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integer_relation.lisp
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;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; Integer relations package
;;
;; Copyright: 2006, Andrej Vodopivec
;; Licence: GPL
;;
;; Integer relation package implements pslq algorithm to find integer
;; relations between bigfloat numbers.
;;
;; Description:
;;
;; For a given vector x of floating point numbers we want to find a
;; vector of integers m such that m.x=0 (in given precision).
;;
;; Uses PLSQ algorithm:
;; D.H.Bailey: Integer Relation Detection and Lattice Reduction.
;;
;;;;;;;;;;;;;;;;;;
(in-package :maxima)
(macsyma-module integer_relation)
($put '$integer_relation 1.0 '$version)
(defmfun $integer_relation (l)
(if (not ($listp l))
(merror "Argument to `integer_relation' not a list."))
(let* ((n ($length l)) (l (cdr l)) (a (make-array n)) ($float2bf t))
(loop for i from 0 to (1- n) do
(let ((elm (car l)))
(if ($floatnump elm)
(setf (aref a i) elm)
(progn
(setq elm ($bfloat elm))
(if (not ($freeof '$%i elm))
(return-from $integer_relation nil))
(if (not ($bfloatp elm))
(merror "Non-bfloat element in argument to `integer_relation':~%~M.~%" elm))
(setf (aref a i) elm))))
(setq l (cdr l)))
(let ((ans (integer-relations a n)))
(if ans
`((mlist simp) ,@ans)
nil))))
(defmvar $pslq_precision nil)
(defmvar $pslq_treshold nil)
(defmvar $pslq_depth nil)
(defmvar $pslq_status 0)
(defmvar $pslq_fail_norm nil)
(defun mabs (x)
(if (mlsp x 0) (m- x) x))
(defun nearest-integer (x)
(let ((nx ($entier x)))
(if (mlsp 0.5 (m- x nx))
(1+ nx)
nx)))
(defun integer-relations (x n)
(let ((A (make-array `(,n ,n) :initial-element 0))
(B (make-array `(,n ,n) :initial-element 0))
(H (make-array `(,n ,(1- n)) :initial-element 0))
(gamma (sqrt (/ 4 3)))
(s (make-array `(,n) :initial-element 0))
(y (make-array `(,n) :initial-element 0))
($pslq_precision (if $pslq_precision $pslq_precision (power 10 (- $fpprec 2))))
($pslq_treshold (if $pslq_treshold $pslq_treshold (power 10 (- 2 $fpprec))))
($pslq_depth (if $pslq_depth $pslq_depth (* 20 n)))
(tt))
;; Initialize A, B and s
(loop for k from 0 to (1- n) do
(setf (aref A k k) 1)
(setf (aref B k k) 1)
(loop for j from k to (1- n) do
(setf (aref s k) (m+ (aref s k) (power (aref x j) 2))))
(setf (aref s k) ($sqrt (aref s k))))
(setf tt (aref s 0))
;; Initialize y, normalize s
(loop for k from 0 to (1- n) do
(setf (aref y k) (m// (aref x k) tt))
(setf (aref s k) (m// (aref s k) tt)))
;; Initialize H
(loop for i from 0 to (1- n) do
(if (< i (- n 1))
(setf (aref H i i) (m// (aref s (1+ i)) (aref s i))))
(loop for j from 0 to (1- i) do
(setf (aref H i j) (m- (m// (m* (aref y i) (aref y j))
(m* (aref s j) (aref s (1+ j))))))))
;; Perform reduction on H, update A, B, y
(loop for i from 1 to (- n 1) do
(loop for j from (1- i) downto 0 do
(setq tt (nearest-integer (m// (aref H i j) (aref H j j))))
(setf (aref y j) (m+ (aref y j) (m* tt (aref y i))))
(loop for k from 0 to j do
(setf (aref H i k) (m- (aref H i k) (m* tt (aref H j k)))))
(loop for k from 0 to (1- n) do
(setf (aref A i k) (m- (aref A i k) (m* tt (aref A j k))))
(setf (aref B k j) (m+ (aref B k j) (m* tt (aref B k i)))))))
(do ((r 1 (1+ r))) ((= r $pslq_depth))
(let ((m 0) (mm 0) (s 1))
;; Find maximal value in H
(loop for i from 0 to (m- n 2) do
(setq s (* gamma s))
(if (mlsp mm (m* s (mabs (aref H i i))))
(progn
(setf m i)
(setf mm (m* (expt gamma i) (mabs (aref H i i)))))))
;; Swap entries in y
(rotatef (aref y m) (aref y (1+ m)))
;; Swap rows in A and columns in B
(loop for i from 0 to (- n 1) do
(rotatef (aref A m i) (aref A (1+ m) i))
(rotatef (aref B i m) (aref B i (1+ m))))
;; Swap rows in H
(loop for i from 0 to (- n 2) do
(rotatef (aref H m i) (aref H (1+ m) i)))
;; Update H
(if (< m (- n 2))
(let* ((t0 ($sqrt (m+ (power (aref H m m) 2) (power (aref H m (1+ m)) 2))))
(t1 (m// (aref H m m) t0))
(t2 (m// (aref H m (1+ m)) t0))
(t3) (t4))
(loop for i from m to (1- n) do
(setq t3 (aref H i m))
(setq t4 (aref H i (1+ m)))
(setf (aref H i m) (m+ (m* t1 t3) (m* t2 t4)))
(setf (aref H i (1+ m)) (m- (m* t1 t4) (m* t2 t3))))))
;; Block reduction on H
(loop for i from (1+ m) to (1- n) do
(loop for j from (min (1- i) (1+ m)) downto 0 do
(setq tt (nearest-integer (m// (aref H i j) (aref H j j))))
(setf (aref y j) (m+ (aref y j) (m* tt (aref y i))))
(loop for k from 0 to j do
(setf (aref H i k) (m- (aref H i k) (m* tt (aref H j k)))))
(loop for k from 0 to (1- n) do
(setf (aref A i k) (m- (aref A i k) (m* tt (aref A j k))))
(setf (aref B k j) (m+ (aref B k j) (m* tt (aref B k i)))))))
;; Find the bound M
(let ((maxNorm 0))
(loop for j from 0 to (1- n) do
(let ((absHj 0))
(loop for i from 0 to (- n 2) do
(if (mlsp absHj (mabs (aref H j i)))
(setq absHj (aref H j i))))
(if (mlsp maxNorm absHj)
(setq maxNorm absHj))))
(setq $pslq_fail_norm (m// 1 maxNorm))
;; Check to see if we have a relation
(loop for j from 0 to (1- n) do
(if (mlsp (mabs (aref y j)) $pslq_treshold)
(progn
(let ((ans ()))
(loop for i from 0 to (1- n) do
(setq ans (append ans `(,(aref B i j)))))
(setq $pslq_status 1)
(return-from integer-relations ans)))))
;; Check to see if we exsausted the precision
(loop for i from 0 to (1- n) do
(loop for j from 0 to (1- n) do
(if (mlsp $pslq_precision (mabs (aref A i j)))
(progn
(setq $pslq_status 2)
(return-from integer-relations nil)))))
)))
(setq $pslq_status 3)
(return-from integer-relations nil) ))