c 24/8/99 c This is the original fgam.f file c It needs to be compiled and loaded into R in order to smooth. c All of this is automatically in Splus subroutine vbsplvd ( t, k, x, left, a, dbiatx, nderiv ) implicit double precision(a-h,o-z) calls bsplvb calculates value and deriv.s of all b-splines which do not vanish at x c c****** i n p u t ****** c t the knot array, of length left+k (at least) c k the order of the b-splines to be evaluated c x the point at which these values are sought c left an integer indicating the left endpoint of the interval of c interest. the k b-splines whose support contains the interval c (t(left), t(left+1)) c are to be considered. c a s s u m p t i o n - - - it is assumed that c t(left) .lt. t(left+1) c division by zero will result otherwise (in b s p l v b ). c also, the output is as advertised only if c t(left) .le. x .le. t(left+1) . c nderiv an integer indicating that values of b-splines and their c derivatives up to but not including the nderiv-th are asked c for. ( nderiv is replaced internally by the integer in (1,k) c closest to it.) c c****** w o r k a r e a ****** c a an array of order (k,k), to contain b-coeff.s of the derivat- c ives of a certain order of the k b-splines of interest. c c****** o u t p u t ****** c dbiatx an array of order (k,nderiv). its entry (i,m) contains c value of (m-1)st derivative of (left-k+i)-th b-spline of c order k for knot sequence t , i=m,...,k; m=1,...,nderiv. c c****** m e t h o d ****** c values at x of all the relevant b-splines of order k,k-1,..., c k+1-nderiv are generated via bsplvb and stored temporarily c in dbiatx . then, the b-coeffs of the required derivatives of the c b-splines of interest are generated by differencing, each from the c preceding one of lower order, and combined with the values of b- c splines of corresponding order in dbiatx to produce the desired c values. c integer k,left,nderiv, i,ideriv,il,j,jlow,jp1mid,kp1,kp1mm * ,ldummy,m,mhigh double precision a(k,k),dbiatx(k,nderiv),t(1),x double precision factor,fkp1mm,sum mhigh = max0(min0(nderiv,k),1) c mhigh is usually equal to nderiv. kp1 = k+1 call bsplvb(t,kp1-mhigh,1,x,left,dbiatx) if (mhigh .eq. 1) go to 99 c the first column of dbiatx always contains the b-spline values c for the current order. these are stored in column k+1-current c order before bsplvb is called to put values for the next c higher order on top of it. ideriv = mhigh do 15 m=2,mhigh jp1mid = 1 do 11 j=ideriv,k dbiatx(j,ideriv) = dbiatx(jp1mid,1) 11 jp1mid = jp1mid + 1 ideriv = ideriv - 1 call bsplvb(t,kp1-ideriv,2,x,left,dbiatx) 15 continue c c at this point, b(left-k+i, k+1-j)(x) is in dbiatx(i,j) for c i=j,...,k and j=1,...,mhigh ('=' nderiv). in particular, the c first column of dbiatx is already in final form. to obtain cor- c responding derivatives of b-splines in subsequent columns, gene- c rate their b-repr. by differencing, then evaluate at x. c jlow = 1 do 20 i=1,k do 19 j=jlow,k 19 a(j,i) = 0d0 jlow = i 20 a(i,i) = 1d0 c at this point, a(.,j) contains the b-coeffs for the j-th of the c k b-splines of interest here. c do 40 m=2,mhigh kp1mm = kp1 - m fkp1mm = dble(kp1mm) il = left i = k c c for j=1,...,k, construct b-coeffs of (m-1)st derivative of c b-splines from those for preceding derivative by differencing c and store again in a(.,j) . the fact that a(i,j) = 0 for c i .lt. j is used.sed. do 25 ldummy=1,kp1mm factor = fkp1mm/(t(il+kp1mm) - t(il)) c the assumption that t(left).lt.t(left+1) makes denominator c in factor nonzero. do 24 j=1,i 24 a(i,j) = (a(i,j) - a(i-1,j))*factor il = il - 1 25 i = i - 1 c c for i=1,...,k, combine b-coeffs a(.,i) with b-spline values c stored in dbiatx(.,m) to get value of (m-1)st derivative of c i-th b-spline (of interest here) at x , and store in c dbiatx(i,m). storage of this value over the value of a b-spline c of order m there is safe since the remaining b-spline derivat- c ive of the same order do not use this value due to the fact c that a(j,i) = 0 for j .lt. i . 30 do 40 i=1,k sum = 0. jlow = max0(i,m) do 35 j=jlow,k 35 sum = a(j,i)*dbiatx(j,m) + sum 40 dbiatx(i,m) = sum 99 return end subroutine bsplvb ( t, jhigh, index, x, left, biatx ) implicit double precision(a-h,o-z) calculates the value of all possibly nonzero b-splines at x of order c c jout = dmax( jhigh , (j+1)*(index-1) ) c c with knot sequence t . c c****** i n p u t ****** c t.....knot sequence, of length left + jout , assumed to be nonde- c creasing. a s s u m p t i o n . . . . c t(left) .lt. t(left + 1) . c d i v i s i o n b y z e r o will result if t(left) = t(left+1) c jhigh, c index.....integers which determine the order jout = max(jhigh, c (j+1)*(index-1)) of the b-splines whose values at x are to c be returned. index is used to avoid recalculations when seve- c ral columns of the triangular array of b-spline values are nee- c ded (e.g., in bvalue or in vbsplvd ). precisely, c if index = 1 , c the calculation starts from scratch and the entire triangular c array of b-spline values of orders 1,2,...,jhigh is generated c order by order , i.e., column by column . c if index = 2 , c only the b-spline values of order j+1, j+2, ..., jout are ge- c nerated, the assumption being that biatx , j , deltal , deltar c are, on entry, as they were on exit at the previous call. c in particular, if jhigh = 0, then jout = j+1, i.e., just c the next column of b-spline values is generated. c c w a r n i n g . . . the restriction jout .le. jmax (= 20) is im- c posed arbitrarily by the dimension statement for deltal and c deltar below, but is n o w h e r e c h e c k e d for . c c x.....the point at which the b-splines are to be evaluated. c left.....an integer chosen (usually) so that c t(left) .le. x .le. t(left+1) . c c****** o u t p u t ****** c biatx.....array of length jout , with biatx(i) containing the val- c ue at x of the polynomial of order jout which agrees with c the b-spline b(left-jout+i,jout,t) on the interval (t(left), c t(left+1)) . c c****** m e t h o d ****** c the recurrence relation c c x - t(i) t(i+j+1) - x c b(i,j+1)(x) = -----------b(i,j)(x) + ---------------b(i+1,j)(x) c t(i+j)-t(i) t(i+j+1)-t(i+1) c c is used (repeatedly) to generate the (j+1)-vector b(left-j,j+1)(x), c ...,b(left,j+1)(x) from the j-vector b(left-j+1,j)(x),..., c b(left,j)(x), storing the new values in biatx over the old. the c facts that c b(i,1) = 1 if t(i) .le. x .lt. t(i+1) c and that c b(i,j)(x) = 0 unless t(i) .le. x .lt. t(i+j) c are used. the particular organization of the calculations follows al- c gorithm (8) in chapter x of the text. c parameter(jmax = 20) integer index,jhigh,left, i,j,jp1 double precision biatx(jhigh),t(1),x, deltal(jmax) double precision deltar(jmax),saved,term c dimension biatx(jout), t(left+jout) current fortran standard makes it impossible to specify the length of c t and of biatx precisely without the introduction of otherwise c superfluous additional arguments. data j/1/ c save j,deltal,deltar (valid in fortran 77) c go to (10,20), index 10 j = 1 biatx(1) = 1d0 if (j .ge. jhigh) go to 99 c 20 jp1 = j + 1 deltar(j) = t(left+j) - x deltal(j) = x - t(left+1-j) saved = 0d0 do 26 i=1,j term = biatx(i)/(deltar(i) + deltal(jp1-i)) biatx(i) = saved + deltar(i)*term 26 saved = deltal(jp1-i)*term biatx(jp1) = saved j = jp1 if (j .lt. jhigh) go to 20 c 99 return end double precision function bvalue ( t, bcoef, n, k, x, jderiv ) implicit double precision(a-h,o-z) calls vinterv c calculates value at x of jderiv-th derivative of spline from b-repr. c the spline is taken to be continuous from the right. c c****** i n p u t ****** c t, bcoef, n, k......forms the b-representation of the spline f to c be evaluated. specifically, c t.....knot sequence, of length n+k, assumed nondecreasing. c bcoef.....b-coefficient sequence, of length n . c n.....length of bcoef and dimension of s(k,t), c a s s u m e d positive . c k.....order of the spline . c c w a r n i n g . . . the restriction k .le. kmax (=20) is imposed c arbitrarily by the dimension statement for aj, dm, dm below, c but is n o w h e r e c h e c k e d for. c c x.....the point at which to evaluate . c jderiv.....integer giving the order of the derivative to be evaluated c a s s u m e d to be zero or positive. c c****** o u t p u t ****** c bvalue.....the value of the (jderiv)-th derivative of f at x . c c****** m e t h o d ****** c the nontrivial knot interval (t(i),t(i+1)) containing x is lo- c cated with the aid of vinterv . the k b-coeffs of f relevant for c this interval are then obtained from bcoef (or taken to be zero if c not explicitly available) and are then differenced jderiv times to c obtain the b-coeffs of (d**jderiv)f relevant for that interval. c precisely, with j = jderiv, we have from x.(12) of the text that c c (d**j)f = sum ( bcoef(.,j)*b(.,k-j,t) ) c c where c / bcoef(.), , j .eq. 0 c / c bcoef(.,j) = / bcoef(.,j-1) - bcoef(.-1,j-1) c / ----------------------------- , j .gt. 0 c / (t(.+k-j) - t(.))/(k-j) c c then, we use repeatedly the fact that c c sum ( a(.)*b(.,m,t)(x) ) = sum ( a(.,x)*b(.,m-1,t)(x) ) c with c (x - t(.))*a(.) + (t(.+m-1) - x)*a(.-1) c a(.,x) = --------------------------------------- c (x - t(.)) + (t(.+m-1) - x) c c to write (d**j)f(x) eventually as a linear combination of b-splines c of order 1 , and the coefficient for b(i,1,t)(x) must then c be the desired number (d**j)f(x). (see x.(17)-(19) of text). c parameter(kmax = 20) integer jderiv,k,n, i,ilo,imk,j,jc,jcmin,jcmax,jj,km1,mflag,nmi double precision bcoef(n),t(1),x double precision aj(kmax),dm(kmax),dp(kmax),fkmj c dimension t(n+k) current fortran standard makes it impossible to specify the length of c t precisely without the introduction of otherwise superfluous c additional arguments. bvalue = 0. if (jderiv .ge. k) go to 99 c c *** find i s.t. 1 .le. i .lt. n+k and t(i) .lt. t(i+1) and c t(i) .le. x .lt. t(i+1) . if no such i can be found, x lies c outside the support of the spline f and bvalue = 0. c (the asymmetry in this choice of i makes f rightcontinuous) if( (x.ne.t(n+1)) .or. (t(n+1).ne.t(n+k)) ) go to 700 i = n go to 701 700 call vinterv ( t, n+k, x, i, mflag ) if (mflag .ne. 0) go to 99 701 continue c *** if k = 1 (and jderiv = 0), bvalue = bcoef(i). km1 = k - 1 if (km1 .gt. 0) go to 1 bvalue = bcoef(i) go to 99 c c *** store the k b-spline coefficients relevant for the knot interval c (t(i),t(i+1)) in aj(1),...,aj(k) and compute dm(j) = x - t(i+1-j), c dp(j) = t(i+j) - x, j=1,...,k-1 . set any of the aj not obtainable c from input to zero. set any t.s not obtainable equal to t(1) or c to t(n+k) appropriately. 1 jcmin = 1 imk = i - k if (imk .ge. 0) go to 8 jcmin = 1 - imk do 5 j=1,i 5 dm(j) = x - t(i+1-j) do 6 j=i,km1 aj(k-j) = 0. 6 dm(j) = dm(i) go to 10 8 do 9 j=1,km1 9 dm(j) = x - t(i+1-j) c 10 jcmax = k nmi = n - i if (nmi .ge. 0) go to 18 jcmax = k + nmi do 15 j=1,jcmax 15 dp(j) = t(i+j) - x do 16 j=jcmax,km1 aj(j+1) = 0. 16 dp(j) = dp(jcmax) go to 20 18 do 19 j=1,km1 19 dp(j) = t(i+j) - x c 20 do 21 jc=jcmin,jcmax 21 aj(jc) = bcoef(imk + jc) c c *** difference the coefficients jderiv times. if (jderiv .eq. 0) go to 30 do 23 j=1,jderiv kmj = k-j fkmj = dble(kmj) ilo = kmj do 23 jj=1,kmj aj(jj) = ((aj(jj+1) - aj(jj))/(dm(ilo) + dp(jj)))*fkmj 23 ilo = ilo - 1 c c *** compute value at x in (t(i),t(i+1)) of jderiv-th derivative, c given its relevant b-spline coeffs in aj(1),...,aj(k-jderiv). 30 if (jderiv .eq. km1) go to 39 jdrvp1 = jderiv + 1 do 33 j=jdrvp1,km1 kmj = k-j ilo = kmj do 33 jj=1,kmj aj(jj) = (aj(jj+1)*dm(ilo) + aj(jj)*dp(jj))/(dm(ilo)+dp(jj)) 33 ilo = ilo - 1 39 bvalue = aj(1) c 99 return end subroutine vinterv ( xt, lxt, x, left, mflag ) implicit double precision(a-h,o-z) computes left = max( i ; 1 .le. i .le. lxt .and. xt(i) .le. x ) . c c****** i n p u t ****** c xt.....a double precision sequence, of length lxt , assumed to be nondecreasing c lxt.....number of terms in the sequence xt . c x.....the point whose location with respect to the sequence xt is c to be determined. c c****** o u t p u t ****** c left, mflag.....both integers, whose value is c c 1 -1 if x .lt. xt(1) c i 0 if xt(i) .le. x .lt. xt(i+1) c lxt 1 if xt(lxt) .le. x c c in particular, mflag = 0 is the 'usual' case. mflag .ne. 0 c indicates that x lies outside the halfopen interval c xt(1) .le. y .lt. xt(lxt) . the asymmetric treatment of the c interval is due to the decision to make all pp functions cont- c inuous from the right. c c****** m e t h o d ****** c the program is designed to be efficient in the common situation that c it is called repeatedly, with x taken from an increasing or decrea- c sing sequence. this will happen, e.g., when a pp function is to be c graphed. the first guess for left is therefore taken to be the val- c ue returned at the previous call and stored in the l o c a l varia- c ble ilo . a first check ascertains that ilo .lt. lxt (this is nec- c essary since the present call may have nothing to do with the previ- c ous call). then, if xt(ilo) .le. x .lt. xt(ilo+1), we set left = c ilo and are done after just three comparisons. c otherwise, we repeatedly double the difference istep = ihi - ilo c while also moving ilo and ihi in the direction of x , until c xt(ilo) .le. x .lt. xt(ihi) , c after which we use bisection to get, in addition, ilo+1 = ihi . c left = ilo is then returned. c integer left,lxt,mflag, ihi,ilo,istep,middle double precision x,xt(lxt) data ilo /1/ c save ilo (a valid fortran statement in the new 1977 standard) ihi = ilo + 1 if (ihi .lt. lxt) go to 20 if (x .ge. xt(lxt)) go to 110 if (lxt .le. 1) go to 90 ilo = lxt - 1 ihi = lxt c 20 if (x .ge. xt(ihi)) go to 40 if (x .ge. xt(ilo)) go to 100 c c **** now x .lt. xt(ilo) . decrease ilo to capture x . 30 istep = 1 31 ihi = ilo ilo = ihi - istep if (ilo .le. 1) go to 35 if (x .ge. xt(ilo)) go to 50 istep = istep*2 go to 31 35 ilo = 1 if (x .lt. xt(1)) go to 90 go to 50 c **** now x .ge. xt(ihi) . increase ihi to capture x . 40 istep = 1 41 ilo = ihi ihi = ilo + istep if (ihi .ge. lxt) go to 45 if (x .lt. xt(ihi)) go to 50 istep = istep*2 go to 41 45 if (x .ge. xt(lxt)) go to 110 ihi = lxt c c **** now xt(ilo) .le. x .lt. xt(ihi) . narrow the interval. 50 middle = (ilo + ihi)/2 if (middle .eq. ilo) go to 100 c note. it is assumed that middle = ilo in case ihi = ilo+1 . if (x .lt. xt(middle)) go to 53 ilo = middle go to 50 53 ihi = middle go to 50 c**** set output and return. 90 mflag = -1 left = 1 return 100 mflag = 0 left = ilo return 110 mflag = 1 left = lxt return end c ===================================================================== c These two subroutines, dpbfa8 and dpbsl8, are called by sslvrg. c Note: a rational cholesky version of these functions are available, c called vdpbfa7 and vdpbsl7 c T.Yee 7/10/99 c 1/7/02 c T.Yee has renamed dbpbfa to dbpbfa8 and dpbsl to dpbsl8, to ensure uniqueness subroutine dpbfa8(abd,lda,n,m,info) integer lda,n,m,info double precision abd(lda,1) c c dpbfa8 factors a double precision symmetric positive definite c matrix stored in band form. c c dpbfa8 is usually called by dpbco, but it can be called c directly with a saving in time if rcond is not needed. c c on entry c c abd double precision(lda, n) c the matrix to be factored. the columns of the upper c triangle are stored in the columns of abd and the c diagonals of the upper triangle are stored in the c rows of abd . see the comments below for details. c c lda integer c the leading dimension of the array abd . c lda must be .ge. m + 1 . c c n integer c the order of the matrix a . c c m integer c the number of diagonals above the main diagonal. c 0 .le. m .lt. n . c c on return c c abd an upper triangular matrix r , stored in band c form, so that a = trans(r)*r . c c info integer c = 0 for normal return. c = k if the leading minor of order k is not c positive definite. c c band storage c c if a is a symmetric positive definite band matrix, c the following program segment will set up the input. c c m = (band width above diagonal) c do 20 j = 1, n c i1 = max0(1, j-m) c do 10 i = i1, j c k = i-j+m+1 c abd(k,j) = a(i,j) c 10 continue c 20 continue c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c blas ddot c fortran max0,dsqrt c c internal variables c double precision ddot8,t double precision s integer ik,j,jk,k,mu c begin block with ...exits to 40 c c do 30 j = 1, n info = j s = 0.0d0 ik = m + 1 jk = max0(j-m,1) mu = max0(m+2-j,1) if (m .lt. mu) go to 20 do 10 k = mu, m t = abd(k,j) - ddot8(k-mu,abd(ik,jk),1,abd(mu,j),1) t = t/abd(m+1,jk) abd(k,j) = t s = s + t*t ik = ik - 1 jk = jk + 1 10 continue 20 continue s = abd(m+1,j) - s c ......exit if (s .le. 0.0d0) go to 40 abd(m+1,j) = dsqrt(s) 30 continue info = 0 40 continue return end subroutine dpbsl8(abd,lda,n,m,b) integer lda,n,m double precision abd(lda,1),b(1) c c dpbsl8 solves the double precision symmetric positive definite c band system a*x = b c using the factors computed by dpbco or dpbfa8. c c on entry c c abd double precision(lda, n) c the output from dpbco or dpbfa8. c c lda integer c the leading dimension of the array abd . c c n integer c the order of the matrix a . c c m integer c the number of diagonals above the main diagonal. c c b double precision(n) c the right hand side vector. c c on return c c b the solution vector x . c c error condition c c a division by zero will occur if the input factor contains c a zero on the diagonal. technically this indicates c singularity but it is usually caused by improper subroutine c arguments. it will not occur if the subroutines are called c correctly and info .eq. 0 . c c to compute inverse(a) * c where c is a matrix c with p columns c call dpbco(abd,lda,n,rcond,z,info) c if (rcond is too small .or. info .ne. 0) go to ... c do 10 j = 1, p c call dpbsl8(abd,lda,n,c(1,j)) c 10 continue c c linpack. this version dated 08/14/78 . c cleve moler, university of new mexico, argonne national lab. c c subroutines and functions c c blas daxpy,ddot c fortran min0 c c internal variables c double precision ddot8,t integer k,kb,la,lb,lm c c solve trans(r)*y = b c do 10 k = 1, n lm = min0(k-1,m) la = m + 1 - lm lb = k - lm t = ddot8(lm,abd(la,k),1,b(lb),1) b(k) = (b(k) - t)/abd(m+1,k) 10 continue c c solve r*x = y c do 20 kb = 1, n k = n + 1 - kb lm = min0(k-1,m) la = m + 1 - lm lb = k - lm b(k) = b(k)/abd(m+1,k) t = -b(k) call daxpy8(lm,t,abd(la,k),1,b(lb),1) 20 continue return end

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