/* sbart.f -- translated by f2c (version 20010821).
 * ------- and f2c-clean,v 1.9 2000/01/13
 *
 * According to the GAMFIT sources, this was derived from code by
 * Finbarr O'Sullivan.
 */

#include "modreg.h"
#include <math.h>
#include <Rmath.h>

/* sbart() : The cubic spline smoother
   -------
 Calls	 sgram	(sg0,sg1,sg2,sg3,knot,nk)
	 stxwx	(xs,ys,ws,n,knot,nk,xwy,hs0,hs1,hs2,hs3)
	 sslvrg (penalt,dofoff,xs,ys,ws,ssw,n,knot,nk,	coef,sz,lev,crit,icrit,
		 lambda, xwy, hs0,hs1,hs2,hs3, sg0,sg1,sg2,sg3,
		 abd,p1ip,p2ip,ld4,ldnk,ier)

 is itself called from	 qsbart() [./qsbart.f]	 which has only one work array

 Now allows to pass 'lambda' (not just 'spar') via spar[0] == *spar  iff  *isetup = 2
*/
void F77_SUB(sbart)
    (double *penalt, double *dofoff,
     double *xs, double *ys, double *ws, double *ssw,
     int *n, double *knot, int *nk, double *coef,
     double *sz, double *lev, double *crit,
     int *icrit, double *spar, int *ispar, int *iter,
     double *lspar, double *uspar, double *tol, double *eps, double *Ratio,
     int *isetup,
     double *xwy, double *hs0, double *hs1, double *hs2,
     double *hs3, double *sg0, double *sg1, double *sg2,
     double *sg3, double *abd, double *p1ip, double *p2ip,
     int *ld4, int *ldnk, int *ier)
{

/* A Cubic B-spline Smoothing routine.

   The algorithm minimises:

	(1/n) * sum ws(i)^2 * (ys(i)-sz(i))^2 + lambda* int ( s"(x) )^2 dx

   lambda is a function of the spar which is assumed to be between 0 and 1

 INPUT
 -----
   penalt	A penalty > 1 to be used in the gcv criterion
   dofoff	either `df.offset' for GCV or `df' (to be matched).
   n		number of data points
   ys(n)	vector of length n containing the observations
   ws(n)	vector containing the weights given to each data point
		NB: the code alters the values here.
   xs(n)	vector containing the ordinates of the observations
   ssw          `centered weighted sum of y^2'
   nk		number of b-spline coefficients to be estimated
		nk <= n+2
   knot(nk+4)	vector of knot points defining the cubic b-spline basis.
		To obtain full cubic smoothing splines one might
		have (provided the xs-values are strictly increasing)
   spar		penalised likelihood smoothing parameter
   ispar	indicating if spar is supplied (ispar=1) or to be estimated
   lspar, uspar lower and upper values for spar search;  0.,1. are good values
   tol, eps	used in Golden Search routine
   isetup	setup indicator initially 0 or 2 (if 'spar' is lambda)
	NB: this alters that, and it is a constant in the caller!
   icrit	indicator saying which cross validation score is to be computed
		0: none ;  1: GCV ;  2: CV ;  3: 'df matching'
   ld4		the leading dimension of abd (ie ld4=4)
   ldnk		the leading dimension of p2ip (not referenced)

 OUTPUT
 ------
   coef(nk)	vector of spline coefficients
   sz(n)	vector of smoothed z-values
   lev(n)	vector of leverages
   crit		either ordinary or generalized CV score
   spar         if ispar != 1
   lspar         == lambda (a function of spar and the design if(setup != 1)
   iter		number of iterations needed for spar search (if ispar != 1)
   ier		error indicator
		ier = 0 ___  everything fine
		ier = 1 ___  spar too small or too big
			problem in cholesky decomposition

 Working arrays/matrix
   xwy			X'Wy
   hs0,hs1,hs2,hs3	the non-zero diagonals of the X'WX matrix
   sg0,sg1,sg2,sg3	the non-zero diagonals of the Gram matrix SIGMA
   abd (ld4, nk)	[ X'WX + lambda*SIGMA ] = R'R in banded form; output = R
   p1ip(ld4, nk)	inner products between columns of R^{-1}
   p2ip(ldnk,nk)	all inner products between columns of R inverse
			where  R'R = [X'WX + lambda*SIGMA]  NOT REFERENCED
*/

// "Correct" ./sslvrg.f (line 129):   crit = 3 + (dofoff-df)**2
#define CRIT(FX) (*icrit == 3 ? FX - 3. : FX)
	/* cancellation in (3 + eps) - 3, but still...informative */

#define BIG_f (1e100)

    /* c_Gold is the squared inverse of the golden ratio */
    static const double c_Gold = 0.381966011250105151795413165634;
    /* == (3. - sqrt(5.)) / 2. */

    /* Local variables */
    static double ratio;/* must be static (not needed in R) */

    double a, b, d, e, p, q, r, u, v, w, x;
    double ax, fu, fv, fw, fx, bx, xm;
    double tol1, tol2;

    int i, maxit;
    bool Fparabol = false, tracing = (*ispar < 0), spar_is_lambda = false;

    /* unnecessary initializations to keep  -Wall happy */
    d = 0.; fu = 0.; u = 0.;
    // never computed if(spar_is_lambda)
    ratio = 1.;

/*  Compute SIGMA, X' W X, X' W z, trace ratio, s0, s1.

	SIGMA	-> sg0,sg1,sg2,sg3   -- via sgram() in ./sgram.f
	X' W X	-> hs0,hs1,hs2,hs3   \
	X' W Z	-> xwy               _\ via stxwx() in ./stxwx.f
*/

/* trevor fixed this 4/19/88
 * Note: sbart, i.e. stxwx() and sslvrg() {mostly, not always!}, use
 *	 the square of the weights; the following rectifies that */
    for (i = 0; i < *n; ++i)
	if (ws[i] > 0.)
	    ws[i] = sqrt(ws[i]);

    if (*isetup < 0)
	spar_is_lambda = TRUE;
    else if (*isetup != 1) { // 0 or 2
	/* SIGMA[i,j] := Int  B''(i,t) B''(j,t) dt  {B(k,.) = k-th B-spline} */
	F77_CALL(sgram)(sg0, sg1, sg2, sg3, knot, nk);
	F77_CALL(stxwx)(xs, ys, ws, n,
			knot, nk,
			xwy,
			hs0, hs1, hs2, hs3);
	spar_is_lambda = (*isetup == 2);
	if(!spar_is_lambda) {
	    /* Compute ratio :=  tr(X' W X) / tr(SIGMA) */
	    double t1 = 0., t2 = 0.;
	    for (i = 3 - 1; i < (*nk - 3); ++i) {
		t1 += hs0[i];
		t2 += sg0[i];
	    }
	    ratio = t1 / t2;
	}
	*isetup = 1;
    }
/*     Compute estimate */

// Compute SSPLINE(SPAR), assign result to *crit (and the auxil.variables)
#define SSPLINE_COMP(_SPAR_)						\
    *lspar = spar_is_lambda ? _SPAR_					\
                            : ratio * R_pow(16., (_SPAR_) * 6. - 2.);   \
    F77_CALL(sslvrg)(penalt, dofoff, xs, ys, ws, ssw, n,		\
		     knot, nk,						\
		     coef, sz, lev, crit, icrit, lspar, xwy,		\
		     hs0, hs1, hs2, hs3,				\
		     sg0, sg1, sg2, sg3, abd,				\
		     p1ip, p2ip, ld4, ldnk, ier)

    if (*ispar == 1) { /* Value of spar supplied */
	SSPLINE_COMP(*spar);
	/* got through check 2 */
	*Ratio = ratio;
	return;
    }

/* ELSE ---- spar not supplied --> compute it ! ---------------------------
 */
    ax = *lspar;
    bx = *uspar;

/*
       Use Forsythe Malcom and Moler routine to MINIMIZE criterion
       f denotes the value of the criterion

       an approximation	x  to the point where	f  attains a minimum  on
       the interval  (ax,bx)  is determined.


   INPUT

   ax	 left endpoint of initial interval
   bx	 right endpoint of initial interval
   f	 function subprogram which evaluates  f(x)  for any  x
	 in the interval  (ax,bx)
   tol	 desired length of the interval of uncertainty of the final
	 result ( >= 0 )

   OUTPUT

   fmin	 abcissa approximating the point where	f  attains a minimum
*/

/*
   The method used is a combination of  golden  section  search  and
   successive parabolic interpolation.	convergence is never much slower
   than	 that  for  a  fibonacci search.  if  f	 has a continuous second
   derivative which is positive at the minimum (which is not  at  ax  or
   bx),	 then  convergence  is	superlinear, and usually of the order of
   about  1.324....
	the function  f  is never evaluated at two points closer together
   than	 eps*abs(fmin) + (tol/3), where eps is	approximately the square
   root	 of  the  relative  machine  precision.	  if   f   is a unimodal
   function and the computed values of	 f   are  always  unimodal  when
   separated by at least  eps*abs(x) + (tol/3), then  fmin  approximates
   the abcissa of the global minimum of	 f  on the interval  ax,bx  with
   an error less than  3*eps*abs(fmin) + tol.  if   f	is not unimodal,
   then fmin may approximate a local, but perhaps non-global, minimum to
   the same accuracy.
	this function subprogram is a slightly modified	version	 of  the
   algol  60 procedure	localmin  given in richard brent, algorithms for
   minimization without derivatives, prentice - hall, inc. (1973).

   Double	 a,b,c,d,e,eps,xm,p,q,r,tol1,tol2,u,v,w
   Double	 fu,fv,fw,fx,x
*/

/*  eps is approximately the square root of the relative machine
    precision.

    -	 eps = 1e0
    - 10	 eps = eps/2e0
    -	 tol1 = 1e0 + eps
    -	 if (tol1 > 1e0) go to 10
    -	 eps = sqrt(eps)
    R Version <= 1.3.x had
    eps = .000244     ( = sqrt(5.954 e-8) )
     -- now eps is passed as argument
*/

    /* initialization */

    maxit = *iter;
    *iter = 0;
    a = ax;
    b = bx;
    v = a + c_Gold * (b - a);
    w = v;
    x = v;
    e = 0.;
    SSPLINE_COMP(x);
    fx = *crit;
    fv = fx;
    fw = fx;

/* main loop
   --------- */
    while(*ier == 0) { /* L20: */
	xm = (a + b) * .5;
	tol1 = *eps * fabs(x) + *tol / 3.;
	tol2 = tol1 * 2.;
	++(*iter);

	if(tracing) {
	    if(*iter == 1) {/* write header */
		Rprintf("sbart (ratio = %15.8g) iterations;"
			" initial tol1 = %12.6e :\n"
			"%11s %14s  %9s %11s  Kind %11s %12s\n%s\n",
			ratio, tol1, "spar",
			((*icrit == 1) ? "GCV" :
			 (*icrit == 2) ?  "CV" :
			 (*icrit == 3) ?"(df0-df)^2" :
			 /*else (should not happen) */"?f?"),
			"b - a", "e", "NEW lspar", "crit",
			" ---------------------------------------"
			"----------------------------------------");
	    }
	    Rprintf("%11.8f %14.9g %9.4e %11.5g", x, CRIT(fx), b - a, e);
	    Fparabol = FALSE;
	}

	/* Check the (somewhat peculiar) stopping criterion: note that
	   the RHS is negative as long as the interval [a,b] is not small:*/
	if (fabs(x - xm) <= tol2 - (b - a) * .5 || *iter > maxit)
	    goto L_End;


/* is golden-section necessary */

	if (fabs(e) <= tol1 ||
	    /*  if had Inf then go to golden-section */
	    fx >= BIG_f || fv >= BIG_f || fw >= BIG_f) goto L_GoldenSect;

/* Fit Parabola */
	if(tracing) { Rprintf(" FP"); Fparabol = TRUE; }

	r = (x - w) * (fx - fv);
	q = (x - v) * (fx - fw);
	p = (x - v) * q - (x - w) * r;
	q = (q - r) * 2.;
	if (q > 0.)
	    p = -p;
	q = fabs(q);
	r = e;
	e = d;

/* is parabola acceptable?  Otherwise do golden-section */

	if (fabs(p) >= fabs(.5 * q * r) ||
	    q == 0.)
	    /* above line added by BDR;
	     * [the abs(.) >= abs() = 0 should have branched..]
	     * in FTN: COMMON above ensures q is NOT a register variable */

	    goto L_GoldenSect;

	if (p <= q * (a - x) ||
	    p >= q * (b - x))			goto L_GoldenSect;



/* Parabolic Interpolation step */

	if(tracing) Rprintf(" PI ");
	d = p / q;
	if(!R_FINITE(d))
	    REprintf(" !FIN(d:=p/q): ier=%d, (v,w, p,q)= %g, %g, %g, %g\n",
		     *ier, v,w, p, q);
	u = x + d;

	/* f must not be evaluated too close to ax or bx */
	if (u - a < tol2 ||
	    b - u < tol2)	d = fsign(tol1, xm - x);

	goto L50;
	/*------*/

    L_GoldenSect: /* a golden-section step */

	if(tracing) Rprintf(" GS%s ", Fparabol ? "" : " --");

	if (x >= xm)    e = a - x;
	else/* x < xm*/ e = b - x;
	d = c_Gold * e;


    L50:
	u = x + ((fabs(d) >= tol1) ? d : fsign(tol1, d));
	/*  tol1 check : f must not be evaluated too close to x */

	SSPLINE_COMP(u);
	fu = *crit;
	if(tracing) Rprintf("%11g %12g\n", *lspar, CRIT(fu));
	if(!R_FINITE(fu)) {
	    REprintf("spar-finding: non-finite value %g; using BIG value\n", fu);
	    fu = 2. * BIG_f;
	}

/*  update  a, b, v, w, and x */

	if (fu <= fx) {
	    if (u >= x) a = x; else b = x;

	    v = w; fv = fw;
	    w = x; fw = fx;
	    x = u; fx = fu;
	}
	else {
	    if (u < x)  a = u; else b = u;

	    if (fu <= fw || w == x) {		        /* L70: */
		v = w; fv = fw;
		w = u; fw = fu;
	    } else if (fu <= fv || v == x || v == w) {	/* L80: */
		v = u; fv = fu;
	    }
	}
    }/* end main loop -- goto L20; */

 L_End:
    if(tracing) Rprintf("  >>> %12g %12g\n", *lspar, CRIT(fx));
    *Ratio = ratio;
    *spar = x;
    *crit = fx;
    return;
} /* sbart */
