Fibonacci matrix-exponentiation: Difference between revisions

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(→‎{{header|Haskell}}: Follow new requirments)
(Rename Perl 6 -> Raku, alphabetize, minor clean-up)
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*   [[Matrix-exponentiation operator]]
*   [[Matrix-exponentiation operator]]
<br/>
<br/>

=={{header|C sharp|C#}}==
=={{header|C sharp|C#}}==
===Matrix exponentiation===
===Matrix exponentiation===
Line 1,138: Line 1,139:
F(2^32) = 61999319689381859818 ... 39623735538208076347
F(2^32) = 61999319689381859818 ... 39623735538208076347
F(2^64) = 11175807536929528424 ... 17529800348089840187</pre>
F(2^64) = 11175807536929528424 ... 17529800348089840187</pre>

=={{header|Perl 6}}==
Following the general approach of Sidef, and relying on Perl for <code>fibmod</code> function. Borrowed/adapted routines from [http://rosettacode.org/wiki/Ramanujan%27s_constant Ramanujan's_constant] task to allow <code>FatRat</code> calculations throughout. Does not quite meet task spec, as n^64 results not working yet.
<lang perl6>use Math::Matrix;
use Inline::Perl5;
my $p5 = Inline::Perl5.new();
$p5.use( 'Math::AnyNum' );

constant D = 53; # set the size of FatRat calcluations

# matrix exponentiation
sub fibonacci ($n) {
my $M = Math::Matrix.new( [[1,1],[1,0]] );
($M ** $n)[0][1]
}

# calculation of 𝑒
sub postfix:<!> (Int $n) { (constant f = 1, |[\×] 1..*)[$n] }
sub 𝑒 (--> FatRat) { sum map { FatRat.new(1,.!) }, ^D }

# calculation of π
sub π (--> FatRat) {
my ($a, $n, $g, $z, $pi) = 1, 1, sqrt(1/2.FatRat), 0.25;

for ^5 {
given [ ($a + $g)/2, sqrt $a × $g ] {
$z -= (.[0] - $a)**2 × $n;
$n += $n;
($a, $g) = @$_;
$pi = ($a ** 2 / $z).substr: 0, 2 + D
}
}
$pi.FatRat
}

# square-root: accepts/return FatRat
multi sqrt(FatRat $r --> FatRat) {
FatRat.new: sqrt($r.nude[0] × 10**(D×2) div $r.nude[1]), 10**D
}

# square-root: accepts/return Int
multi sqrt(Int $n --> Int) {
my $guess = 10**($n.chars div 2);
my $iterator = { ( $^x + $n div ($^x) ) div 2 };
my $endpoint = { $^x == $^y|$^z };
min ($guess, $iterator … $endpoint)[*-1, *-2]
}

# arithmetic-geometric mean: accepts/returns FatRat
sub AG-mean(FatRat $a is copy, FatRat $g is copy --> FatRat) {
($a, $g) = ($a + $g)/2, sqrt $a × $g until $a - $g < 10**-D;
$a
}

# override built-in definitions with 'FatRat' versions
constant 𝑒 = &𝑒();
constant π = &π();

# approximation of natural log, accepts any numeric, returns FatRat
sub log-approx ($x --> FatRat) {
constant ln2 = 2 * [+] (((1/3).FatRat**(2*$_+1))/(2*$_+1) for 0..D); # 1/3 = (2-1)/(2+1)
π / (2 × AG-mean(1.FatRat, 2.FatRat**(2-D)/$x)) - D × ln2
}

# power function, with exponent less than zero: accepts/returns FatRat
multi infix:<**> (FatRat $base, FatRat $exp is copy where * < 0 --> FatRat) {
constant ε = 10**-D;
my ($low, $high) = 0.FatRat, 1.FatRat;
my $mid = $high / 2;
my $acc = my $sqr = sqrt($base);
$exp = -$exp;

while (abs($mid - $exp) > ε) {
$sqr = sqrt($sqr);
if ($mid <= $exp) { $low = $mid; $acc ×= $sqr }
else { $high = $mid; $acc ×= 1/$sqr }
$mid = ($low + $high) / 2
}

(1/$acc).substr(0, D).FatRat
}

sub binet_approx (Int $n --> FatRat) {
constant PHI = sqrt(1.25.FatRat) + 0.5 ;
constant IHP = -(sqrt(1.25.FatRat) - 0.5);
$n × log-approx(PHI) - log-approx(PHI - IHP)
}

sub nth_fib_first_k_digits ($n,$k) {
my $f = binet_approx($n);
my $log10 = log-approx(10);
floor( 𝑒**($f - $log10×(floor($f/$log10 + 1))) × 10**$k)
}

my &nth_fib_last_k_digits =
$p5.run('sub { my($n,$k) = @_; Math::AnyNum::fibmod($n, 10**$k) }');

# matrix exponentiation is very inefficient, n^64 not feasible
for (16, 32) -> $n {
my $f = fibonacci(2**$n);
printf "F(2^$n) = %s ... %s\n", substr($f,0,20), $f % 10**20
}

# this way is much faster, but not yet able to handle 2^64 case
for 16, 32 -> $n {
my $first20 = nth_fib_first_k_digits(2**$n, 20);
my $last20 = nth_fib_last_k_digits(2**$n, 20);
printf "F(2^$n) = %s ... %s\n", $first20, $last20
}</lang>
{{out}}
<pre>F(2^16) = 73199214460290552832 ... 97270190955307463227
F(2^32) = 61999319689381859818 ... 39623735538208076347
F(2^16) = 73199214460290552832 ... 97270190955307463227
F(2^32) = 61999319689381859818 ... 39623735538208076347</pre>


=={{header|Phix}}==
=={{header|Phix}}==
Line 1,437: Line 1,324:
</pre>
</pre>
Clearly 2^32 (897 million digits, apparently) is a tad out of bounds, let alone 2^64.
Clearly 2^32 (897 million digits, apparently) is a tad out of bounds, let alone 2^64.

=={{header|Raku}}==
(formerly Perl 6)
Following the general approach of Sidef, and relying on Perl for <code>fibmod</code> function. Borrowed/adapted routines from [http://rosettacode.org/wiki/Ramanujan%27s_constant Ramanujan's_constant] task to allow <code>FatRat</code> calculations throughout. Does not quite meet task spec, as n^64 results not working yet.
<lang perl6>use Math::Matrix;
use Inline::Perl5;
my $p5 = Inline::Perl5.new();
$p5.use( 'Math::AnyNum' );

constant D = 53; # set the size of FatRat calcluations

# matrix exponentiation
sub fibonacci ($n) {
my $M = Math::Matrix.new( [[1,1],[1,0]] );
($M ** $n)[0][1]
}

# calculation of 𝑒
sub postfix:<!> (Int $n) { (constant f = 1, |[\×] 1..*)[$n] }
sub 𝑒 (--> FatRat) { sum map { FatRat.new(1,.!) }, ^D }

# calculation of π
sub π (--> FatRat) {
my ($a, $n, $g, $z, $pi) = 1, 1, sqrt(1/2.FatRat), 0.25;

for ^5 {
given [ ($a + $g)/2, sqrt $a × $g ] {
$z -= (.[0] - $a)**2 × $n;
$n += $n;
($a, $g) = @$_;
$pi = ($a ** 2 / $z).substr: 0, 2 + D
}
}
$pi.FatRat
}

# square-root: accepts/return FatRat
multi sqrt(FatRat $r --> FatRat) {
FatRat.new: sqrt($r.nude[0] × 10**(D×2) div $r.nude[1]), 10**D
}

# square-root: accepts/return Int
multi sqrt(Int $n --> Int) {
my $guess = 10**($n.chars div 2);
my $iterator = { ( $^x + $n div ($^x) ) div 2 };
my $endpoint = { $^x == $^y|$^z };
min ($guess, $iterator … $endpoint)[*-1, *-2]
}

# arithmetic-geometric mean: accepts/returns FatRat
sub AG-mean(FatRat $a is copy, FatRat $g is copy --> FatRat) {
($a, $g) = ($a + $g)/2, sqrt $a × $g until $a - $g < 10**-D;
$a
}

# override built-in definitions with 'FatRat' versions
constant 𝑒 = &𝑒();
constant π = &π();

# approximation of natural log, accepts any numeric, returns FatRat
sub log-approx ($x --> FatRat) {
constant ln2 = 2 * [+] (((1/3).FatRat**(2*$_+1))/(2*$_+1) for 0..D); # 1/3 = (2-1)/(2+1)
π / (2 × AG-mean(1.FatRat, 2.FatRat**(2-D)/$x)) - D × ln2
}

# power function, with exponent less than zero: accepts/returns FatRat
multi infix:<**> (FatRat $base, FatRat $exp is copy where * < 0 --> FatRat) {
constant ε = 10**-D;
my ($low, $high) = 0.FatRat, 1.FatRat;
my $mid = $high / 2;
my $acc = my $sqr = sqrt($base);
$exp = -$exp;

while (abs($mid - $exp) > ε) {
$sqr = sqrt($sqr);
if ($mid <= $exp) { $low = $mid; $acc ×= $sqr }
else { $high = $mid; $acc ×= 1/$sqr }
$mid = ($low + $high) / 2
}

(1/$acc).substr(0, D).FatRat
}

sub binet_approx (Int $n --> FatRat) {
constant PHI = sqrt(1.25.FatRat) + 0.5 ;
constant IHP = -(sqrt(1.25.FatRat) - 0.5);
$n × log-approx(PHI) - log-approx(PHI - IHP)
}

sub nth_fib_first_k_digits ($n,$k) {
my $f = binet_approx($n);
my $log10 = log-approx(10);
floor( 𝑒**($f - $log10×(floor($f/$log10 + 1))) × 10**$k)
}

my &nth_fib_last_k_digits =
$p5.run('sub { my($n,$k) = @_; Math::AnyNum::fibmod($n, 10**$k) }');

# matrix exponentiation is very inefficient, n^64 not feasible
for (16, 32) -> $n {
my $f = fibonacci(2**$n);
printf "F(2^$n) = %s ... %s\n", substr($f,0,20), $f % 10**20
}

# this way is much faster, but not yet able to handle 2^64 case
for 16, 32 -> $n {
my $first20 = nth_fib_first_k_digits(2**$n, 20);
my $last20 = nth_fib_last_k_digits(2**$n, 20);
printf "F(2^$n) = %s ... %s\n", $first20, $last20
}</lang>
{{out}}
<pre>F(2^16) = 73199214460290552832 ... 97270190955307463227
F(2^32) = 61999319689381859818 ... 39623735538208076347
F(2^16) = 73199214460290552832 ... 97270190955307463227
F(2^32) = 61999319689381859818 ... 39623735538208076347</pre>


=={{header|Sidef}}==
=={{header|Sidef}}==
Retrieved from "https://rosettacode.org/wiki/Fibonacci_matrix-exponentiation"