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Task/Formal-power-series/EchoLisp/formal-power-series-1.echolisp

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+(require 'math)
+;; converts a finite polynomial (a_0 a_1 .. a_n) to an infinite serie (a_0 ..a_n 0 0 0 ...)
+(define (poly->stream list)
+	(make-stream (lambda(n) (cons (if (< n (length list)) (list-ref list n) 0) (1+ n))) 0))
+
+;; c = a + b , c_n = a_n + b_n
+(define (s-add a b)
+	(make-stream (lambda (n) (cons (+ (stream-ref a n) (stream-ref b n)) (1+ n))) 0))
+	
+;; c = a * b , c_n = ∑ (0 ..n) a_i * b_n-i
+(define (s-mul-coeff n a b) (sigma (lambda(i) (* (stream-ref a i)(stream-ref b (- n i)))) 0 n))
+
+(define (s-mul a b)
+	(make-stream (lambda(n) (cons (s-mul-coeff n a b) (1+ n))) 0))
+
+;; b = 1/a ; b_0 = 1/a_0, b_n =  - ∑ (1..n) a_i * b_n-i / a_0
+(define (s-inv-coeff n a b)
+			(if (zero? n) (/ (stream-ref a 0))
+			(- (/ (sigma (lambda(i) (* (stream-ref a i)(stream-ref b (- n i)))) 1 n)
+			 (stream-ref a 0)))))
+
+;; note the self keyword which refers to b = (s-inv a)
+(define (s-inv a)
+	(make-stream (lambda(n) (cons (s-inv-coeff n a self ) (1+ n))) 0))
+
+;; b = (s-k-add k a) = k + a_0, a_1, a_2, ...
+(define (s-k-add k a)
+	(make-stream (lambda(n) (cons
+	(if(zero? n) (+ k (stream-ref a 0)) (stream-ref a n)) (1+ n))) 0))
+	
+;; b = (s-neg a) = -a_0,-a_1, ....
+(define (s-neg a)
+	(make-stream (lambda(n) (cons (- (stream-ref a n)) (1+ n))) 0))
+
+;; b = (s-int a) = ∫ a ; b_0 = 0 by convention, b_n = a_n-1/n
+(define (s-int a)
+	(make-stream (lambda(n) (cons (if (zero? n) 0 (/ (stream-ref a (1- n)) n)) (1+ n))) 0))
+	
+;; value of power serie at x, n terms
+(define (s-value a x (n 20))
+	(poly x (take a n)))
+
+;; stream-cons allows mutual delayed references
+;; sin = ∫ cos
+(define sin-x (stream-cons 0 (stream-rest (s-int cos-x))))
+;; cos = 1 - ∫ sin
+(define cos-x (stream-cons 1 (stream-rest (s-k-add 1  (s-neg (s-int sin-x))))))

Task/Formal-power-series/EchoLisp/formal-power-series-2.echolisp

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+(take cos-x 16)
+    → (1 0 -1/2 0 1/24 0 -1/720 0 1/40320 0 -1/3628800 0 1/479001600 0 -1.1470745597729725e-11 0)
+(take sin-x 16)
+    → (0 1 0 -1/6 0 1/120 0 -1/5040 0 1/362880 0 -1/39916800 0 1.6059043836821613e-10 0 -7.647163731819816e-13)
+
+;; compute (cos PI)
+(s-value cos-x PI)
+    → -1.0000000035290808
+
+;; check that 1 / (1 - x) = 1 + x + x^1 + x^2 + ...
+(define fps-1 (poly->stream '( 1 -1)))
+(take fps-1 13)
+   → (1 -1 0 0 0 0 0 0 0 0 0 0 0)
+
+(define inv-fps-1 (s-inv fps-1))
+(take inv-fps-1 13)
+    → (1 1 1 1 1 1 1 1 1 1 1 1 1)
+(s-value inv-fps-1 0.5) ;; check that 1 / (1 - 0.5) = 2
+   → 1.9999980926513672
+(s-value inv-fps-1 0.5 100) ;; 100 terms
+   → 2

Task/Formal-power-series/jq/formal-power-series-1.jq

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+1/(1+.)

Task/Formal-power-series/jq/formal-power-series-10.jq

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+def poly(ary): ary[.] // 0;

Task/Formal-power-series/jq/formal-power-series-11.jq

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+(poly([1,2,3]) + poly([-1,-2,-3]))

Task/Formal-power-series/jq/formal-power-series-12.jq

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+# Multiply two power series, s and t:
+def M(s;t):
+  . as $i | reduce range(0; 1+$i) as $k
+    (0; . + ($k|s) * (($i - $k)|t));
+
+# Derivative of the power series, s:
+def D(s): (. + 1) as $i | $i * ($i|s);
+
+# Integral of the power series, s,
+# with an integration constant equal to 0:
+def I(s):
+  . as $i
+  | if $i == 0 then 0 else (($i-1)|s) /$i end;

Task/Formal-power-series/jq/formal-power-series-13.jq

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+def ps_equal(s; t; k; eps):
+  def abs: if . < 0 then -. else . end;
+  reduce range(0;k) as $i
+    (true;
+     if . then ((($i|s) - ($i|t))|abs) <= eps
+     else .
+     end);

Task/Formal-power-series/jq/formal-power-series-14.jq

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+# evaluate p(x) based on the first k terms of polynomial p, where x is the input
+def ps_eval(p; k):
+  . as $x
+  | reduce range(0;k) as $i
+    # state: [sum, x^i]
+      ([0, 1];
+       .[1] as $xn
+       | ($i|p) as $coeff
+       | [ .[0] + $coeff * $xn, $x * $xn])
+  | .[0];
+
+# If |x| < 1 then ps_evaluate(x) will evaluate to p(x) with high precision
+# if the coefficients of the polynomial are eventually bounded.
+#
+# WARNING: ps_evaluate(p) will not detect divergence and is not intended to
+# produce accurate results unless the terms of p(x) are reasonably well-behaved.
+# For |x| > 1, the result will be null if x^n overflows before convergence is achieved.
+#
+def ps_evaluate(p):
+  def abs: if . < 0 then -. else . end;
+  def eval(p;x):
+    # state: [i, x^i, sum of i terms, delta, prevdelta]
+    recurse(
+      .[0] as $i
+      | .[1] as $xi
+      | .[2] as $sum
+      | .[3] as $delta
+      | .[4] as $prevdelta
+      | if $delta < 1e-17 and $prevdelta < 1e-17
+           and ( $xi < 1e-100
+                 or ( $sum != 0 and
+                      (($delta/$sum) | abs) < 1e-10 and
+                      (($prevdelta/$sum) | abs) < 1e-10) )
+        then empty
+        else
+          ($xi * ($i|p)) as $newdelta
+        | [ $i + 1,
+            x*$xi,
+            $sum+$newdelta,
+            ($newdelta|abs), $delta]
+        end ) ;
+   . as $x
+   | [0, 1, 0, 1, 1]
+   | reduce eval(p; $x) as $vector (0; $vector[2]);

Task/Formal-power-series/jq/formal-power-series-15.jq

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+# Utility functions:
+
+def abs: if . < 0 then -. else . end;
+
+# The power series whose only non-zero coefficient is 1 at x^i:
+def ps_at(i): if . ==  i then 1 else 0 end;
+
+# Create an array consisting of the first . coefficients of the power series, p:
+def ps_to_array(p): . as $in | reduce range(0;$in) as $i ([]; . + [$i|p]);
+
+def pi: 4 * (1|atan);

Task/Formal-power-series/jq/formal-power-series-16.jq

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+# Verify that the first 100 terms of I(cos) and of sin are the same:
+
+ps_equal( I(ps_cos); ps_sin; 100; 1e-15)
+# => true
+
+# Verify that the two power series agree when evaluated at pi:
+
+((pi | ps_evaluate(I(ps_cos))) - (pi | ps_evaluate(ps_sin))) | abs < 1e-15
+# => true

Task/Formal-power-series/jq/formal-power-series-17.jq

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+# Verify that the first 100 terms of cos and (1 - I(sin)) are the same:
+
+ps_equal( ps_cos; ps_at(0) - I(ps_sin); 100; 1e-5)
+# => true
+
+# Verify that the two power series agree at pi:
+
+((pi | ps_evaluate(ps_cos)) - (pi | ps_evaluate(ps_at(0) - I(ps_sin)))) | abs < 1e-15
+# => true

Task/Formal-power-series/jq/formal-power-series-2.jq

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+1/factorial

Task/Formal-power-series/jq/formal-power-series-3.jq

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+def factorial:
+  reduce range(1; . + 1) as $i
+    (1; . * $i);

Task/Formal-power-series/jq/formal-power-series-4.jq

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+def ps_exp: 1/factorial;

Task/Formal-power-series/jq/formal-power-series-5.jq

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+1 | ps_evaluate(ps_exp)

Task/Formal-power-series/jq/formal-power-series-6.jq

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+1 | ps_evaluate(1/factorial)

Task/Formal-power-series/jq/formal-power-series-7.jq

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+def pow(n):
+  . as $x | n as $n
+  | reduce range(0;$n) as $i (1; . * $x);

Task/Formal-power-series/jq/formal-power-series-8.jq

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+1/pow(.)

Task/Formal-power-series/jq/formal-power-series-9.jq

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+# ln(1+x) = x - x^2 / 2 + ...
+def ln_1px:
+  def c: if . % 2 == 0 then -1 else 1 end;
+  . as $i | if $i == 0 then 0 else ($i|c) / $i  end;