chez-scmutils/scmutils/numerics/linear/eigen.scm

#| -*-Scheme-*-

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    Institute of Technology

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|#

;;;; Simple code to find eigenvalues and eigenvectors of small systems.


(define* (matrix->eigenvalues matrix #:optional expand-multiplicities?)
  (if (default-object? expand-multiplicities?)
      (set! expand-multiplicities? #t))
  (let ((x (generate-uninterned-symbol 'x)))
    (let ((poly
	   (m:determinant
	    (g:- matrix (g:* x (m:make-identity (m:dimension matrix)))))))
      (pcf:expression-> (expression poly)
			(lambda (pcf syms)
			  (poly->roots pcf expand-multiplicities?))))))


(define* (real-matrix->eigenvalues-eigenvectors matrix #:optional cutoff)
  (if (default-object? cutoff)
      (set! cutoff (* 1000 *machine-epsilon*)))
  (let ((eigenvalues (matrix->eigenvalues matrix #f)))
    (map (lambda (root)
	   (let ((m (car root))		; multiplicity
		 (x (cdr root)))
	     (svd (g:- matrix (g:* x (m:make-identity (m:dimension matrix))))
		  (lambda (U SIGMA V W)
		    (let ((n (vector-length W))
			  (minw
			   (* cutoff (apply max (map abs (vector->list W))))))
		      (cons x
			    (let lp ((i 0))
			      (cond ((fix:= i n) '())
				    ((< (abs (vector-ref W i)) cutoff)
				     (cons (m:nth-col V i)
					   (lp (fix:+ i 1))))
				    (else
				     (lp (fix:+ i 1)))))))))))
	 eigenvalues)))

(define (matrix->eigenvalues-eigenvectors matrix)
  (let ((eigenvalues (matrix->eigenvalues matrix #f)))
    (map (lambda (root)
	   (let ((m (car root))		; multiplicity
		 (x (cdr root)))
	     (cons x
		   (lu-null-space
		    (g:- matrix
			 (g:* x (m:make-identity (m:dimension matrix))))))))
	 eigenvalues)))

#|
;;; For example, this system has 3 distinct eigenvalues and
;;; corresponding eigenvectors:

(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(2 -1 0)
		     '(-1 2 -1)
		     '(0 -1 2))))
((.585786437626913 #(.5000000000000014 .7071067811865455 .5000000000000014))
 (2. #(-.7071067811865475 0. .7071067811865475))
 (3.414213562373087 #(.5000000000000014 -.7071067811865455 .5000000000000014)))


;;; For systems with multiplicity the multiple eigenvectors appear
;;; with the multiple root:

(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(0 0 0)
		     '(0 1 0)
		     '(0 0 1))))
((0. #(1 0 0)) (1. #(0 0 1) #(0 1 0)))

;;; A real example: the standard map at a hyperbolic point.

(pp (matrix->eigenvalues-eigenvectors
     (a^m_n->mmn
      (let ((K 1))
	((D (lambda (v)
	      (let ((x (ref v 0))
		    (y (ref v 1)))
		(let ((yp  (+ y (* K (sin x)))))
		  (up (+ x yp) yp)))))
	 (up 0 0))))))
((.38196601125010315 #(-.525731112119133 .8506508083520402))
 (2.6180339887498967 #(.8506508083520401 .5257311121191331)))


;;; Pavel's test

(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(1 1)
		     '(0 1))))
((1. #(1 0)))

(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(13 -4 2)
		     '(-4 13 -2)
		     '(2 -2 10))))
((9. #(-.4472135954999579 0. .8944271909999159)
     #(.7071067811865475 .7071067811865475 0.))
 (17.999999999999954
  #(.6666666666666701 -.6666666666666624 .33333333333333504)))

(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(1 2 3)
		     '(4 5 6)
		     '(7 8 9))))

((0.
  #(.4082482904638631 -.8164965809277261 .4082482904638631))
 (-1.1168439698070243
  #(-.7858302387420639 -.08675133925663416 .6123275602288135))
 (16.116843969807025
  #(.2319706872462861 .5253220933012344 .8186734993561811)))


(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(2 0 0 0)
		     '(1 2 0 0)
		     '(0 0 2 0)
		     '(0 0 0 2))))
((2. #(0. 0. 0. 1.) #(0. 0. 1. 0.) #(0. 1. 0. 0.)))


(pp (matrix->eigenvalues-eigenvectors
     (matrix-by-rows '(2 0 0 0)
		     '(1 2 0 0)
		     '(0 0 2 0)
		     '(0 0 1 2))))
((2. #(0. 0. 0. 1.) #(0. 1. 0. 0.)))

|#

#|
;;; Bug!

(define A
  (matrix-by-rows (list 8 0 3/16)
                  (list 0 8 3/16)
                  (list 3/16 3/16 8)))

(pp (matrix->eigenvalues-eigenvectors A))
((7.7348349570559485) (7.9999999999982006) (8.265165042945835))

#|
;; Eigenvects missing -- answer should have been (matlab sez)

((7.7348349570559485 #(1 0 1))
 (7.9999999999982006 #(.7071 1 -.7071))
 (8.265165042945835 #(.7071 -1 -.7071)))
|#


(lu-null-space
 (g:- A
      (g:* 8 (m:make-identity (m:dimension A)))))
;Value: (#(-.7071067811865475 .7071067811865475 0.))

(lu-null-space
 (g:- A
      (g:* 7.9999999999982006 (m:make-identity (m:dimension A)))))
;Value: ()


(real-matrix->eigenvalues-eigenvectors A)
;Value: ((7.7348349570559485) (7.9999999999982006) (8.265165042945835))

(pp (real-matrix->eigenvalues-eigenvectors A (* *machine-epsilon* 1e4)))
((7.7348349570559485 #(-.5 -.49999999999999994 .7071067811865475))
 (7.9999999999982006 #(.7071067811865475 -.7071067811865475 -3.465221769689776e-27))
 (8.265165042945835 #(.49999999999999994 .49999999999999967 .7071067811865474)))


;;; Indeed 
(/ (* A #(-1/2 -1/2 (/ (sqrt 2) 2))) 7.7348349570559485)
;Value: #(-.49999999999994155 -.49999999999994155 .707106781186465)


;;; And, indeed:

(set! heuristic-zero-test-bugger-factor 1e-11)

(pp (matrix->eigenvalues-eigenvectors A))
((7.7348349570559485
  #(-.5000000000008521 -.5000000000008521 .7071067811853424))
 (7.9999999999982006 #(-.7071067811865475 .7071067811865475 0.))
 (8.265165042945835
  #(.49999999999917066 .49999999999917066 .7071067811877204)))


;;; Problem is roots are not good enuf for null-space test


(pe (determinant (- A (* 'lam (m:identity-like A)))))
(+ (* -1 (expt lam 3)) (* 24 (expt lam 2)) (* -24567/128 lam) 8183/16)

((lambda (lam) 
   (+ (* -1 (expt lam 3)) (* 24 (expt lam 2)) (* -24567/128 lam) 8183/16))
 8)
;Value: 0

((lambda (lam) 
   (+ (* -1 (expt lam 3)) (* 24 (expt lam 2)) (* -24567/128 lam) 8183/16))
 7.9999999999982006)
;Value: -1.1368683772161603e-13

;;; ugh!

;;; but I can generally not do better:
(zbrent (lambda (lam) 
	  (+ (* -1 (expt lam 3)) (* 24 (expt lam 2)) (* -24567/128 lam) 8183/16))
	7.9999999999982006
	8.0003)
;Value: (#t 8.000000000001434 3)

(- 8.000000000001434 8)
;Value: 1.4335199693960021e-12
|#