}
|
||||
}
|
||||
// checkPolynomial(PolynomialsUtils.createLegendrePolynomial(40), 274877906944l,
|
||||
// "34461632205.0"
|
||||
// + " - 28258538408100.0 x^2"
|
||||
// + " + 3847870979902950.0 x^4"
|
||||
// + " - 207785032914759300.0 x^6"
|
||||
// + " + 5929294332103310025.0 x^8"
|
||||
// + " - 103301483474866556880.0 x^10"
|
||||
// + " + 1197358103913226000200.0 x^12"
|
||||
// + " - 9763073770369381232400.0 x^14"
|
||||
// + " + 58171647881784229843050.0 x^16"
|
||||
// + " - 260061484647976556945400.0 x^18"
|
||||
// + " + 888315281771246239250340.0 x^20"
|
||||
// + " - 2345767627188139419665400.0 x^22"
|
||||
// + " + 4819022625419112503443050.0 x^24"
|
||||
// + " - 7710436200670580005508880.0 x^26"
|
||||
// + " + 9566652323054238154983240.0 x^28"
|
||||
// + " - 9104813935044723209570256.0 x^30"
|
||||
// + " + 6516550296251767619752905.0 x^32"
|
||||
// + " - 3391858621221953912598660.0 x^34"
|
||||
// + " + 1211378079007840683070950.0 x^36"
|
||||
// + " - 265365894974690562152100.0 x^38"
|
||||
// + " + 26876802183334044115405.0 x^40");
|
||||
}
|
||||
|
||||
private void checkPolynomial(PolynomialFunction p, long denominator, String reference) {
|
||||
|
|
|
|||
Loading…
Reference in New Issue