Bernstein basis polynomials: Difference between revisions

=={{header|julia}}==

{{trans|Python}}

<syntaxhighlight lang="julia">""" rosettacode.org/wiki/Bernstein_basis_polynomials """

""" Subprogram (1). """

monomial_to_bernstein_degree2(mc) = ((a0, a1, a2) = mc; (a0, a0 + ((1/2) * a1), a0 + a1 + a2))

""" Subprogram (2). """

function evaluate_bernstein_degree2(bernstein_coefficients, t)

(b0, b1, b2) = bernstein_coefficients

# de Casteljau’s algorithm.

s = 1 - t

b01 = (s * b0) + (t * b1)

b12 = (s * b1) + (t * b2)

b012 = (s * b01) + (t * b12)

return b012

end

""" Subprogram (3). """

function monomial_to_bernstein_degree3(monomial_coefficients)

(a0, a1, a2, a3) = monomial_coefficients

return (a0, a0 + ((1/3) * a1),

a0 + ((2/3) * a1) + ((1/3) * a2),

a0 + a1 + a2 + a3)

end

""" Subprogram (4). """

function evaluate_bernstein_degree3(bernstein_coefficients, t)

(b0, b1, b2, b3) = bernstein_coefficients

# de Casteljau’s algorithm.

s = 1 - t

b01 = (s * b0) + (t * b1)

b12 = (s * b1) + (t * b2)

b23 = (s * b2) + (t * b3)

b012 = (s * b01) + (t * b12)

b123 = (s * b12) + (t * b23)

b0123 = (s * b012) + (t * b123)

return b0123

end

""" Subprogram (5). """

function bernstein_degree2_to_degree3(bernstein_coefficients)

(b0, b1, b2) = bernstein_coefficients

return (b0, ((1/3) * b0) + ((2/3) * b1),

((2/3) * b1) + ((1/3) * b2), b2)

end

evaluate_monomial_degree2(mc, t) = ((a0, a1, a2) = mc; a0 + (t * (a1 + (t * a2))))

evaluate_monomial_degree3(mc, t) = ((a0, a1, a2, a3) = mc; a0 + (t * (a1 + (t * (a2 + (t * a3))))))

#

# For the following polynomials, use Subprogram (1) to find

# coefficients in the degree-2 Bernstein basis:

#

# p(x) = 1

# q(x) = 1 + 2x + 3x²

#

# Display the results.

#

pmono2 = (1, 0, 0)

qmono2 = (1, 2, 3)

pbern2 = monomial_to_bernstein_degree2(pmono2)

qbern2 = monomial_to_bernstein_degree2(qmono2)

println("Subprogram (1) examples:")

println(" mono ", pmono2, " --> bern ", pbern2)

println(" mono ", qmono2, " --> bern ", qbern2)

#

# 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.

println("Subprogram (2) examples:")

for x in (0.25, 7.50)

println(" p(", x, ") = ", evaluate_bernstein_degree2(pbern2, x),

" ( mono: ", evaluate_monomial_degree2(pmono2, x), ")")

end

for x in (0.25, 7.50)

println(" q(", x, ") = ", evaluate_bernstein_degree2(qbern2, x),

" ( mono: ", evaluate_monomial_degree2(qmono2, x), ")")

end

#

# 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.

#

pmono3 = (1, 0, 0, 0)

qmono3 = (1, 2, 3, 0)

rmono3 = (1, 2, 3, 4)

pbern3 = monomial_to_bernstein_degree3(pmono3)

qbern3 = monomial_to_bernstein_degree3(qmono3)

rbern3 = monomial_to_bernstein_degree3(rmono3)

println("Subprogram (3) examples:")

println(" mono ", pmono3, " --> bern ", pbern3)

println(" mono ", qmono3, " --> bern ", qbern3)

println(" mono ", rmono3, " --> bern ", rbern3)

#

# 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.

println("Subprogram (4) examples:")

for x in (0.25, 7.50)

println(" p(", x, ") = " , evaluate_bernstein_degree3(pbern3, x),

" ( mono: " , evaluate_monomial_degree3(pmono3, x), ")")

end

for x in (0.25, 7.50)

println(" q(", x, ") = ", evaluate_bernstein_degree3(qbern3, x),

" ( mono: ", evaluate_monomial_degree3(qmono3, x), ")")

end

for x in (0.25, 7.50)

println(" r(", x, ") = ", evaluate_bernstein_degree3(rbern3, x),

" ( mono: ", evaluate_monomial_degree3(rmono3, x), ")")

end

#

# 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.

#

println("Subprogram (5) examples:")

pbern3a = bernstein_degree2_to_degree3(pbern2)

qbern3a = bernstein_degree2_to_degree3(qbern2)

println(" bern ", pbern2, " --> bern ", pbern3a)

println(" bern ", qbern2, " --> bern ", qbern3a)

</syntaxhighlight>{{out}} Same as Python example.