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monotonic-spline.lisp
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monotonic-spline.lisp
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(in-package #:interpolation)
(defclass <monotonic-spline-data> ()
((xs-vector :accessor xs-vector :initarg :xs)
(ys-vector :accessor ys-vector :initarg :ys)
(ms-vector :accessor ms-vector :initarg :ms)
(prev-k :accessor prev-k :initform nil)
))
(defmethod monotonic-spline ((xs vector) (ys vector))
(let* ((nel (array-total-size xs))
(nelm1 (1- nel)))
(assert (>= (array-total-size ys) nel))
(let ((dels (make-array nelm1))
(ms (make-array nel)))
;; compute secants
(labels ((dely/delx (x1 x2 y1 y2)
(/ (- y2 y1) (- x2 x1))))
(um:nlet iter ((ix 0)
(xprev (aref xs 0))
(yprev (aref ys 0)))
(let ((ixp1 (1+ ix)))
(when (< ixp1 nel)
(let ((xnext (aref xs ixp1))
(ynext (aref ys ixp1)))
(setf (aref dels ix) (dely/delx xprev xnext yprev ynext))
(go-iter ixp1 xnext ynext)) ))))
;; compute slopes
(labels ((avg (a b)
(* 0.5d0 (+ a b))))
(um:nlet iter ((ix 1)
(prev (setf (aref ms 0) (aref dels 0))))
(if (< ix nelm1)
;; midpoint slopes use average of secants
(let ((next (aref dels ix)))
(setf (aref ms ix) (avg prev next))
(go-iter (1+ ix) next))
;; endpoint slopes use one-sided diffs
(setf (aref ms ix) prev))))
;; fixup slopes to preserve monotonicity
(um:nlet iter ((ix 0)
(msprev (aref ms 0)))
(let ((ixp1 (1+ ix)))
(when (< ixp1 nel)
(let ((delk (aref dels ix)))
(if (zerop delk)
(progn
(setf (aref ms ix) 0d0
(aref ms ixp1) 0d0)
(go-iter ixp1 0d0))
;; else
(let* ((msixp1 (aref ms ixp1))
(alpha (/ msprev delk))
(beta (/ msixp1 delk))
(sumsq (+ (* alpha alpha) (* beta beta))))
(when (> sumsq 9d0)
;; (print "adjusting to 3-circle")
(let ((tau (/ 3d0 (sqrt sumsq))))
(setf (aref ms ix) (* tau msprev)
msixp1 (* tau msixp1)
(aref ms ixp1) msixp1)))
(go-iter ixp1 msixp1)) )))))
(make-instance '<monotonic-spline-data>
:xs xs
:ys ys
:ms ms) )))
(defmethod monotonic-splint ((mspl <monotonic-spline-data>) (x real)
&optional (prev-index (prev-k mspl)))
(with-accessors ((xs xs-vector )
(ys ys-vector )
(ms ms-vector)
(prev-k prev-k )) mspl
(let* ((klo (locate xs x prev-index))
(khi (1+ klo)))
(setf prev-k klo)
(when (>= khi (array-total-size xs))
;; check for off right-end of table
(decf klo)
(decf khi))
(let* ((xlower (aref xs klo))
(xupper (aref xs khi))
(ylower (aref ys klo))
(yupper (aref ys khi))
(mlower (aref ms klo))
(mupper (aref ms khi))
(h (- xupper xlower))
(tee (progn
(assert (not (zerop h))) ;; xs must be distinct
(/ (- x xlower) h))))
;; cubic Hermite spline interpolation in x [0,1)
(labels ((h00 (x)
(let ((xx (- 1d0 x)))
(* (+ 1d0 x x) xx xx)))
(h10 (x)
(let ((xx (- 1d0 x)))
(* x xx xx)))
(h01 (x)
(* x x (- 3d0 x x)))
(h11 (x)
(* x x (- x 1d0))))
(+ (* ylower (h00 tee))
(* h mlower (h10 tee))
(* yupper (h01 tee))
(* h mupper (h11 tee))) )))))
#|
(defun sub-array (arr start nel)
(make-array nel
:element-type (array-element-type arr)
:displaced-to arr
:displaced-index-offset start))
(defun tand (x)
(tan (/ (* pi x) 180)))
(defun der (fn x)
(let ((dx (* 0.001 x)))
(/ (- (funcall fn (+ x dx))
(funcall fn (- x dx)))
(* 2 dx))))
(defun tstplot (fn dom npts &key
(title "Interpolation Error")
(xtitle "X")
(ytitle "Error")
(yrange '(-11 11)))
(let* ((x0 (first dom))
(xn (second dom))
(xs (vm:vectorwise ((x (vm:framp (1- npts))))
(+ (* (1+ x) (/ (- xn x0) npts)) x0)))
(ys (vm:vectorwise ((x xs))
(funcall fn x)))
(spl (spline xs ys :natural :natural))
(spl2 (spline xs ys (der fn (bref xs 1)) (der fn (bref xs (1- npts)))))
(splm (monotonic-spline xs ys))
(pxs (vm:vectorwise ((x (vm:framp 500)))
(+ (* x (/ (- xn x0) 500)) x0))))
(plt:plot 'xx pxs (ms-vector splm) :clear t :symbol :circle :plot-joined t)
(plt:plot 'plt pxs (map 'vector (lambda (x) (- (splint spl x)
(funcall fn x)))
pxs)
:thick 2
:clear t
:title title
:xtitle xtitle
:ytitle ytitle
:yrange yrange
:legend "natural cubic spline")
(plt:plot 'plt2 pxs (map 'vector (lambda (x)
(funcall fn x))
pxs)
:clear t
:thick 2)
(plt:plot 'plt2 pxs (map 'vector (lambda (x)
(splint spl x))
pxs)
:color :red
:thick 2
:legend "natural cubic spline")
(plt:plot 'plt2 pxs (map 'vector (lambda (x)
(splint spl2 x))
pxs)
:color :magenta
:thick 2
:legend "der match cubic spline")
(plt:plot 'plt pxs (map 'vector (lambda (x) (- (splint spl2 x)
(funcall fn x)))
pxs)
:color :cyan
:thick 2
:legend "der match cubic spline")
(labels ((int (xs ys x n)
(let* ((k (locate-subtable xs x (1+ n)))
(xs (sub-array xs k (1+ n)))
(ys (sub-array ys k (1+ n))))
(ratint xs ys x)
)))
(plt:plot 'plt pxs (map 'vector (lambda (x) (- (int xs ys x 3)
(funcall fn x)))
pxs)
:color :red
:thick 2
:legend "3rd order rational"))
(labels ((int (xs ys x n)
(let* ((k (locate-subtable xs x (1+ n)))
(xs (sub-array xs k (1+ n)))
(ys (sub-array ys k (1+ n))))
(polint xs ys x))))
(plt:plot 'plt pxs (map 'vector (lambda (x) (- (int xs ys x 3)
(funcall fn x)))
pxs)
:color :blue
:thick 2
:legend "3rd order polynomial"))
(plt:plot 'plt pxs (map 'vector (lambda (x) (- (monotonic-splint splm x)
(funcall fn x)))
pxs)
:color :green
:thick 2
:legend "monotonic spline")
))
(tstplot #'tand '(80 90) 10
:title "Interpolation Error for Tan(x) [tbl:(81, 82, .. 89)]"
:xtitle "X [deg]"
:yrange '(-4 4))
(tstplot #'(lambda (x)
(expt (/ (- 1d0 x)) 2))
'(0 1)
10
:yrange '(-15 15)
:title "Interpolation Error for 1/(1-x)^2 [tbl:(0, 0.1, ... 0.9)]")
(tstplot #'(lambda (x)
(expt (/ (- 1d0 x)) 1))
'(0 0.99)
10
:yrange '(-0.5 0.5)
:title "Interpolation Error for 1/(1-x) [tbl:(0.1, 0.2, ... 0.9)]")
(tstplot #'(lambda (x)
(exp x))
'(0 1)
10
:yrange '(-0.0015 0.0015)
:title "Interpolation Error for Exp(x) [tbl:(0.1, 0.2, .. 0.9)]")
(tstplot #'(lambda (x)
(expt x 1))
'(0 1)
10
:yrange '(-0.0001 0.0001)
:title "Interpolation Error for x [tbl:(0.1, 0.2, .. 0.9)]")
(tstplot #'(lambda (x)
(expt x 2))
'(0 1)
10
:yrange '(-0.0015 0.0015)
:title "Interpolation Error for x^2 [tbl:(0.1, 0.2, .. 0.9)]")
(tstplot #'(lambda (x)
(expt x 3))
'(0 1)
10
:yrange '(-0.004 0.004)
:title "Interpolation Error for x^3 [tbl:(0.1, 0.2, .. 0.9)]")
(tstplot #'(lambda (x)
(expt x 4))
'(0 1)
10
:yrange '(-0.005 0.005)
:title "Interpolation Error for x^4 [tbl:(0.1, 0.2, .. 0.9)]")
(tstplot #'(lambda (x)
(expt x 5))
'(0 1)
10
:yrange '(-0.01 0.01)
:title "Interpolation Error for x^5 [tbl:(0.1, 0.2, .. 0.9)]")
|#