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Faulhaber's formula: Difference between revisions

Faulhaber's formula (view source)

Revision as of 16:00, 16 January 2022

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→‎{{header|Phix}}: added syntax colouring, marked p2js compatible
m (Updated description and link for Fōrmulæ solution)
m (→‎{{header|Phix}}: added syntax colouring, marked p2js compatible)
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=={{header|Phix}}==
{{trans|C#}}
<!--<lang Phix>include builtins\pfrac.e -- (0.8.0+phixonline)-->
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
 
<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>
function bernoulli(integer n)
sequence a = {}
<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>
for m=0 to n do
<span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{}</span>
a = append(a,{1,m+1})
<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>
for j=m to 1 by -1 do
<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>
a[j] = frac_mul({j,1},frac_sub(a[j+1],a[j]))
<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>
end for
<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>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
if n!=1 then return a[1] end if
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
return frac_uminus(a[1])
<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>
end function
<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>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function binomial(integer n, k)
if n<0 or k<0 or n<k then ?9/0 end if
<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>
if n=0 or k=0 then return 1 end if
<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>
integer num = 1,
<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>
denom = 1
<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>
for i=k+1 to n do
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
num *= i
<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>
end for
<span style="color: #000000;">num</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span>
for i=2 to n-k do
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
denom *= i
<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>
end for
<span style="color: #000000;">denom</span> <span style="color: #0000FF;">*=</span> <span style="color: #000000;">i</span>
return num / denom
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">num</span> <span style="color: #0000FF;">/</span> <span style="color: #000000;">denom</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
procedure faulhaber(integer p)
string res = sprintf("%d : ", p)
<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>
frac q = {1, p+1}
<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>
for j=0 to p do
<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>
frac bj = bernoulli(j)
<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>
if frac_ne(bj,frac_zero) then
<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>
frac coeff = frac_mul({binomial(p+1,j),p+1},bj)
<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>
string s = frac_sprint(coeff)
<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>
if j=0 then
<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>
if s="1" then
<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>
s = ""
<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>
end if
<span style="color: #000000;">s</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span>
else
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
if s[1]='-' then
<span s[1..1] style= "color: - #008080;">else</span>
<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>
else
<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>
s[1..0] = " + "
end if<span style="color: #008080;">else</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;">0</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">" + "</span>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
res &= s&"n"
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
integer pwr = p+1-j
<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>
if pwr>1 then
<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>
res &= sprintf("^%d", pwr)
<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>
end if
<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>
end if
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
printf(1,"%s\n",{res})
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
end procedure
<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>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
for i=0 to 9 do
faulhaber(i)
<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>
end for</lang>
<span style="color: #000000;">faulhaber</span><span style="color: #0000FF;">(</span><span style="color: #000000;">i</span><span style="color: #0000FF;">)</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
<!--</lang>-->
{{out}}
<pre>
Petelomax
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