Faulhaber's formula: Difference between revisions
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m (Updated description and link for Fōrmulæ solution) |
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=={{header|Phix}}== |
=={{header|Phix}}== |
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{{trans|C#}} |
{{trans|C#}} |
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<lang Phix> |
<!--<lang Phix>(phixonline)--> |
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<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> |
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<span style="color: #008080;">include</span> <span style="color: #000000;">builtins</span><span style="color: #0000FF;">\</span><span style="color: #000000;">pfrac</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> <span style="color: #000080;font-style:italic;">-- (0.8.0+)</span> |
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function bernoulli(integer n) |
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sequence a = {} |
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<span style="color: #008080;">function</span> <span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">)</span> |
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for m=0 to n do |
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<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span> |
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a = append(a,{1,m+1}) |
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<span style="color: #008080;">for</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span> |
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for j=m to 1 by -1 do |
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<span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">append</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">})</span> |
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a[j] = frac_mul({j,1},frac_sub(a[j+1],a[j])) |
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<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">m</span> <span style="color: #008080;">to</span> <span style="color: #000000;">1</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span> |
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end for |
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<span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_mul</span><span style="color: #0000FF;">({</span><span style="color: #000000;">j</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span><span style="color: #000000;">frac_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">j</span><span style="color: #0000FF;">]))</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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if n!=1 then return a[1] end if |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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return frac_uminus(a[1]) |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">!=</span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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end function |
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<span style="color: #008080;">return</span> <span style="color: #000000;">frac_uminus</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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function binomial(integer n, k) |
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if n<0 or k<0 or n<k then ?9/0 end if |
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<span style="color: #008080;">function</span> <span style="color: #000000;">binomial</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span> |
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if n=0 or k=0 then return 1 end if |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">k</span><span style="color: #0000FF;"><</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">n</span><span style="color: #0000FF;"><</span><span style="color: #000000;">k</span> <span style="color: #008080;">then</span> <span style="color: #0000FF;">?</span><span style="color: #000000;">9</span><span style="color: #0000FF;">/</span><span style="color: #000000;">0</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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integer num = 1, |
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<span style="color: #008080;">if</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">or</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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denom = 1 |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">num</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> |
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for i=k+1 to n do |
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<span style="color: #000000;">denom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span> |
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num *= i |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span> <span style="color: #008080;">do</span> |
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end for |
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<span style="color: #000000;">num</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span> |
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for i=2 to n-k do |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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denom *= i |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">2</span> <span style="color: #008080;">to</span> <span style="color: #000000;">n</span><span style="color: #0000FF;">-</span><span style="color: #000000;">k</span> <span style="color: #008080;">do</span> |
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end for |
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<span style="color: #000000;">denom</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span> |
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return num / denom |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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end function |
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<span style="color: #008080;">return</span> <span style="color: #000000;">num</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">denom</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">function</span> |
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procedure faulhaber(integer p) |
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string res = sprintf("%d : ", p) |
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<span style="color: #008080;">procedure</span> <span style="color: #000000;">faulhaber</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> |
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frac q = {1, p+1} |
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<span style="color: #004080;">string</span> <span style="color: #000000;">res</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"%d : "</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">)</span> |
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for j=0 to p do |
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<span style="color: #000000;">frac</span> <span style="color: #000000;">q</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;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">}</span> |
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frac bj = bernoulli(j) |
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<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">p</span> <span style="color: #008080;">do</span> |
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if frac_ne(bj,frac_zero) then |
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<span style="color: #000000;">frac</span> <span style="color: #000000;">bj</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">bernoulli</span><span style="color: #0000FF;">(</span><span style="color: #000000;">j</span><span style="color: #0000FF;">)</span> |
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frac coeff = frac_mul({binomial(p+1,j),p+1},bj) |
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<span style="color: #008080;">if</span> <span style="color: #000000;">frac_ne</span><span style="color: #0000FF;">(</span><span style="color: #000000;">bj</span><span style="color: #0000FF;">,</span><span style="color: #000000;">frac_zero</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">then</span> |
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string s = frac_sprint(coeff) |
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<span style="color: #000000;">frac</span> <span style="color: #000000;">coeff</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_mul</span><span style="color: #0000FF;">({</span><span style="color: #000000;">binomial</span><span style="color: #0000FF;">(</span><span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j</span><span style="color: #0000FF;">),</span><span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">},</span><span style="color: #000000;">bj</span><span style="color: #0000FF;">)</span> |
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if j=0 then |
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<span style="color: #004080;">string</span> <span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">frac_sprint</span><span style="color: #0000FF;">(</span><span style="color: #000000;">coeff</span><span style="color: #0000FF;">)</span> |
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if s="1" then |
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<span style="color: #008080;">if</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> |
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s = "" |
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<span style="color: #008080;">if</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">=</span><span style="color: #008000;">"1"</span> <span style="color: #008080;">then</span> |
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end if |
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<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> |
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else |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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if s[1]='-' then |
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<span style="color: #008080;">else</span> |
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<span style="color: #008080;">if</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: #008000;">'-'</span> <span style="color: #008080;">then</span> |
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else |
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<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;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">" - "</span> |
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s[1..0] = " + " |
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<span style="color: #008080;">else</span> |
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<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;">0</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">" + "</span> |
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end if |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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res &= s&"n" |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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integer pwr = p+1-j |
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<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">&</span><span style="color: #008000;">"n"</span> |
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if pwr>1 then |
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<span style="color: #004080;">integer</span> <span style="color: #000000;">pwr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">p</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">-</span><span style="color: #000000;">j</span> |
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res &= sprintf("^%d", pwr) |
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<span style="color: #008080;">if</span> <span style="color: #000000;">pwr</span><span style="color: #0000FF;">></span><span style="color: #000000;">1</span> <span style="color: #008080;">then</span> |
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end if |
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<span style="color: #000000;">res</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"^%d"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">pwr</span><span style="color: #0000FF;">)</span> |
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end if |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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end for |
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<span style="color: #008080;">end</span> <span style="color: #008080;">if</span> |
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printf(1,"%s\n",{res}) |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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end procedure |
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<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;">"%s\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">res</span><span style="color: #0000FF;">})</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span> |
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for i=0 to 9 do |
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faulhaber(i) |
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<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">0</span> <span style="color: #008080;">to</span> <span style="color: #000000;">9</span> <span style="color: #008080;">do</span> |
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end for</lang> |
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<span style="color: #000000;">faulhaber</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span> |
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<span style="color: #008080;">end</span> <span style="color: #008080;">for</span> |
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<!--</lang>--> |
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{{out}} |
{{out}} |
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<pre> |
<pre> |