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+;;; calc-nlfit.el --- nonlinear curve fitting for Calc
+
+;; Copyright (C) 2007 Free Software Foundation, Inc.
+
+;; Maintainer: Jay Belanger <jay.p.belanger@gmail.com>
+
+;; This file is part of GNU Emacs.
+
+;; GNU Emacs is free software; you can redistribute it and/or modify
+;; it under the terms of the GNU General Public License as published by
+;; the Free Software Foundation; either version 3, or (at your option)
+;; any later version.
+
+;; GNU Emacs is distributed in the hope that it will be useful,
+;; but WITHOUT ANY WARRANTY; without even the implied warranty of
+;; MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
+;; GNU General Public License for more details.
+
+;; You should have received a copy of the GNU General Public License
+;; along with GNU Emacs; see the file COPYING.  If not, write to the
+;; Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor,
+;; Boston, MA 02110-1301, USA.
+
+;;; Commentary:
+
+;; This code uses the Levenberg-Marquardt method, as described in
+;; _Numerical Analysis_ by H. R. Schwarz, to fit data to 
+;; nonlinear curves.  Currently, the only the following curves are
+;; supported:
+;; The logistic S curve, y=a/(1+exp(b*(t-c)))
+;;   Here, y is usually interpreted as the population of some
+;;   quantity at time t.  So we will think of the data as consisting
+;;   of quantities q0, q1, ..., qn and their respective times
+;;   t0, t1, ..., tn.
+
+;; The logistic bell curve, y=A*exp(B*(t-C))/(1+exp(B*(t-C)))^2
+;;   Note that this is the derivative of the formula for the S curve.
+;;   We get A=-a*b, B=b and C=c.  Here, y is interpreted as the rate 
+;;   of growth of a population at time t.  So we will think of the 
+;;   data as consisting of rates p0, p1, ..., pn and their 
+;;   respective times t0, t1, ..., tn.
+
+;; The Hubbert Linearization, y/x=A*(1-x/B)
+;;   Here, y is thought of as the rate of growth of a population
+;;   and x represents the actual population.  This is essentially 
+;;   the differential equation describing the actual population.
+
+;; The Levenberg-Marquardt method is an iterative process: it takes
+;; an initial guess for the parameters and refines them.  To get an
+;; initial guess for the parameters, we'll use a method described by
+;; Luis de Sousa in "Hubbert's Peak Mathematics".  The idea is that
+;; given quantities Q and the corresponding rates P, they should
+;; satisfy P/Q= mQ+a.  We can use the parameter a for an
+;; approximation for the parameter a in the S curve, and
+;; approximations for b and c are found using least squares on the
+;; linearization log((a/y)-1) = log(bb) + cc*t of
+;; y=a/(1+bb*exp(cc*t)), which is equivalent to the above s curve
+;; formula, and then tranlating it to b and c.  From this, we can
+;; also get approximations for the bell curve parameters.
+
+;;; Code:
+
+(require 'calc-arith)
+
+(defun math-nlfit-least-squares (xdata ydata &optional sdata sigmas)
+  "Return the parameters A and B for the best least squares fit y=a+bx."
+  (let* ((n (length xdata))
+         (s2data (if sdata 
+                     (mapcar 'calcFunc-sqr sdata)
+                  (make-list n 1)))
+         (S (if sdata 0 n))
+         (Sx 0)
+         (Sy 0)
+         (Sxx 0)
+         (Sxy 0)
+         D)
+    (while xdata
+      (let ((x (car xdata))
+            (y (car ydata))
+            (s (car s2data)))
+        (setq Sx  (math-add Sx (if s (math-div x s) x)))
+        (setq Sy  (math-add Sy (if s (math-div y s) y)))
+        (setq Sxx (math-add Sxx (if s (math-div (math-mul x x) s)
+                                  (math-mul x x))))
+        (setq Sxy (math-add Sxy (if s (math-div (math-mul x y) s)
+                                  (math-mul x y))))
+        (if sdata
+            (setq S (math-add S (math-div 1 s)))))
+      (setq xdata (cdr xdata))
+      (setq ydata (cdr ydata))
+      (setq s2data (cdr s2data)))
+    (setq D (math-sub (math-mul S Sxx) (math-mul Sx Sx)))
+    (let ((A (math-div (math-sub (math-mul Sxx Sy) (math-mul Sx Sxy)) D))
+          (B (math-div (math-sub (math-mul S Sxy) (math-mul Sx Sy)) D)))
+      (if sigmas
+          (let ((C11 (math-div Sxx D))
+                (C12 (math-neg (math-div Sx D)))
+                (C22 (math-div S D)))
+            (list (list 'sdev A (calcFunc-sqrt C11))
+                  (list 'sdev B (calcFunc-sqrt C22))
+                  (list 'vec
+                        (list 'vec C11 C12)
+                        (list 'vec C12 C22))))
+        (list A B)))))
+
+;;; The methods described by de Sousa require the cumulative data qdata
+;;; and the rates pdata.  We will assume that we are given either
+;;; qdata and the corresponding times tdata, or pdata and the corresponding
+;;; tdata.  The following two functions will find pdata or qdata, 
+;;; given the other..
+
+;;; First, given two lists; one of values q0, q1, ..., qn and one of 
+;;; corresponding times t0, t1, ..., tn; return a list 
+;;; p0, p1, ..., pn of the rates of  change of the qi with respect to t.
+;;; p0 is the right hand derivative (q1 - q0)/(t1 - t0).
+;;; pn is the left hand derivative (qn - q(n-1))/(tn - t(n-1)).
+;;; The other pis are the averages of the two:
+;;;      (1/2)((qi - q(i-1))/(ti - t(i-1)) + (q(i+1) - qi)/(t(i+1) - ti)).
+
+(defun math-nlfit-get-rates-from-cumul (tdata qdata)
+  (let ((pdata (list
+                (math-div 
+                 (math-sub (nth 1 qdata)
+                           (nth 0 qdata))
+                 (math-sub (nth 1 tdata)
+                           (nth 0 tdata))))))
+    (while (> (length qdata) 2)
+      (setq pdata
+            (cons
+             (math-mul
+              '(float 5 -1)
+              (math-add
+               (math-div
+                (math-sub (nth 2 qdata)
+                          (nth 1 qdata))
+                (math-sub (nth 2 tdata)
+                          (nth 1 tdata)))
+               (math-div
+                (math-sub (nth 1 qdata)
+                          (nth 0 qdata))
+                (math-sub (nth 1 tdata)
+                          (nth 0 tdata)))))
+             pdata))
+      (setq qdata (cdr qdata)))
+    (setq pdata
+          (cons
+           (math-div
+            (math-sub (nth 1 qdata)
+                      (nth 0 qdata))
+            (math-sub (nth 1 tdata)
+                      (nth 0 tdata)))
+           pdata))
+    (reverse pdata)))
+
+;;; Next, given two lists -- one of rates p0, p1, ..., pn and one of 
+;;; corresponding times t0, t1, ..., tn -- and an initial values q0,
+;;;  return a list q0, q1, ..., qn of the cumulative values.
+;;; q0 is the initial value given.
+;;; For i>0, qi is computed using the trapezoid rule:
+;;;     qi = q(i-1) + (1/2)(pi + p(i-1))(ti - t(i-1))
+
+(defun math-nlfit-get-cumul-from-rates (tdata pdata q0)
+  (let* ((qdata (list q0)))
+    (while (cdr pdata)
+      (setq qdata
+            (cons
+             (math-add (car qdata)
+                       (math-mul
+                        (math-mul 
+                         '(float 5 -1)
+                         (math-add (nth 1 pdata) (nth 0 pdata)))
+                        (math-sub (nth 1 tdata)
+                                  (nth 0 tdata))))
+             qdata))
+      (setq pdata (cdr pdata))
+      (setq tdata (cdr tdata)))
+    (reverse qdata)))
+
+;;; Given the qdata, pdata and tdata, find the parameters
+;;; a, b and c that fit q = a/(1+b*exp(c*t)).
+;;; a is found using the method described by de Sousa.  
+;;; b and c are found using least squares on the linearization
+;;; log((a/q)-1) = log(b) + c*t
+;;; In some cases (where the logistic curve may well be the wrong
+;;; model), the computed a will be less than or equal to the maximum
+;;; value of q in qdata; in which case the above linearization won't work.
+;;; In this case, a will be replaced by a number slightly above 
+;;; the maximum value of q.
+
+(defun math-nlfit-find-qmax (qdata pdata tdata)
+  (let* ((ratios (mapcar* 'math-div pdata qdata))
+         (lsdata (math-nlfit-least-squares ratios tdata))
+         (qmax (math-max-list (car qdata) (cdr qdata)))
+         (a (math-neg (math-div (nth 1 lsdata) (nth 0 lsdata)))))
+    (if (math-lessp a qmax)
+        (math-add '(float 5 -1) qmax)
+      a)))
+
+(defun math-nlfit-find-logistic-parameters (qdata pdata tdata)
+  (let* ((a (math-nlfit-find-qmax qdata pdata tdata))
+         (newqdata
+          (mapcar (lambda (q) (calcFunc-ln (math-sub (math-div a q) 1)))
+                  qdata))
+         (bandc (math-nlfit-least-squares tdata newqdata)))
+    (list
+     a
+     (calcFunc-exp (nth 0 bandc))
+     (nth 1 bandc))))
+
+;;; Next, given the pdata and tdata, we can find the qdata if we know q0.
+;;; We first try to find q0, using the fact that when p takes on its largest
+;;; value, q is half of its maximum value.  So we'll find the maximum value
+;;; of q given various q0, and use bisection to approximate the correct q0.
+
+;;; First, given pdata and tdata, find what half of qmax would be if q0=0.
+
+(defun math-nlfit-find-qmaxhalf (pdata tdata)
+  (let ((pmax (math-max-list (car pdata) (cdr pdata)))
+        (qmh 0))
+    (while (math-lessp (car pdata) pmax)
+      (setq qmh
+            (math-add qmh
+                      (math-mul
+                       (math-mul 
+                        '(float 5 -1)
+                        (math-add (nth 1 pdata) (nth 0 pdata)))
+                       (math-sub (nth 1 tdata)
+                                 (nth 0 tdata)))))
+      (setq pdata (cdr pdata))
+      (setq tdata (cdr tdata)))
+    qmh))
+
+;;; Next, given pdata and tdata, approximate q0.
+
+(defun math-nlfit-find-q0 (pdata tdata)
+  (let* ((qhalf (math-nlfit-find-qmaxhalf pdata tdata))
+         (q0 (math-mul 2 qhalf))
+         (qdata (math-nlfit-get-cumul-from-rates tdata pdata q0)))
+    (while (math-lessp (math-nlfit-find-qmax 
+                        (mapcar
+                         (lambda (q) (math-add q0 q))
+                         qdata)
+                        pdata tdata)
+                       (math-mul
+                        '(float 5 -1)
+                        (math-add
+                         q0
+                         qhalf)))
+      (setq q0 (math-add q0 qhalf)))
+    (let* ((qmin (math-sub q0 qhalf))
+           (qmax q0)
+           (qt (math-nlfit-find-qmax
+                (mapcar
+                 (lambda (q) (math-add q0 q))
+                 qdata)
+                pdata tdata))
+           (i 0))
+      (while (< i 10)
+        (setq q0 (math-mul '(float 5 -1) (math-add qmin qmax)))
+        (if (math-lessp 
+             (math-nlfit-find-qmax
+              (mapcar
+               (lambda (q) (math-add q0 q))
+               qdata)
+              pdata tdata)
+             (math-mul '(float 5 -1) (math-add qhalf q0)))
+            (setq qmin q0)
+          (setq qmax q0))
+        (setq i (1+ i)))
+      (math-mul '(float 5 -1) (math-add qmin qmax)))))
+
+;;; To improve the approximations to the parameters, we can use 
+;;; Marquardt method as described in Schwarz's book.
+
+;;; Small numbers used in the Givens algorithm
+(defvar math-nlfit-delta '(float 1 -8))
+
+(defvar math-nlfit-epsilon '(float 1 -5))
+
+;;; Maximum number of iterations
+(defvar math-nlfit-max-its 100)
+
+;;; Next, we need some functions for dealing with vectors and
+;;; matrices.  For convenience, we'll work with Emacs lists
+;;; as vectors, rather than Calc's vectors.
+
+(defun math-nlfit-set-elt (vec i x)
+  (setcar (nthcdr (1- i) vec) x))
+
+(defun math-nlfit-get-elt (vec i)
+  (nth (1- i) vec))
+
+(defun math-nlfit-make-matrix (i j)
+  (let ((row (make-list j 0))
+        (mat nil)
+        (k 0))
+    (while (< k i)
+      (setq mat (cons (copy-list row) mat))
+      (setq k (1+ k)))
+    mat))
+
+(defun math-nlfit-set-matx-elt (mat i j x)
+  (setcar (nthcdr (1- j) (nth (1- i) mat)) x))
+
+(defun math-nlfit-get-matx-elt (mat i j)
+  (nth (1- j) (nth (1- i) mat)))
+
+;;; For solving the linearized system.
+;;; (The Givens method, from Schwarz.)
+
+(defun math-nlfit-givens (C d)
+  (let* ((C (copy-tree C))
+         (d (copy-tree d))
+         (n (length (car C)))
+         (N (length C))
+         (j 1)
+         (r (make-list N 0))
+         (x (make-list N 0))
+         w
+         gamma
+         sigma
+         rho)
+    (while (<= j n)
+      (let ((i (1+ j)))
+        (while (<= i N)
+          (let ((cij (math-nlfit-get-matx-elt C i j))
+                (cjj (math-nlfit-get-matx-elt C j j)))
+            (when (not (math-equal 0 cij))
+                (if (math-lessp (calcFunc-abs cjj) 
+                                (math-mul math-nlfit-delta (calcFunc-abs cij)))
+                    (setq w (math-neg cij)
+                          gamma 0
+                          sigma 1
+                          rho 1)
+                  (setq w (math-mul
+                           (calcFunc-sign cjj)
+                           (calcFunc-sqrt 
+                            (math-add
+                             (math-mul cjj cjj)
+                             (math-mul cij cij))))
+                        gamma (math-div cjj w)
+                        sigma (math-neg (math-div cij w)))
+                  (if (math-lessp (calcFunc-abs sigma) gamma)
+                      (setq rho sigma)
+                    (setq rho (math-div (calcFunc-sign sigma) gamma))))
+              (setq cjj w
+                    cij rho)
+              (math-nlfit-set-matx-elt C j j w)
+              (math-nlfit-set-matx-elt C i j rho)
+              (let ((k (1+ j)))
+                (while (<= k n)       
+                  (let* ((cjk (math-nlfit-get-matx-elt C j k))
+                         (cik (math-nlfit-get-matx-elt C i k))
+                         (h (math-sub 
+                             (math-mul gamma cjk) (math-mul sigma cik))))
+                    (setq cik (math-add
+                               (math-mul sigma cjk)
+                               (math-mul gamma cik)))
+                    (setq cjk h)
+                    (math-nlfit-set-matx-elt C i k cik)
+                    (math-nlfit-set-matx-elt C j k cjk)
+                    (setq k (1+ k)))))
+              (let* ((di (math-nlfit-get-elt d i))
+                     (dj (math-nlfit-get-elt d j))
+                     (h (math-sub
+                         (math-mul gamma dj)
+                         (math-mul sigma di))))
+                (setq di (math-add
+                          (math-mul sigma dj)
+                          (math-mul gamma di)))
+                (setq dj h)
+                (math-nlfit-set-elt d i di)
+                (math-nlfit-set-elt d j dj))))
+          (setq i (1+ i))))
+      (setq j (1+ j)))
+    (let ((i n))
+      (while (>= i 1)
+        (math-nlfit-set-elt r i 0)
+        (setq s (math-nlfit-get-elt d i))
+        (let ((k (1+ i)))
+          (while (<= k n)
+            (setq s (math-add s (math-mul (math-nlfit-get-matx-elt C i k)
+                            (math-nlfit-get-elt x k))))
+            (setq k (1+ k))))
+        (math-nlfit-set-elt x i 
+                            (math-neg 
+                             (math-div s 
+                                       (math-nlfit-get-matx-elt C i i))))
+        (setq i (1- i))))
+    (let ((i (1+ n)))
+      (while (<= i N)
+        (math-nlfit-set-elt r i (math-nlfit-get-elt d i))
+        (setq i (1+ i))))
+    (let ((j n))
+      (while (>= j 1)
+        (let ((i N))
+          (while (>= i (1+ j))
+            (setq rho (math-nlfit-get-matx-elt C i j))
+            (if (math-equal rho 1)
+                (setq gamma 0
+                      sigma 1)
+              (if (math-lessp (calcFunc-abs rho) 1)
+                  (setq sigma rho
+                        gamma (calcFunc-sqrt 
+                               (math-sub 1 (math-mul sigma sigma))))
+                (setq gamma (math-div 1 (calcFunc-abs rho))
+                      sigma (math-mul (calcFunc-sign rho)
+                                       (calcFunc-sqrt
+                                        (math-sub 1 (math-mul gamma gamma)))))))
+            (let ((ri (math-nlfit-get-elt r i))
+                  (rj (math-nlfit-get-elt r j)))
+              (setq h (math-add (math-mul gamma rj)
+                         (math-mul sigma ri)))
+              (setq ri (math-sub
+                        (math-mul gamma ri)
+                        (math-mul sigma rj)))
+              (setq rj h)
+              (math-nlfit-set-elt r i ri)
+              (math-nlfit-set-elt r j rj))
+            (setq i (1- i))))
+        (setq j (1- j))))
+
+    x))
+
+(defun math-nlfit-jacobian (grad xlist parms &optional slist)
+  (let ((j nil))
+    (while xlist 
+      (let ((row (apply grad (car xlist) parms)))
+        (setq j
+              (cons
+               (if slist
+                   (mapcar (lambda (x) (math-div x (car slist))) row)
+                 row)
+               j)))
+      (setq slist (cdr slist))
+      (setq xlist (cdr xlist)))
+    (reverse j)))
+
+(defun math-nlfit-make-ident (l n)
+  (let ((m (math-nlfit-make-matrix n n))
+        (i 1))
+    (while (<= i n)
+      (math-nlfit-set-matx-elt m i i l)
+      (setq i (1+ i)))
+    m))
+
+(defun math-nlfit-chi-sq (xlist ylist parms fn &optional slist)
+  (let ((cs 0))
+    (while xlist
+      (let ((c
+             (math-sub
+              (apply fn (car xlist) parms)
+              (car ylist))))
+        (if slist
+            (setq c (math-div c (car slist))))
+        (setq cs
+              (math-add cs
+                 (math-mul c c))))
+      (setq xlist (cdr xlist))
+      (setq ylist (cdr ylist))
+      (setq slist (cdr slist)))
+    cs))
+
+(defun math-nlfit-init-lambda (C)
+  (let ((l 0)
+        (n (length (car C)))
+        (N (length C)))
+    (while C
+      (let ((row (car C)))
+        (while row
+          (setq l (math-add l (math-mul (car row) (car row))))
+          (setq row (cdr row))))
+      (setq C (cdr C)))
+    (calcFunc-sqrt (math-div l (math-mul n N)))))
+
+(defun math-nlfit-make-Ctilda (C l)
+  (let* ((n (length (car C)))
+         (bot (math-nlfit-make-ident l n)))
+    (append C bot)))
+
+(defun math-nlfit-make-d (fn xdata ydata parms &optional sdata)
+  (let ((d nil))
+    (while xdata
+      (setq d (cons
+               (let ((dd (math-sub (apply fn (car xdata) parms)
+                                   (car ydata))))
+                 (if sdata (math-div dd (car sdata)) dd))
+               d))
+      (setq xdata (cdr xdata))
+      (setq ydata (cdr ydata))
+      (setq sdata (cdr sdata)))
+    (reverse d)))
+    
+(defun math-nlfit-make-dtilda (d n)
+  (append d (make-list n 0)))
+
+(defun math-nlfit-fit (xlist ylist parms fn grad &optional slist)
+  (let*
+      ((C (math-nlfit-jacobian grad xlist parms slist))
+       (d (math-nlfit-make-d fn xlist ylist parms slist))
+       (chisq (math-nlfit-chi-sq xlist ylist parms fn slist))
+       (lambda (math-nlfit-init-lambda C))
+       (really-done nil)
+       (iters 0))
+    (while (and
+            (not really-done)
+            (< iters math-nlfit-max-its))
+      (setq iters (1+ iters))
+      (let ((done nil))
+        (while (not done)
+          (let* ((Ctilda (math-nlfit-make-Ctilda C lambda))
+                 (dtilda (math-nlfit-make-dtilda d (length (car C))))
+                 (zeta (math-nlfit-givens Ctilda dtilda))
+                 (newparms (mapcar* 'math-add (copy-tree parms) zeta))
+                 (newchisq (math-nlfit-chi-sq xlist ylist newparms fn slist)))
+            (if (math-lessp newchisq chisq)
+                (progn
+                  (if (math-lessp 
+                       (math-div 
+                        (math-sub chisq newchisq) newchisq) math-nlfit-epsilon)
+                      (setq really-done t))
+                  (setq lambda (math-div lambda 10))
+                  (setq chisq newchisq)
+                  (setq parms newparms)
+                  (setq done t))
+              (setq lambda (math-mul lambda 10)))))
+        (setq C (math-nlfit-jacobian grad xlist parms slist))
+        (setq d (math-nlfit-make-d fn xlist ylist parms slist))))
+    (list chisq parms)))
+
+;;; The functions that describe our models, and their gradients.
+
+(defun math-nlfit-s-logistic-fn (x a b c)
+  (math-div a (math-add 1 (math-mul b (calcFunc-exp (math-mul c x))))))
+
+(defun math-nlfit-s-logistic-grad (x a b c)
+  (let* ((ep (calcFunc-exp (math-mul c x)))
+         (d (math-add 1 (math-mul b ep)))
+         (d2 (math-mul d d)))
+    (list
+     (math-div 1 d)
+     (math-neg (math-div (math-mul a ep) d2))
+     (math-neg (math-div (math-mul a (math-mul b (math-mul x ep))) d2)))))
+
+(defun math-nlfit-b-logistic-fn (x a c d)
+  (let ((ex (calcFunc-exp (math-mul c (math-sub x d)))))
+    (math-div
+     (math-mul a ex)
+     (math-sqr 
+      (math-add
+       1 ex)))))
+
+(defun math-nlfit-b-logistic-grad (x a c d)
+  (let* ((ex (calcFunc-exp (math-mul c (math-sub x d))))
+        (ex1 (math-add 1 ex))
+        (xd (math-sub x d)))
+    (list
+     (math-div
+      ex
+      (math-sqr ex1))
+     (math-sub
+      (math-div
+       (math-mul a (math-mul xd ex))
+       (math-sqr ex1))
+      (math-div
+       (math-mul 2 (math-mul a (math-mul xd (math-sqr ex))))
+       (math-pow ex1 3)))
+     (math-sub
+      (math-div
+       (math-mul 2 (math-mul a (math-mul c (math-sqr ex))))
+       (math-pow ex1 3))
+      (math-div
+       (math-mul a (math-mul c ex))
+       (math-sqr ex1))))))
+
+;;; Functions to get the final covariance matrix and the sdevs
+
+(defun math-nlfit-find-covar (grad xlist pparms)
+  (let ((j nil))
+    (while xlist 
+      (setq j (cons (cons 'vec (apply grad (car xlist) pparms)) j))
+      (setq xlist (cdr xlist)))
+    (setq j (cons 'vec (reverse j)))
+    (setq j
+          (math-mul
+           (calcFunc-trn j) j))
+    (calcFunc-inv j)))
+
+(defun math-nlfit-get-sigmas (grad xlist pparms chisq)
+  (let* ((sgs nil)
+         (covar (math-nlfit-find-covar grad xlist pparms))
+         (n (1- (length covar)))
+         (N (length xlist))
+         (i 1))
+    (when (> N n)
+      (while (<= i n)
+        (setq sgs (cons (calcFunc-sqrt (nth i (nth i covar))) sgs))
+        (setq i (1+ i)))
+      (setq sgs (reverse sgs)))
+    (list sgs covar)))
+   
+;;; Now the Calc functions
+
+(defun math-nlfit-s-logistic-params (xdata ydata)
+  (let ((pdata (math-nlfit-get-rates-from-cumul xdata ydata)))
+    (math-nlfit-find-logistic-parameters ydata pdata xdata)))
+
+(defun math-nlfit-b-logistic-params (xdata ydata)
+  (let* ((q0 (math-nlfit-find-q0 ydata xdata))
+         (qdata (math-nlfit-get-cumul-from-rates xdata ydata q0))
+         (abc (math-nlfit-find-logistic-parameters qdata ydata xdata))
+         (B (nth 1 abc))
+         (C (nth 2 abc))
+         (A (math-neg
+             (math-mul
+              (nth 0 abc)
+              (math-mul B C))))
+         (D (math-neg (math-div (calcFunc-ln B) C)))
+         (A (math-div A B)))
+    (list A C D)))
+
+;;; Some functions to turn the parameter lists and variables
+;;; into the appropriate functions.
+
+(defun math-nlfit-s-logistic-solnexpr (pms var)
+  (let ((a (nth 0 pms))
+        (b (nth 1 pms))
+        (c (nth 2 pms)))
+    (list '/ a
+            (list '+
+                  1
+             (list '*
+                   b
+                   (calcFunc-exp
+                    (list '*
+                          c
+                          var)))))))
+
+(defun math-nlfit-b-logistic-solnexpr (pms var)
+  (let ((a (nth 0 pms))
+        (c (nth 1 pms))
+        (d (nth 2 pms)))
+    (list '/
+          (list '*
+                a
+                (calcFunc-exp
+                 (list '*
+                       c
+                       (list '- var d))))
+          (list '^
+                (list '+
+                      1
+                      (calcFunc-exp
+                       (list '*
+                             c
+                             (list '- var d))))
+                2))))
+
+(defun math-nlfit-enter-result (n prefix vals)
+  (setq calc-aborted-prefix prefix)
+  (calc-pop-push-record-list n prefix vals)
+  (calc-handle-whys))
+
+(defun math-nlfit-fit-curve (fn grad solnexpr initparms &optional sdv)
+  (calc-slow-wrapper
+   (let* ((sdevv (or (eq sdv 'calcFunc-efit) (eq sdv 'calcFunc-xfit)))
+          (calc-display-working-message nil)
+          (data (calc-top 1))
+          (xdata (cdr (car (cdr data))))
+          (ydata (cdr (car (cdr (cdr data)))))
+          (sdata (if (math-contains-sdev-p ydata)
+                     (mapcar (lambda (x) (math-get-sdev x t)) ydata)
+                   nil))
+          (ydata (mapcar (lambda (x) (math-get-value x)) ydata))
+          (zzz (progn (setq j1 xdata j2 ydata j3 sdata) 1))
+
+          (calc-curve-varnames nil)
+          (calc-curve-coefnames nil)
+          (calc-curve-nvars 1)
+          (fitvars (calc-get-fit-variables 1 3))
+          (var (nth 1 calc-curve-varnames))
+          (parms (cdr calc-curve-coefnames))
+          (parmguess
+           (funcall initparms xdata ydata))
+          (fit (math-nlfit-fit xdata ydata parmguess fn grad sdata))
+          (finalparms (nth 1 fit))
+          (sigmacovar 
+           (if sdevv
+               (math-nlfit-get-sigmas grad xdata finalparms (nth 0 fit))))
+          (sigmas 
+           (if sdevv
+               (nth 0 sigmacovar)))
+          (finalparms 
+           (if sigmas
+               (mapcar* (lambda (x y) (list 'sdev x y)) finalparms sigmas)
+             finalparms))
+          (soln (funcall solnexpr finalparms var)))
+     (let ((calc-fit-to-trail t)
+           (traillist nil))
+       (while parms
+         (setq traillist (cons (list 'calcFunc-eq (car parms) (car finalparms))
+                               traillist))
+         (setq finalparms (cdr finalparms))
+         (setq parms (cdr parms)))
+       (setq traillist (calc-normalize (cons 'vec (nreverse traillist))))
+       (cond ((eq sdv 'calcFunc-efit)
+              (math-nlfit-enter-result 1 "efit" soln))
+             ((eq sdv 'calcFunc-xfit)
+              (let (sln)
+                (setq sln
+                      (list 'vec 
+                            soln 
+                            traillist
+                            (nth 1 sigmacovar)
+                            '(vec)
+                            (nth 0 fit)
+                            (let ((n (length xdata))
+                                  (m (length finalparms)))
+                              (if (and sdata (> n m))
+                                  (calcFunc-utpc (nth 0 fit) 
+                                                 (- n m))
+                                '(var nan var-nan)))))
+                (math-nlfit-enter-result 1 "xfit" sln)))
+             (t
+              (math-nlfit-enter-result 1 "fit" soln)))
+       (calc-record traillist "parm")))))
+
+(defun calc-fit-s-shaped-logistic-curve (arg)
+  (interactive "P")
+  (math-nlfit-fit-curve 'math-nlfit-s-logistic-fn
+                        'math-nlfit-s-logistic-grad
+                        'math-nlfit-s-logistic-solnexpr
+                        'math-nlfit-s-logistic-params
+                        arg))
+
+(defun calc-fit-bell-shaped-logistic-curve (arg)
+  (interactive "P")
+  (math-nlfit-fit-curve 'math-nlfit-b-logistic-fn
+                        'math-nlfit-b-logistic-grad
+                        'math-nlfit-b-logistic-solnexpr
+                        'math-nlfit-b-logistic-params
+                        arg))
+
+(defun calc-fit-hubbert-linear-curve (&optional sdv)
+  (calc-slow-wrapper
+   (let* ((sdevv (or (eq sdv 'calcFunc-efit) (eq sdv 'calcFunc-xfit)))
+          (calc-display-working-message nil)
+          (data (calc-top 1))
+          (qdata (cdr (car (cdr data))))
+          (pdata (cdr (car (cdr (cdr data)))))
+          (sdata (if (math-contains-sdev-p pdata)
+                     (mapcar (lambda (x) (math-get-sdev x t)) pdata)
+                   nil))
+          (pdata (mapcar (lambda (x) (math-get-value x)) pdata))
+          (poverqdata (mapcar* 'math-div pdata qdata))
+          (parmvals (math-nlfit-least-squares qdata poverqdata sdata sdevv))
+          (finalparms (list (nth 0 parmvals)
+                            (math-neg
+                             (math-div (nth 0 parmvals)
+                                       (nth 1 parmvals)))))
+          (calc-curve-varnames nil)
+          (calc-curve-coefnames nil)
+          (calc-curve-nvars 1)
+          (fitvars (calc-get-fit-variables 1 2))
+          (var (nth 1 calc-curve-varnames))
+          (parms (cdr calc-curve-coefnames))
+          (soln (list '* (nth 0 finalparms)
+                      (list '- 1
+                            (list '/ var (nth 1 finalparms))))))
+     (let ((calc-fit-to-trail t)
+           (traillist nil))
+       (setq traillist
+             (list 'vec
+                   (list 'calcFunc-eq (nth 0 parms) (nth 0 finalparms))
+                   (list 'calcFunc-eq (nth 1 parms) (nth 1 finalparms))))
+       (cond ((eq sdv 'calcFunc-efit)
+              (math-nlfit-enter-result 1 "efit" soln))
+             ((eq sdv 'calcFunc-xfit)
+              (let (sln
+                    (chisq
+                     (math-nlfit-chi-sq
+                      qdata poverqdata
+                      (list (nth 1 (nth 0 finalparms))
+                            (nth 1 (nth 1 finalparms)))
+                      (lambda (x a b)
+                        (math-mul a 
+                                  (math-sub
+                                   1
+                                   (math-div x b))))
+                      sdata)))
+                (setq sln
+                      (list 'vec 
+                            soln 
+                            traillist
+                            (nth 2 parmvals)
+                            (list
+                             'vec
+                             '(calcFunc-fitdummy 1)
+                             (list 'calcFunc-neg
+                                   (list '/
+                                         '(calcFunc-fitdummy 1)
+                                         '(calcFunc-fitdummy 2))))
+                            chisq
+                            (let ((n (length qdata)))
+                              (if (and sdata (> n 2))
+                                  (calcFunc-utpc 
+                                   chisq
+                                   (- n 2))
+                                '(var nan var-nan)))))
+                (math-nlfit-enter-result 1 "xfit" sln)))
+             (t
+              (math-nlfit-enter-result 1 "fit" soln)))
+       (calc-record traillist "parm")))))
+
+(provide 'calc-nlfit)