Trying to learn METAFONT (act.)

Pythagorean addition $\sqrt{a^2+b^2}$

Estava a "tentar" aprender METAFONT, o significado das aspas tornar-se-á claro em poucos instantes, e dei com o seguinte comentário escrito pelo próprio Knuth

Pythagorean addition $latex2png equation$ is implemented by an elegant iterative scheme due to Cleve Moler and Donald Morrison IBM Journal of Research and Development 27 (1983), 577—581. It modifies |a| and |b| in such a way that their Pythagorean sum remains invariant, while the smaller argument decreases.

E, justifica-se as aspas, tive de descobrir o artigo (não encontrei outro sítio para o ler) e implementar o código em Lisp.

(defun pythag (a b &optional tol)
  "Pythagorean addition $\sqrt{a^2+b^2}$ is implemented by an elegant
iterative scheme due to Cleve Moler and Donald Morrison IBM Journal
of Research and Development 27 (1983), 577--581. It modifies |a| and |b|
in such a way that their Pythagorean sum remains invariant, while the
smaller argument decreases."
  (if (>= a b)
      (pythag b a tol)
      (if (<= a (if tol
                    tol
                    (expt 10.0 -16)))
          b
          (let* ((r (expt (/ a b) 2))
                 (s (/ r (+ 4 r)))
                 (b (+ b (* 2 s b)))
                 (a (* s a)))
            (pythag a b tol)))))

> (pythag 1.0 2.0 .000001)
2.236068
> (expt  5 .5)
2.236068

e com a operação inversa

(defun pythag (pm a b &optional tol)
  "Pythagorean addition $\sqrt{a^2\pm b^2}$ is implemented by an elegant
iterative scheme due to Cleve Moler and Donald Morrison IBM Journal
of Research and Development 27 (1983), 577--581. It modifies |a| and |b|
in such a way that their Pythagorean sum remains invariant, while the
smaller argument decreases. pm -> +1 gives sqrt(a^2+b^2); pm -> -1 gives sqrt(a^2-b^2);"
  (if (>= a b)
      (pythag pm b a tol)
      (if (<= a (if tol
                    tol
                    (expt 10.0 -16)))
          b
          (let* ((r (* pm (expt (/ a b) 2)))
                 (s (/ r (+ 4 r)))
                 (b (+ b (* 2 s b)))
                 (a (* s a)))
            (pythag pm a b tol)))))

Voltando ao METAFONT.

Palavras chave/keywords: Metafont, LaTeX, Pythagorean addition, lisp

Criado/Created: NaN

Última actualização/Last updated: 10-10-2022 [14:26]

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