Bernstein basis polynomials: Difference between revisions
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mono [1 1 1] --> bern [1 1 1 1]
mono [1 2 6] --> bern [1 1.6666666666667 3.3333333333333 6]
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=={{header|Java}}==
<syntaxhighlight lang="java">
import java.util.Arrays;
import java.util.List;
public final class BernsteinBasisPolynomials {
public static void main(String[] args) {
/**
* For the following polynomials, use Subprogram (1) to find coefficients in the degree-2 Bernstein basis:
*
* p(x) = 1
* q(x) = 1 + 2x + 3x²
*/
QuadraticCoefficients pMonomial2 = new QuadraticCoefficients(1.0, 0.0, 0.0);
QuadraticCoefficients qMonomial2 = new QuadraticCoefficients(1.0, 2.0, 3.0);
QuadraticCoefficients pBernstein2 = monomialToBernsteinDegree2(pMonomial2);
QuadraticCoefficients qBernstein2 = monomialToBernsteinDegree2(qMonomial2);
System.out.println("Subprogram (1) examples:");
System.out.println(" monomial " + pMonomial2 + " --> bernstein " + pBernstein2);
System.out.println(" monomial " + qMonomial2 + " --> bernstein " + qBernstein2);
/**
* Use Subprogram (2) to evaluate p(x) and q(x) at x = 0.25, 7.50. Display the results.
* Optionally also display results from evaluating in the original monomial basis.
*/
System.out.println("Subprogram (2) examples:");
for ( double x : List.of( 0.25, 7.50 ) ) {
System.out.println(" p(" + x + ") = " + evaluateBernsteinDegree2(pBernstein2, x) +
" ( mono: " + evaluateMonomialDegree2(pMonomial2, x) + " )");
}
for ( double x : List.of( 0.25, 7.50 ) ) {
System.out.println(" q(" + x + ") = " + evaluateBernsteinDegree2(qBernstein2, x) +
" ( mono: " + evaluateMonomialDegree2(qMonomial2, x) + " )");
}
/**
* For the following polynomials, use Subprogram (3) to find coefficients in the degree-3 Bernstein basis:
*
* p(x) = 1
* q(x) = 1 + 2x + 3x²
* r(x) = 1 + 2x + 3x² + 4x³
*
* Display the results.
*/
CubicCoefficients pMonomial3 = new CubicCoefficients(1.0, 0.0, 0.0, 0.0);
CubicCoefficients qMonomial3 = new CubicCoefficients(1.0, 2.0, 3.0, 0.0);
CubicCoefficients rMonomial3 = new CubicCoefficients(1.0, 2.0, 3.0, 4.0);
CubicCoefficients pBernstein3 = monomialToBernsteinDegree3(pMonomial3);
CubicCoefficients qBernstein3 = monomialToBernsteinDegree3(qMonomial3);
CubicCoefficients rBernstein3 = monomialToBernsteinDegree3(rMonomial3);
System.out.println("Subprogram (3) examples:");
System.out.println(" monomial " + pMonomial3 + " --> bernstein " + pBernstein3);
System.out.println(" monomial " + qMonomial3 + " --> bernstein " + qBernstein3);
System.out.println(" monomial " + rMonomial3 + " --> bernstein " + rBernstein3);
/**
* Use Subprogram (4) to evaluate p(x), q(x), and r(x) at x = 0.25, 7.50. Display the results.
* Optionally also display results from evaluating in the original monomial basis.
*/
System.out.println("Subprogram (4) examples:");
for ( double x : List.of( 0.25, 7.50 ) ) {
System.out.println(" p(" + x + ") = " + evaluateBernsteinDegree3(pBernstein3, x) +
" ( mono: " + evaluateMonomialDegree3(pMonomial3, x) + " )");
}
for ( double x : List.of( 0.25, 7.50 ) ) {
System.out.println(" p(" + x + ") = " + evaluateBernsteinDegree3(qBernstein3, x) +
" ( mono: " + evaluateMonomialDegree3(qMonomial3, x) + " )");
}
for ( double x : List.of( 0.25, 7.50 ) ) {
System.out.println(" p(" + x + ") = " + evaluateBernsteinDegree3(rBernstein3, x) +
" ( mono: " + evaluateMonomialDegree3(rMonomial3, x) + " )");
}
/**
* For the following polynomials, using the result of Subprogram (1) applied to the polynomial,
* use Subprogram (5) to get coefficients for the degree-3 Bernstein basis:
*
* p(x) = 1
* q(x) = 1 + 2x + 3x²
*
* Display the results.
*/
System.out.println("Subprogram (5) examples:");
CubicCoefficients pBernstein3a = bernsteinDegree2ToDegree3(pBernstein2);
CubicCoefficients qBernstein3a = bernsteinDegree2ToDegree3(qBernstein2);
System.out.println(" bernstein " + pBernstein2 + " --> bernstein " + pBernstein3a);
System.out.println(" bernstein " + qBernstein2 + " --> bernstein " + qBernstein3a);
}
private static record QuadraticCoefficients(double q0, double q1, double q2) {
@Override
public String toString() {
return Arrays.asList(q0, q1, q2).toString();
}
}
private static record CubicCoefficients(double c0, double c1, double c2, double c3) {
@Override
public String toString() {
return Arrays.asList(c0, c1, c2, c3).toString();
}
}
// Subprogram (1)
private static QuadraticCoefficients monomialToBernsteinDegree2(QuadraticCoefficients monomial) {
return new QuadraticCoefficients( monomial.q0,
monomial.q0 + ( monomial.q1 / 2.0 ),
monomial.q0 + monomial.q1 + monomial.q2 );
}
// Subprogram (2)
private static double evaluateBernsteinDegree2(QuadraticCoefficients bernstein, double t) {
// de Casteljau’s algorithm
final double s = 1 - t;
final double b01 = ( s * bernstein.q0 ) + ( t * bernstein.q1 );
final double b12 = ( s * bernstein.q1 ) + ( t * bernstein.q2 );
return ( s * b01 ) + ( t * b12 );
}
// Subprogram (3)
private static CubicCoefficients monomialToBernsteinDegree3(CubicCoefficients monomial) {
return new CubicCoefficients( monomial.c0,
monomial.c0 + ( monomial.c1 / 3.0 ),
monomial.c0 + ( 2.0 * monomial.c1 / 3.0 ) + ( monomial.c2 / 3.0 ),
monomial.c0 + monomial.c1 + monomial.c2 + monomial.c3 );
}
// Subprogram (4)
private static double evaluateBernsteinDegree3(CubicCoefficients bernstein, double t) {
// de Casteljau’s algorithm
final double s = 1 - t;
final double b01 = ( s * bernstein.c0 ) + ( t * bernstein.c1 );
final double b12 = ( s * bernstein.c1 ) + ( t * bernstein.c2 );
final double b23 = ( s * bernstein.c2 ) + ( t * bernstein.c3 );
final double b012 = ( s * b01 ) + ( t * b12 );
final double b123 = ( s * b12 ) + ( t * b23 );
return ( s * b012 ) + ( t * b123 );
}
// Subprogram (5)
private static CubicCoefficients bernsteinDegree2ToDegree3(QuadraticCoefficients bernstein) {
return new CubicCoefficients( bernstein.q0,
( bernstein.q0 / 3.0 ) + ( 2.0 * bernstein.q1 / 3.0 ),
( 2.0 * bernstein.q1 / 3.0 ) + ( bernstein.q2 / 3.0 ),
bernstein.q2 );
}
private static double evaluateMonomialDegree2(QuadraticCoefficients monomial, double t) {
// Horner’s rule
return monomial.q0 + ( t * ( monomial.q1 + ( t * monomial.q2 ) ) );
}
private static double evaluateMonomialDegree3(CubicCoefficients monomial, double t) {
// Horner’s rule
return monomial.c0 + ( t * ( monomial.c1 + ( t * ( monomial.c2 + ( t * monomial.c3 ) ) ) ) );
}
}
</syntaxhighlight>
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<pre>
Subprogram (1) examples:
monomial [1.0, 0.0, 0.0] --> bernstein [1.0, 1.0, 1.0]
monomial [1.0, 2.0, 3.0] --> bernstein [1.0, 2.0, 6.0]
Subprogram (2) examples:
p(0.25) = 1.0 ( mono: 1.0 )
p(7.5) = 1.0 ( mono: 1.0 )
q(0.25) = 1.6875 ( mono: 1.6875 )
q(7.5) = 184.75 ( mono: 184.75 )
Subprogram (3) examples:
monomial [1.0, 0.0, 0.0, 0.0] --> bernstein [1.0, 1.0, 1.0, 1.0]
monomial [1.0, 2.0, 3.0, 0.0] --> bernstein [1.0, 1.6666666666666665, 3.333333333333333, 6.0]
monomial [1.0, 2.0, 3.0, 4.0] --> bernstein [1.0, 1.6666666666666665, 3.333333333333333, 10.0]
Subprogram (4) examples:
p(0.25) = 1.0 ( mono: 1.0 )
p(7.5) = 1.0 ( mono: 1.0 )
p(0.25) = 1.6874999999999998 ( mono: 1.6875 )
p(7.5) = 184.75000000000034 ( mono: 184.75 )
p(0.25) = 1.7499999999999998 ( mono: 1.75 )
p(7.5) = 1872.25 ( mono: 1872.25 )
Subprogram (5) examples:
bernstein [1.0, 1.0, 1.0] --> bernstein [1.0, 1.0, 1.0, 1.0]
bernstein [1.0, 2.0, 6.0] --> bernstein [1.0, 1.6666666666666665, 3.333333333333333, 6.0]
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