Bernstein basis polynomials: Difference between revisions

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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">monomial_to_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">m_to_bern2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">a01</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">2</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000000;">a02</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a2</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a01</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a02</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">evaluate_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">m_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000000;">a01</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000080;font-style:italic;">-- de Casteljau’s algorithm.</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">a02</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">b01</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b0</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">a03</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a3</span>
<span style="color: #000000;">b12</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">),</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a01</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a02</span><span style="color: #0000FF;">,</span><span style="color: #000000;">a03</span><span style="color: #0000FF;">}</span>
⚫
<span style="color: #000000;">b012</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b01</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b12</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">b012</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">monomial_to_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">bern2_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b01</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b0</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b12</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span>
<span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b01</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b12</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">evaluate_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">eval_bern2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #000080;font-style:italic;">// using de Casteljau’s algorithm</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000080;font-style:italic;">// de Casteljau’s algorithm.</span>
<span style="color: #004080;">atom</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> <span style="color: #0000FF;">-</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000080;font-style:italic;">// first mid-points</span>
⚫
<span style="color: #000000;">b01</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b0</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b12</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b01</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span>
<span style="color: #000000;">b23</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b3</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b12</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b2</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000080;font-style:italic;">// second mid-point is on the curve</span>
⚫
<span style="color: #000000;">b012</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b01</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b12</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b123</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b12</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b23</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">b012</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b01</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b12</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">b012</span>
⚫
<span style="color: #000000;">b0123</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">s</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b012</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b123</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">b0123</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">bernstein_degree2_to_degree3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">eval_bern3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #000080;font-style:italic;">// using de Casteljau’s algorithm</span>
⚫
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span>
<span style="color: #008080;">return</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b0</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">),</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">b0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">b3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_coefficients</span><span style="color: #0000FF;">,</span>
<span style="color: #0000FF;">((</span><span style="color: #000000;">2</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b1</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">((</span><span style="color: #000000;">1</span><span style="color: #0000FF;">/</span><span style="color: #000000;">3</span><span style="color: #0000FF;">)</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">),</span> <span style="color: #000000;">b2</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span>
<span style="color: #000080;font-style:italic;">// first mid-points</span>
<span style="color: #000000;">b01</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b1</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000000;">b12</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b2</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000000;">b23</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b3</span><span style="color: #0000FF;">,</span>
<span style="color: #000080;font-style:italic;">// second mid-points</span>
⚫
<span style="color: #000000;">b012</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b01</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b12</span><span style="color: #0000FF;">,</span>
⚫
<span style="color: #000000;">b123</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b12</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b23</span><span style="color: #0000FF;">,</span>
<span style="color: #000080;font-style:italic;">// third mid-point is on the curve</span>
⚫
<span style="color: #000000;">b0123</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b012</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">b123</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">b0123</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">evaluate_monomial_degree2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">eval_m2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a2</span><span style="color: #0000FF;">)</span> <span style="color: #000080;font-style:italic;">-- Horner’s rule</span>
⚫
<span style="color: #000080;font-style:italic;">/* Horner’s rule. */</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">)))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">evaluate_monomial_degree3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">eval_m3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">monomial_coefficients</span><span style="color: #0000FF;">,</span> <span style="color: #004080;">atom</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
<span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_coefficients</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*(</span><span style="color: #000000;">a2</span> <span style="color: #0000FF;">+</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">*</span><span style="color: #000000;">a3</span><span style="color: #0000FF;">))</span> <span style="color: #000080;font-style:italic;">-- Horner’s rule</span>
⚫
<span style="color: #000080;font-style:italic;">--/* Horner’s rule. --*/</span>
⚫
<span style="color: #008080;">return</span> <span style="color: #000000;">a0</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">a1</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">a2</span> <span style="color: #0000FF;">+</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">t</span> <span style="color: #0000FF;">*</span> <span style="color: #000000;">a3</span><span style="color: #0000FF;">)))))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">pmono2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- p(x) = 1</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">pm2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- p(x) = 1</span>
<span style="color: #000000;">qmono2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- q(x) = 1 + 2x + 3x²</span>
<span style="color: #000000;">pm3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- (ditto in degree 3)</span>
<span style="color: #000000;">pbern2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_to_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pmono2</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">qm2</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- q(x) = 1 + 2x + 3x²</span>
<span style="color: #000000;">qbern2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_to_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qmono2</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">qm3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- (ditto in degree 3)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Subprogram (1) examples:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #000000;">rm3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- r(x) = 1 + 2x + 3x² + 4x³</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" mono %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pmono2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pbern2</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">pb2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_to_bern2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pm2</span><span style="color: #0000FF;">),</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" mono %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qmono2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qbern2</span><span style="color: #0000FF;">})</span>
<span style="color: #000000;">pb3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pm3</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">qb2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_to_bern2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qm2</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">qb3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qm3</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">rb3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm3</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" m_to_bern2(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pm2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pb2</span><span style="color: #0000FF;">})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" m_to_bern2(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qm2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qb2</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" m_to_bern3(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pb3</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" m_to_bern3(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qb3</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" m_to_bern3(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">rm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rb3</span><span style="color: #0000FF;">})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" bern2_to_bern3(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pb2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bern2_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pb2</span><span style="color: #0000FF;">)})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" bern2_to_bern3(%v) --&gt; %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qb2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">bern2_to_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qb2</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Subprogram (2) examples:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Evaluating bernstein degree 2 examples:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span> <span style="color: #008080;">in</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0.25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7.50</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span> <span style="color: #008080;">in</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0.25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7.50</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" p(%g) = %g (mono %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">evaluate_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pbern2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" p(%.2f) = %8g (mono: %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">eval_bern2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pb2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span><span style="color: #000000;">eval_m2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pm2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #000000;">evaluate_monomial_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pmono2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" q(%.2f) = %8g (mono: %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">eval_bern2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qb2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span><span style="color: #000000;">eval_m2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qm2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" q(%g) = %g (mono %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">evaluate_bernstein_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qbern2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">evaluate_monomial_degree2</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qmono2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">constant</span> <span style="color: #000000;">pmono3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- p(x) = 1</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Evaluating bernstein degree 3 examples:\n"</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #000000;">qmono3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- q(x) = 1 + 2x + 3x²</span>
<span style="color: #000000;">rmono3</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">4</span><span style="color: #0000FF;">},</span> <span style="color: #000080;font-style:italic;">-- r(x) = 1 + 2x + 3x² + 4x³</span>
⚫
<span style="color: #000000;">pbern3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_to_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pmono3</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">qbern3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_to_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qmono3</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">rbern3</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">monomial_to_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rmono3</span><span style="color: #0000FF;">)</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Subprogram (3) examples:\n"</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" mono %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pbern3</span><span style="color: #0000FF;">})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" mono %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qbern3</span><span style="color: #0000FF;">})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" mono %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">rmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">rbern3</span><span style="color: #0000FF;">})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Subprogram (4) examples:\n"</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span> <span style="color: #008080;">in</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0.25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7.50</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
<span style="color: #008080;">for</span> <span style="color: #000000;">x</span> <span style="color: #008080;">in</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">0.25</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">7.50</span><span style="color: #0000FF;">}</span> <span style="color: #008080;">do</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" p(%g) = %g (mono %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">evaluate_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pbern3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" p(%.2f) = %8g (mono: %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">eval_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pb3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span><span style="color: #000000;">eval_m3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #000000;">evaluate_monomial_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" q(%.2f) = %8g (mono: %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">eval_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qb3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span><span style="color: #000000;">eval_m3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" q(%g) = %g (mono %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">evaluate_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qbern3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" r(%.2f) = %8g (mono: %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">eval_bern3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rb3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span><span style="color: #000000;">eval_m3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rm3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #000000;">evaluate_monomial_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" r(%g) = %g (mono %g)\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">evaluate_bernstein_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rbern3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">),</span>
<span style="color: #000000;">evaluate_monomial_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rmono3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">x</span><span style="color: #0000FF;">)})</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"Subprogram (5) examples:\n"</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #008080;">constant</span> <span style="color: #000000;">pbern3a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_degree2_to_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">pbern2</span><span style="color: #0000FF;">),</span>
⚫
<span style="color: #000000;">qbern3a</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernstein_degree2_to_degree3</span><span style="color: #0000FF;">(</span><span style="color: #000000;">qbern2</span><span style="color: #0000FF;">)</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" bern %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">pbern2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">pbern3a</span><span style="color: #0000FF;">})</span>
⚫
<span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">" bern %v --&gt; bern %v\n"</span><span style="color: #0000FF;">,</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">qbern2</span><span style="color: #0000FF;">,</span><span style="color: #000000;">qbern3a</span><span style="color: #0000FF;">})</span>
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<pre>
<pre>
⚫
m_to_bern2({1,0,0}) --> {1,1,1}
Subprogram (1) examples:
mono {1,0,0} --> bern {1,1,1}
m_to_bern2({1,2,3}) --> {1,2,6}
mono {1,2,3} --> bern {1,2,6}
m_to_bern3({1,0,0,0}) --> {1,1,1,1}
⚫
m_to_bern3({1,2,3,0}) --> {1,1.666666667,3.333333333,6}
Subprogram (2) examples:
⚫
m_to_bern3({1,2,3,4}) --> {1,1.666666667,3.333333333,10}
⚫
p(0.25) = 1 (mono 1)
⚫
bern2_to_bern3({1,1,1}) --> {1,1,1,1}
⚫
q(0.25) = 1.6875 (mono 1.6875)
⚫
bern2_to_bern3({1,2,6}) --> {1,1.666666667,3.333333333,6}
⚫
p(7.5) = 1 (mono 1)
Evaluating bernstein degree 2 examples:
⚫
q(7.5) = 184.75 (mono 184.75)
⚫
p(0.25) = 1 (mono: 1)
Subprogram (3) examples:
⚫
q(0.25) = 1.6875 (mono: 1.6875)
⚫
mono {1,0,0,0} --> bern {1,1,1,1}
⚫
p(7.50) = 1 (mono: 1)
⚫
mono {1,2,3,0} --> bern {1,1.666666667,3.333333333,6}
⚫
q(7.50) = 184.75 (mono: 184.75)
⚫
mono {1,2,3,4} --> bern {1,1.666666667,3.333333333,10}
Subprogram (4) examples:
Evaluating bernstein degree 3 examples:
p(0.25) = 1 (mono 1)
p(0.25) = 1 (mono: 1)
q(0.25) = 1.6875 (mono 1.6875)
q(0.25) = 1.6875 (mono: 1.6875)
r(0.25) = 1.75 (mono 1.75)
r(0.25) = 1.75 (mono: 1.75)
p(7.5) = 1 (mono 1)
p(7.50) = 1 (mono: 1)
q(7.5) = 184.75 (mono 184.75)
q(7.50) = 184.75 (mono: 184.75)
r(7.5) = 1872.25 (mono 1872.25)
r(7.50) = 1872.25 (mono: 1872.25)
Subprogram (5) examples:
⚫
bern {1,1,1} --> bern {1,1,1,1}
⚫
bern {1,2,6} --> bern {1,1.666666667,3.333333333,6}
</pre>
</pre>


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