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Multidimensional Newton-Raphson method: Difference between revisions

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Multidimensional Newton-Raphson method (view source)

Revision as of 23:09, 30 January 2022

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→‎{{header|Phix}}: added syntax colouring, made p2js compatible
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[[Matrix_multiplication#Phix]]<br>
See std distro for a complete runnable version.
<!--<lang Phix>(phixonline)-->
<lang Phix>-- demo\rosetta\Multidimensional_Newton-Raphson_method.exw
<span style="color: #000080;font-style:italic;">-- demo\rosetta\Multidimensional_Newton-Raphson_method.exw</span>
function solve(sequence fs, jacob, guesses)
<span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span>
integer size := length(fs),
<span style="color: #008080;">function</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">)</span>
maxIter := 12,
<span style="color: #004080;">integer</span> <span style="color: #000000;">size</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">length</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">),</span>
iter := 0
<span style="color: #000000;">maxIter</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">12</span><span style="color: #0000FF;">,</span>
sequence gu1, g, t, f, g1,
<span style="color: #000000;">iter</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
gu2 := guesses,
<span style="color: #004080;">sequence</span> <span style="color: #000000;">gu1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">g1</span><span style="color: #0000FF;">,</span>
jac := repeat(repeat(0,size),size)
<span style="color: #000000;">gu2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">,</span>
atom tol := 1e-8
<span style="color: #000000;">jac</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span><span style="color: #000000;">size</span><span style="color: #0000FF;">),</span><span style="color: #000000;">size</span><span style="color: #0000FF;">)</span>
while true do
<span style="color: #004080;">atom</span> <span style="color: #000000;">tol</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1e-8</span>
gu1 := gu2
<span style="color: #008080;">while</span> <span style="color: #004600;">true</span> <span style="color: #008080;">do</span>
g := matrix_transpose({gu1})
<span style="color: #000000;">gu1</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">gu2</span>
t := repeat(0, size)
<span style="color: #000000;">g</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">matrix_transpose</span><span style="color: #0000FF;">({</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
for i=1 to size do
<span style="color: #000000;">t</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">size</span><span style="color: #0000FF;">)</span>
t[i] := call_func(fs[i],{gu1})
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
end for
<span style="color: #000000;">t</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">call_func</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">],{</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
f := matrix_transpose({t})
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
for i=1 to size do
<span style="color: #000000;">f</span> <span style="color: #0000FF;">:=</span> <span style="color: #000000;">matrix_transpose</span><span style="color: #0000FF;">({</span><span style="color: #000000;">t</span><span style="color: #0000FF;">})</span>
for j=1 to size do
<span style="color: #008080;">for</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
jac[i][j] := call_func(jacob[i][j],{gu1})
<span style="color: #008080;">for</span> <span style="color: #000000;">j</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">size</span> <span style="color: #008080;">do</span>
end for
<span style="color: #000000;">jac</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</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: #7060A8;">call_func</span><span style="color: #0000FF;">(</span><span style="color: #000000;">jacob</span><span style="color: #0000FF;">[</span><span style="color: #000000;">i</span><span style="color: #0000FF;">][</span><span style="color: #000000;">j</span><span style="color: #0000FF;">],{</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">})</span>
end for
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
g1 := sq_sub(g,matrix_mul(inverse(jac),f))
<span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
gu2 := vslice(g1,1)
<span style="color: #000000;">g1</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">,</span><span style="color: #000000;">matrix_mul</span><span style="color: #0000FF;">(</span><span style="color: #000000;">inverse</span><span style="color: #0000FF;">(</span><span style="color: #000000;">jac</span><span style="color: #0000FF;">),</span><span style="color: #000000;">f</span><span style="color: #0000FF;">))</span>
iter += 1
<span style="color: #000000;">gu2</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">vslice</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
bool any := find(true,sq_gt(sq_sub(sq_abs(gu2),gu1),tol))!=0
<span style="color: #000000;">iter</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
if not any or iter >= maxIter then exit end if
<span style="color: #004080;">bool</span> <span style="color: #000000;">any</span> <span style="color: #0000FF;">:=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #004600;">true</span><span style="color: #0000FF;">,</span><span style="color: #7060A8;">sq_gt</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_sub</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">sq_abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">gu2</span><span style="color: #0000FF;">),</span><span style="color: #000000;">gu1</span><span style="color: #0000FF;">),</span><span style="color: #000000;">tol</span><span style="color: #0000FF;">))!=</span><span style="color: #000000;">0</span>
end while
<span style="color: #008080;">if</span> <span style="color: #008080;">not</span> <span style="color: #000000;">any</span> <span style="color: #008080;">or</span> <span style="color: #000000;">iter</span> <span style="color: #0000FF;">>=</span> <span style="color: #000000;">maxIter</span> <span style="color: #008080;">then</span> <span style="color: #008080;">exit</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
return gu2
<span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
end function
<span style="color: #008080;">return</span> <span style="color: #000000;">gu2</span>
 
<span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
function f1(sequence v) atom {x,y} = v return -x*x+x+0.5-y end function
function f2(sequence v) atom {x,y} = v return y+5*x*y-x*x end function
function f3(sequence v) atom {x,y,z} = v return 9*x*x+36*y*y+4*z*z-36 end function
function f4(sequence v) atom {x,y,z} = v return x*x-2*y*y-20*z end function
function f5(sequence v) atom {x,y,z} = v return x*x-y*y+z*z end function
 
function j1(sequence v) atom {x} = v return -2*x+1 end function
function j2(sequence /*v*/) return -1 end function
function j3(sequence v) atom {x,y} = v return 5*y-2*x end function
function j4(sequence v) atom {x} = v return 1+5*x end function
function j11(sequence v) atom {x} = v return 18*x end function
function j12(sequence v) atom {?,y} = v return 72*y end function
function j13(sequence v) atom {?,?,z} = v return 8*z end function
function j21(sequence v) atom {x} = v return 2*x end function
function j22(sequence v) atom {?,y} = v return -4*y end function
function j23(sequence /*v*/) return -20 end function
function j31(sequence v) atom {x} = v return 2*x end function
function j32(sequence v) atom {?,y} = v return -2*y end function
function j33(sequence v) atom {?,?,z} = v return 2*z end function
 
procedure main()
sequence fs, jacob, guesses
/*
solve the two non-linear equations:
y = -x^2 + x + 0.5
y + 5xy = x^2
given initial guesses of x = y = 1.2
<span style="color: #008080;">function</span> <span style="color: #000000;">f1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">0.5</span><span style="color: #0000FF;">-</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
Example taken from:
<span style="color: #008080;">function</span> <span style="color: #000000;">f2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
<span style="color: #008080;">function</span> <span style="color: #000000;">f3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">9</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">36</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span><span style="color: #0000FF;">-</span><span style="color: #000000;">36</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f4</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">20</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">f5</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">x</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">-</span><span style="color: #000000;">y</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">+</span><span style="color: #000000;">z</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j1</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
Expected results: x = 1.23332, y = 0.2122
<span style="color: #008080;">function</span> <span style="color: #000000;">j2</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000080;font-style:italic;">/*v*/</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
*/
<span style="color: #008080;">function</span> <span style="color: #000000;">j3</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span><span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
fs = {routine_id("f1"),routine_id("f2")}
<span style="color: #008080;">function</span> <span style="color: #000000;">j4</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">5</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
jacob = {{routine_id("j1"),routine_id("j2")},
<span style="color: #008080;">function</span> <span style="color: #000000;">j11</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">18</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
{routine_id("j3"),routine_id("j4")}}
<span style="color: #008080;">function</span> <span style="color: #000000;">j12</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">72</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
guesses := {1.2, 1.2}
<span style="color: #008080;">function</span> <span style="color: #000000;">j13</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,?,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">8</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
printf(1,"Approximate solutions are x = %.7f, y = %.7f\n\n", solve(fs, jacob, guesses))
<span style="color: #008080;">function</span> <span style="color: #000000;">j21</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j22</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">4</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j23</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000080;font-style:italic;">/*v*/</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">20</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j31</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">x</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">x</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j32</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">y</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">function</span> <span style="color: #000000;">j33</span><span style="color: #0000FF;">(</span><span style="color: #004080;">sequence</span> <span style="color: #000000;">v</span><span style="color: #0000FF;">)</span> <span style="color: #004080;">atom</span> <span style="color: #0000FF;">{?,?,</span><span style="color: #000000;">z</span><span style="color: #0000FF;">}</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">v</span> <span style="color: #008080;">return</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #000000;">z</span> <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
<span style="color: #008080;">procedure</span> <span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
/*
<span style="color: #004080;">sequence</span> <span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span>
solve the three non-linear equations:
<span style="color: #000080;font-style:italic;">/*
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 -solve 2y^2the two non-linear 20z = 0equations:
x^2 y = - yx^2 + z^2x =+ 0.5
given initial guesses of x = y =+ 1.0 and z5xy = 0.0x^2
given initial guesses of x = y = 1.2
Example taken from:
http://www.fixoncloud.com/Home/LoginValidate/OneProblemComplete_Detailed.php?problemid=286
Expected results: x = 1.23332, y = 0.2122
*/</span>
<span style="color: #000000;">fs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">f2</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">jacob</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">j1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j2</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j3</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j4</span><span style="color: #0000FF;">}}</span>
<span style="color: #000000;">guesses</span> <span style="color: #0000FF;">:=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">1.2</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">1.2</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;">"Approximate solutions are x = %.7f, y = %.7f\n\n"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">))</span>
<span style="color: #000080;font-style:italic;">/*
solve the three non-linear equations:
9x^2 + 36y^2 + 4z^2 - 36 = 0
x^2 - 2y^2 - 20z = 0
x^2 - y^2 + z^2 = 0
given initial guesses of x = y = 1.0 and z = 0.0
Example taken from:
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/</span>
<span style="color: #000000;">fs</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">f3</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f4</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">f5</span><span style="color: #0000FF;">}</span>
<span style="color: #000000;">jacob</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{{</span><span style="color: #000000;">j11</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j12</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j13</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j21</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j22</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j23</span><span style="color: #0000FF;">},</span>
<span style="color: #0000FF;">{</span><span style="color: #000000;">j31</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j32</span><span style="color: #0000FF;">,</span><span style="color: #000000;">j33</span><span style="color: #0000FF;">}}</span>
<span style="color: #000000;">guesses</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;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">0</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;">"Approximate solutions are x = %.7f, y = %.7f, z = %.7f\n"</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">solve</span><span style="color: #0000FF;">(</span><span style="color: #000000;">fs</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">jacob</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">guesses</span><span style="color: #0000FF;">))</span>
<span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
Example taken from:
<span style="color: #000000;">main</span><span style="color: #0000FF;">()</span>
http://mathfaculty.fullerton.edu/mathews/n2003/FixPointNewtonMod.html (exercise 5)
<!--</lang>-->
Expected results: x = 0.893628, y = 0.894527, z = -0.0400893
*/
fs = {routine_id("f3"), routine_id("f4"), routine_id("f5")}
jacob = {{routine_id("j11"),routine_id("j12"),routine_id("j13")},
{routine_id("j21"),routine_id("j22"),routine_id("j23")},
{routine_id("j31"),routine_id("j32"),routine_id("j33")}}
guesses = {1, 1, 0}
printf(1,"Approximate solutions are x = %.7f, y = %.7f, z = %.7f\n", solve(fs, jacob, guesses))
 
end procedure
main()</lang>
{{out}}
<pre>
Petelomax
7,804

edits

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