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{
This file is part of the Numlib package.
Copyright (c) 1986-2000 by
Kees van Ginneken, Wil Kortsmit and Loek van Reij of the
Computational centre of the Eindhoven University of Technology
FPC port Code by Marco van de Voort (marco@freepascal.org)
documentation by Michael van Canneyt (Michael@freepascal.org)
Undocumented unit. B- and other Splines. Not imported by the other units
afaik.
See the file COPYING.FPC, included in this distribution,
for details about the copyright.
This program 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.
**********************************************************************}
unit spl;
{$I direct.inc}
interface
uses typ, math, sle;
function spl1bspv(q: ArbInt; var kmin1, c1: ArbFloat; x: ArbFloat; var term: ArbInt): ArbFloat;
function spl2bspv(qx, qy: ArbInt; var kxmin1, kymin1, c11: ArbFloat; x, y: ArbFloat; var term: ArbInt): ArbFloat;
procedure spl1bspf(M, Q: ArbInt; var XYW1: ArbFloat;
var Kmin1, C1, residu: ArbFloat;
var term: ArbInt);
procedure spl2bspf(M, Qx, Qy: ArbInt; var XYZW1: ArbFloat;
var Kxmin1, Kymin1, C11, residu: ArbFloat;
var term: ArbInt);
procedure spl1nati(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
procedure spl1naki(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
procedure spl1cmpi(n: ArbInt; var xyc1: ArbFloat; dy1, dyn: ArbFloat;
var term: ArbInt);
procedure spl1peri(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
function spl1pprv(n: ArbInt; var xyc1: ArbFloat; t: ArbFloat; var term: ArbInt): ArbFloat;
procedure spl1nalf(n: ArbInt; var xyw1: ArbFloat; lambda:ArbFloat;
var xac1, residu: ArbFloat; var term: ArbInt);
function spl2natv(n: ArbInt; var xyg0: ArbFloat; u, v: ArbFloat): ArbFloat;
procedure spl2nalf(n: ArbInt; var xyzw1: ArbFloat; lambda:ArbFloat;
var xyg0, residu: ArbFloat; var term: ArbInt);
{ term = 1: succes,
term = 2: set linear equations is not "PD"
term = 4: Approx. number of points? On a line.
term = 3: wrong input n<3 or a weight turned out to be <=0 }
implementation
{$goto on}
type
Krec = record K1, K2, K3, K4, K5, K6 : ArbFloat end;
function spl1bspv(q: ArbInt; var kmin1, c1: ArbFloat; x: ArbFloat; var term: ArbInt): ArbFloat;
var c : arfloat1 absolute c1;
k : arfloat_1 absolute kmin1;
D1, D2, D3,
E2, E3, E4, E5: ArbFloat;
pk : ^Krec;
l, r, m : ArbInt;
begin
spl1bspv := NaN;
term := 3; { q >=4 ! }
if q<4 then exit; { at least 1 interval }
if (x<k[2]) or (x>k[q-1]) then exit; { x inside the interval }
term := 1; { Let's hope the params are good :-)}
l := 2; r := q-1;
while l+1<r do { after this loop goes: }
begin { k[l]<=x<=k[l+1] with }
m := (l+r) div 2; { k[l] < k[l+1] }
if x>=k[m] then l := m else r := m
end;
pk := @k[l-2]; { the (de) Boor algoritm .. }
with pk^ do
begin
E2 := X - K2; E3 := X - K3; E4 := K4 - X; E5 := K5 - X;
D2 := C[l]; D3 := C[l+1];
D1 := ((X-K1)*D2+E4*C[l-1])/(K4-K1);
D2 := (E2*D3+E5*D2)/(K5-K2);
D3 := (E3*C[l+2]+(K6-X)*D3)/(K6-K3);
D1 := (E2*D2+E4*D1)/(K4-K2);
D2 := (E3*D3+E5*D2)/(K5-K3);
spl1bspv := (E3*D2+E4*D1)/(K4-K3)
end;
end;
function spl2bspv(qx, qy: ArbInt; var kxmin1, kymin1, c11: ArbFloat; x, y: ArbFloat; var term: ArbInt): ArbFloat;
var pd: ^arfloat1;
i, iy: ArbInt;
c: arfloat1 absolute c11;
begin
GetMem(pd, qx*SizeOf(ArbFloat));
i := 0;
iy := 1;
repeat
i := i + 1;
pd^[i] := spl1bspv(qy, kymin1, c[iy], y, term);
Inc(iy, qy)
until (i=qx) or (term<>1);
if term=1
then spl2bspv := spl1bspv(qx, kxmin1, pd^[1], x, term)
else spl2bspv := NaN;
FreeMem(pd, qx*SizeOf(ArbFloat));
end;
(* Bron: NAG LIBRARY SUBROUTINE E02BAF *)
function Imin(x, y: ArbInt): ArbInt;
begin if x<y then Imin := x else Imin := y end;
type ar4 = array[1..$ffe0 div (4*SizeOf(ArbFloat)),1..4] of ArbFloat;
ar3 = array[1..$ffe0 div (3*SizeOf(ArbFloat)),1..3] of ArbFloat;
r_3 = record x, y, w: ArbFloat end;
r3Ar= array[1..$ffe0 div SizeOf(r_3)] of r_3;
r_4 = record x, y, z, w: ArbFloat end;
r4Ar= array[1..$ffe0 div SizeOf(r_4)] of r_4;
r4 = array[1..4] of ArbFloat;
r2 = array[1..2] of ArbFloat;
r4x = record xy: R2; alfa, d: ArbFloat end;
r4xAr= array[1..$ffe0 div SizeOf(r4x)] of r4x;
nsp2rec = array[0..$ff80 div (3*SizeOf(ArbFloat))] of
record xy: R2; gamma: ArbFloat end;
procedure spl1bspf(M, Q: ArbInt; var XYW1: ArbFloat;
var Kmin1, C1, residu: ArbFloat;
var term: ArbInt);
var work1: ^arfloat1;
work2: ^ar4;
c : arfloat1 absolute c1;
k : arfloat_1 absolute kmin1;
xyw : r3Ar absolute XYW1;
r, j, jmax, l, lplus1, i, iplusj, jold, jrev,
jplusl, iu, lplusu : ArbInt;
s, k0, k4, sigma,
d, d4, d5, d6, d7, d8, d9,
e2, e3, e4, e5,
n1, n2, n3,
relemt, dprime, cosine, sine,
acol, arow, crow, ccol, ss : ArbFloat;
pk : ^Krec;
label einde;
(*
DOUBLE PRECISION C(NCAP7), K(NCAP7), W(M), WORK1(M),
* WORK2(4,NCAP7), X(M), Y(M)
.. Local Scalars ..
DOUBLE PRECISION ACOL, AROW, CCOL, COSINE, CROW, D, D4, D5, D6,
* D7, D8, D9, DPRIME, E2, E3, E4, E5, K0, K1, K2,
* K3, K4, K5, K6, N1, N2, N3, RELEMT, S, SIGMA,
* SINE, WI, XI
INTEGER I, IERROR, IPLUSJ, IU, J, JOLD, JPLUSL, JREV, L,
* L4, LPLUS1, LPLUSU, NCAP, NCAP3, NCAPM1, R
*)
begin
term := 3;
if q<4 then exit;
if m<q then exit;
(*
CHECK THAT THE VALUES OF M AND NCAP7 ARE REASONABLE
IF (NCAP7.LT.8 .OR. M.LT.NCAP7-4) GO TO 420
NCAP = NCAP7 - 7
NCAPM1 = NCAP - 1
NCAP3 = NCAP + 3
IN ORDER TO DEFINE THE FULL B-SPLINE BASIS, AUGMENT THE
PRESCRIBED INTERIOR KNOTS BY KNOTS OF MULTIPLICITY FOUR
AT EACH END OF THE DATA RANGE.
*)
for j:=-1 to 2 do k[j] := xyw[1].x;
for j:=q-1 to q+2 do k[j] := xyw[m].x;
if (k[3]<=xyw[1].x) or (k[q-2]>=xyw[m].x) then exit;
(*
CHECK THAT THE KNOTS ARE ORDERED AND ARE INTERIOR
TO THE DATA INTERVAL.
*)
j := 3; while (k[j]<=k[j+1]) and (j<q-2) do Inc(j);
if j<q-2 then exit;
(*
CHECK THAT THE WEIGHTS ARE STRICTLY POSITIVE.
*)
j := 1;
while (xyw[j].w>0) and (j<m) do Inc(j);
if xyw[j].w<=0 then exit;
(*
CHECK THAT THE DATA ABSCISSAE ARE ORDERED, THEN FORM THE
ARRAY WORK1 FROM THE ARRAY X. THE ARRAY WORK1 CONTAINS
THE
SET OF DISTINCT DATA ABSCISSAE.
*)
GetMem(Work1, m*SizeOf(ArbFloat));
GetMem(Work2, q*4*SizeOf(ArbFloat));
r := 1; work1^[1] := xyw[1].x;
j := 1;
while (j<m) do
begin
Inc(j);
if xyw[j].x>work1^[r]
then begin Inc(r); work1^[r] := xyw[j].x end
else if xyw[j].x<work1^[r] then goto einde;
end;
if r<q then goto einde;
(*
CHECK THE FIRST S AND THE LAST S SCHOENBERG-WHITNEY
CONDITIONS ( S = MIN(NCAP - 1, 4) ).
*)
jmax := Imin(q-4,4);
j := 1;
while (j<=jmax) do
begin
if (work1^[j]>=k[j+2]) or (k[q-j-1]>=work1^[r-j+1]) then goto einde;
Inc(j)
end;
(*
CHECK ALL THE REMAINING SCHOENBERG-WHITNEY CONDITIONS.
*)
Dec(r, 4); i := 4; j := 5;
while j<=q-4 do
begin
K0 := K[j+2]; K4 := K[J-2];
repeat Inc(i) until (Work1^[i]>k4);
if (I>R) or (WORK1^[I]>=K0) then goto einde;
Inc(j)
end;
(*
INITIALISE A BAND TRIANGULAR SYSTEM (I.E. A
MATRIX AND A RIGHT HAND SIDE) TO ZERO. THE
PROCESSING OF EACH DATA POINT IN TURN RESULTS
IN AN UPDATING OF THIS SYSTEM. THE SUBSEQUENT
SOLUTION OF THE RESULTING BAND TRIANGULAR SYSTEM
YIELDS THE COEFFICIENTS OF THE B-SPLINES.
*)
FillChar(Work2^, q*4*SizeOf(ArbFloat), 0);
FillChar(c, q*SizeOf(ArbFloat), 0);
SIGMA := 0; j := 0; jold := 0;
for i:=1 to m do
with xyw[i] do
begin
(*
FOR THE DATA POINT (X(I), Y(I)) DETERMINE AN INTERVAL
K(J + 3) .LE. X .LT. K(J + 4) CONTAINING X(I). (IN THE
CASE J + 4 .EQ. NCAP THE SECOND EQUALITY IS RELAXED TO
INCLUDE
EQUALITY).
*)
while (x>=k[j+2]) and (j<=q-4) do Inc(j);
if j<>jold then
begin
pk := @k[j-1];
with pk^ do
begin
D4 := 1/(K4-K1); D5 := 1/(K5-K2); D6 := 1/(K6-K3);
D7 := 1/(K4-K2); D8 := 1/(K5-K3); D9 := 1/(K4-K3)
end;
JOLD := J;
end;
(*
COMPUTE AND STORE IN WORK1(L) (L = 1, 2, 3, 4) THE VALUES
OF
THE FOUR NORMALIZED CUBIC B-SPLINES WHICH ARE NON-ZERO AT
X=X(I).
*) with pk^ do
begin
E5 := k5 - X;
E4 := K4 - X;
E3 := X - K3;
E2 := X - K2;
N1 := W*D9;
N2 := E3*N1*D8;
N1 := E4*N1*D7;
N3 := E3*N2*D6;
N2 := (E2*N1+E5*N2)*D5;
N1 := E4*N1*D4;
WORK1^[4] := E3*N3;
WORK1^[3] := E2*N2 + (K6-X)*N3;
WORK1^[2] := (X-K1)*N1 + E5*N2;
WORK1^[1] := E4*N1;
CROW := Y*W;
end;
(*
ROTATE THIS ROW INTO THE BAND TRIANGULAR SYSTEM USING PLANE
ROTATIONS.
*)
for lplus1:=1 to 4 do
begin L := LPLUS1 - 1;
RELEMT := WORK1^[LPLUS1];
if relemt<>0 then
begin JPLUSL := J + L;
D := WORK2^[JPLUSL,1];
IF (ABS(RELEMT)>=D)
then DPRIME := ABS(RELEMT)*SQRT(1+sqr(D/RELEMT))
else DPRIME := D*SQRT(1+sqr(RELEMT/D));
WORK2^[JPLUSL,1] := DPRIME;
COSINE := D/DPRIME; SINE := RELEMT/DPRIME;
for iu :=2 to 4-l do
begin
LPLUSU := L + IU;
ACOL := WORK2^[JPLUSL,iu];
AROW := WORK1^[LPLUSU];
WORK2^[JPLUSL,iu] := COSINE*ACOL + SINE*AROW;
WORK1^[LPLUSU] := COSINE*AROW - SINE*ACOL
end;
CCOL := C[JPLUSL];
C[JPLUSL] := COSINE*CCOL + SINE*CROW;
CROW := COSINE*CROW - SINE*CCOL
end;
end;
SIGMA := SIGMA + sqr(CROW)
end;
residu := SIGMA;
(*
SOLVE THE BAND TRIANGULAR SYSTEM FOR THE B-SPLINE
COEFFICIENTS. IF A DIAGONAL ELEMENT IS ZERO, AND HENCE
THE TRIANGULAR SYSTEM IS SINGULAR, THE IMPLICATION IS
THAT THE SCHOENBERG-WHITNEY CONDITIONS ARE ONLY JUST
SATISFIED. THUS IT IS APPROPRIATE TO EXIT IN THIS
CASE WITH THE SAME VALUE (IFAIL=5) OF THE ERROR
INDICATOR.
*)
term := 2;
L := -1;
for jrev:=1 to q do
begin
J := q - JREV + 1; D := WORK2^[J,1];
if d=0 then goto einde;
IF l<3 then L := L + 1;
S := C[j];
for i:=1 to l do
begin
IPLUSJ := I + J;
S := S - WORK2^[j,i+1]*C[IPLUSJ];
end;
C[J] := S/D
end;
term:=1;
einde:
FreeMem(Work2, q*4*SizeOf(ArbFloat));
FreeMem(Work1, m*SizeOf(ArbFloat))
end;
procedure spl2bspf(M, Qx, Qy: ArbInt; var XYZW1: ArbFloat;
var Kxmin1, Kymin1, C11, residu: ArbFloat;
var term: ArbInt);
(* !!!!!!!! Test input !!!!!!!!!! *)
(*
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc
c part 1: determination of the number of knots and their position. c
c **************************************************************** c
c given a set of knots we compute the least-squares spline sinf(x,y), c
c and the corresponding weighted sum of squared residuals fp=f(p=inf). c
c if iopt=-1 sinf(x,y) is the requested approximation. c
c if iopt=0 or iopt=1 we check whether we can accept the knots: c
c if fp <=s we will continue with the current set of knots. c
c if fp > s we will increase the number of knots and compute the c
c corresponding least-squares spline until finally fp<=s. c
c the initial choice of knots depends on the value of s and iopt. c
c if iopt=0 we first compute the least-squares polynomial of degree c
c 3 in x and 3 in y; nx=nminx=2*3+2 and ny=nminy=2*3+2. c
c fp0=f(0) denotes the corresponding weighted sum of squared c
c residuals c
c if iopt=1 we start with the knots found at the last call of the c
c routine, except for the case that s>=fp0; then we can compute c
c the least-squares polynomial directly. c
c eventually the independent variables x and y (and the corresponding c
c parameters) will be switched if this can reduce the bandwidth of the c
c system to be solved. c
cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc *)
function Min(a, b:ArbInt): ArbInt;
begin if a<b then Min := a else Min := b end;
procedure WisselR(var x, y: ArbFloat);
var h: ArbFloat; begin h := x; x := y; y := h end;
procedure Wisseli(var x, y: ArbInt);
var h: ArbInt; begin h := x; x := y; y := h end;
procedure fprota(var cos1, sin1, a, b: ArbFloat);
var store: ArbFloat;
begin
store := b; b := cos1*b+sin1*a; a := cos1*a-sin1*store
end;
procedure fpgivs(var piv, ww, cos1, sin1: ArbFloat);
var store, dd: ArbFloat;
begin
store := abs(piv);
if store>=ww
then dd := store*sqrt(1+sqr(ww/piv))
else dd := ww*sqrt(1+sqr(piv/ww));
cos1 := ww/dd; sin1 := piv/dd; ww := dd
end;
procedure fpback(var a11, z1: ArbFloat; n, k: ArbInt; var c1: ArbFloat);
(*
subroutine fpback calculates the solution of the system of
equations a*c = z with a a n x n upper triangular matrix
of bandwidth k.
ArbFloat a(.,k)
*)
var a: arfloat1 absolute a11;
z: arfloat1 absolute z1;
c: arfloat1 absolute c1;
i, l: ArbInt;
store : ArbFloat;
begin
for i:=n downto 1 do
begin
store := z[i];
for l:=min(n+1-i,k)-1 downto 1 do store := store-c[i+l]*a[(i-1)*k+l+1];
c[i] := store/a[(i-1)*k+1]
end;
end;
procedure fpbspl(var kmin1: ArbFloat; x: ArbFloat; l: ArbInt; var h: r4);
(*
subroutine fpbspl evaluates the 4 non-zero b-splines of
degree 3 at t(l) <= x < t(l+1) using the stable recurrence
relation of de boor and cox.
*)
var k : arfloat_1 absolute kmin1;
f : ArbFloat;
hh: array[1..3] of ArbFloat;
i, j, li, lj : ArbInt;
begin
h[1] := 1;
for j:=1 to 3 do
begin
for i:=1 to j do hh[i] := h[i];
h[1] := 0;
for i:=1 to j do
begin
li := l+i; lj := li-j;
f := hh[i]/(k[li]-k[lj]);
h[i] := h[i]+f*(k[li]-x);
h[i+1] := f*(x-k[lj])
end;
end;
end;
procedure fporde(m, qx, qy: ArbInt; var xyzw1, kxmin1, kymin1: ArbFloat;
var nummer1, index1: ArbInt);
var xi, yi : ArbFloat;
i, im, num,
k, l : ArbInt;
xyzw : r4Ar absolute xyzw1;
kx : arfloat_1 absolute kxmin1;
ky : arfloat_1 absolute kymin1;
nummer : arint1 absolute nummer1;
index : arint1 absolute index1;
begin
for i:=1 to (qx-3)*(qy-3) do index[i] := 0;
for im:=1 to m do
with xyzw[im] do
begin
l := 2; while (x>=kx[l+1]) and (l<qx-2) do Inc(l);
k := 2; while (y>=ky[k+1]) and (k<qy-2) do Inc(k);
num := (l-2)*(qy-3)+k-1;
nummer[im] := index[num]; index[num] := im
end;
end;
label einde;
var x0, x1, y0, y1, eps, cos1, sin1, dmax, sigma,
wi, zi, hxi, piv : ArbFloat;
i, j, l, l1, l2, lx, ly, nreg, ncof, jrot,
inpanel, i1, j1,
iband, num, irot : ArbInt;
xyzw : r4Ar absolute xyzw1;
kx, ky : ^arfloat_1;
a, f, h : ^arfloat1;
c : arfloat1 absolute c11;
nummer, index : ^arint1;
hx, hy : r4;
ichang, fullrank : boolean;
begin
eps := 10*macheps;
(* find the position of the additional knots which are needed for the
b-spline representation of s(x,y) *)
iband := 1+min(3*qy+3,3*qx+3);
if qy>qx then
begin
ichang := true;
kx := @kymin1; ky := @kxmin1;
for i:=1 to m do with xyzw[i] do Wisselr(x, y);
WisselI(qx, qy)
end else
begin
ichang := false;
kx := @kxmin1; ky := @kymin1;
end;
with xyzw[1] do begin x0 := x; x1 := x; y0 := y; y1 := y end;
for i:=2 to m do with xyzw[i] do
begin if x<x0 then x0 := x; if x>x1 then x1 := x;
if y<y0 then y0 := y; if y>y1 then y1 := y
end;
for i:=-1 to 2 do kx^[i] := x0;
for i:=-1 to 2 do ky^[i] := y0;
for i:=qx-1 to qx+2 do kx^[i] := x1;
for i:=qy-1 to qy+2 do ky^[i] := y1;
(* arrange the data points according to the panel they belong to *)
nreg := (qx-3)*(qy-3);
ncof := qx*qy;
GetMem(nummer, m*SizeOf(ArbInt));
GetMem(index, nreg*SizeOf(ArbInt));
GetMem(h, iband*SizeOf(ArbFloat));
GetMem(a, iband*ncof*SizeOf(ArbFloat));
GetMem(f, ncof*SizeOf(ArbFloat));
fporde(m, qx, qy, xyzw1, kx^[-1], ky^[-1], nummer^[1], index^[1]);
for i:=1 to ncof do f^[i] := 0;
for j:=1 to ncof*iband do a^[j] := 0;
residu := 0;
(* fetch the data points in the new order. main loop for the different panels *)
for num:=1 to nreg do
begin
lx := (num-1) div (qy-3); l1 := lx+2;
ly := (num-1) mod (qy-3); l2 := ly+2;
jrot := lx*qy+ly;
inpanel := index^[num];
while inpanel<>0 do
with xyzw[inpanel] do
begin
wi := w; zi := z*wi;
fpbspl(kx^[-1], x, l1, hx);
fpbspl(ky^[-1], y, l2, hy);
for i:=1 to iband do h^[i] := 0;
i1 := 0;
for i:=1 to 4 do
begin
hxi := hx[i]; j1 := i1;
for j:=1 to 4 do begin Inc(j1); h^[j1] := hxi*hy[j]*wi end;
Inc(i1, qy)
end;
irot := jrot;
for i:=1 to iband do
begin
Inc(irot); piv := h^[i];
if piv<>0 then
begin
fpgivs(piv, a^[(irot-1)*iband+1], cos1, sin1);
fprota(cos1, sin1, zi, f^[irot]);
for j:=i+1 to iband do
fprota(cos1, sin1, h^[j], a^[(irot-1)*iband+j-i+1])
end;
end;
residu := residu+sqr(zi);
inpanel := nummer^[inpanel]
end;
end;
dmax := 0;
i := 1;
while i<ncof*iband do
begin
if dmax<a^[i] then dmax:=a^[i]; Inc(i, iband)
end;
sigma := eps*dmax;
i := 1; fullrank := true;
while fullrank and (i<ncof*iband) do
begin
fullrank := a^[i]>sigma; Inc(i, iband)
end;
term := 2; if not fullrank then goto einde;
term := 1;
fpback(a^[1], f^[1], ncof, iband, c11);
if ichang then
begin
l1 := 1;
for i:=1 to qx do
begin
l2 := i;
for j:=1 to qy do
begin
f^[l2] := c[l1]; Inc(l1); Inc(l2, qx)
end;
end;
for i:=1 to ncof do c[i] := f^[i]
end;
einde:
if ichang then for i:=1 to m do with xyzw[i] do Wisselr(x, y);
FreeMem(f, ncof*SizeOf(ArbFloat));
FreeMem(a, iband*ncof*SizeOf(ArbFloat));
FreeMem(h, iband*SizeOf(ArbFloat));
FreeMem(index, nreg*SizeOf(ArbInt));
FreeMem(nummer, m*SizeOf(ArbInt))
end;
procedure spl1nati(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
var
xyc : r3Ar absolute XYC1;
l, b, d, u, c : ^arfloat1;
h2, h3, s2, s3: ArbFloat;
i, m : ArbInt; { afmeting van op te lossen stelsel }
begin
term:=3;
if n < 2 then exit;
for i:=2 to n do if xyc[i-1].x>=xyc[i].x then exit;
term:=1;
xyc[1].w := 0; xyc[n].w := 0; { c1=cn=0 }
m := n-2;
if m=0 then exit;
getmem(u, n*SizeOf(ArbFloat));
getmem(l, n*Sizeof(ArbFloat));
getmem(d, n*SizeOf(ArbFloat));
getmem(c, n*SizeOf(ArbFloat));
getmem(b, n*SizeOf(ArbFloat));
h3:=xyc[2].x-xyc[1].x;
s3:=(xyc[2].y-xyc[1].y)/h3;
for i:=2 to n-1 do
begin
h2:=h3; h3:=xyc[i+1].x-xyc[i].x;
s2:=s3; s3:=(xyc[i+1].y-xyc[i].y)/h3;
l^[i]:=h2/6;
d^[i]:=(h2+h3)/3;
u^[i]:=h3/6;
b^[i]:=s3-s2
end;
sledtr(m, l^[3], d^[2], u^[2], b^[2], c^[2], term);
for i:=2 to n-1 do xyc[i].w := c^[i];
Freemem(b, n*SizeOf(ArbFloat));
Freemem(c, n*SizeOf(ArbFloat));
Freemem(d, n*SizeOf(ArbFloat));
Freemem(l, n*Sizeof(ArbFloat));
Freemem(u, n*SizeOf(ArbFloat));
end; {spl1nati}
procedure spl1naki(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
var
xyc : r3Ar absolute XYC1;
l, b, d, u, c : ^arfloat1;
h2, h3, s2, s3: ArbFloat;
i, m : ArbInt; { Dimensions of set lin eqs to solve}
begin
term:=3;
if n < 4 then exit;
for i:=2 to n do if xyc[i-1].x>=xyc[i].x then exit;
term:=1;
m := n-2;
getmem(u, n*SizeOf(ArbFloat));
getmem(l, n*Sizeof(ArbFloat));
getmem(d, n*SizeOf(ArbFloat));
getmem(c, n*SizeOf(ArbFloat));
getmem(b, n*SizeOf(ArbFloat));
h3:=xyc[2].x-xyc[1].x;
s3:=(xyc[2].y-xyc[1].y)/h3;
for i:=2 to n-1 do
begin
h2:=h3; h3:=xyc[i+1].x-xyc[i].x;
s2:=s3; s3:=(xyc[i+1].y-xyc[i].y)/h3;
l^[i]:=h2/6;
d^[i]:=(h2+h3)/3;
u^[i]:=h3/6;
b^[i]:=s3-s2
end;
d^[n-1]:=d^[n-1]+h3/6*(1+h3/h2); l^[n-1]:=l^[n-1]-sqr(h3)/(6*h2);
h2:=xyc[2].x-xyc[1].x; h3:=xyc[3].x-xyc[2].x;
d^[2]:=d^[2]+h2/6*(1+h2/h3); u^[2]:=u^[2]-sqr(h2)/(6*h3);
sledtr(m, l^[3], d^[2], u^[2], b^[2], c^[2], term);
for i:=2 to n-1 do xyc[i].w := c^[i];
xyc[1].w := xyc[2].w + (h2/h3)*(xyc[2].w-xyc[3].w);
h2:=xyc[n-1].x-xyc[n-2].x; h3:=xyc[n].x-xyc[n-1].x;
xyc[n].w := xyc[n-1].w + (h3/h2)*(xyc[n-1].w-xyc[n-2].w);
Freemem(b, n*SizeOf(ArbFloat));
Freemem(c, n*SizeOf(ArbFloat));
Freemem(d, n*SizeOf(ArbFloat));
Freemem(l, n*Sizeof(ArbFloat));
Freemem(u, n*SizeOf(ArbFloat));
end; {spl1naki}
procedure spl1cmpi(n: ArbInt; var xyc1: ArbFloat; dy1, dyn: ArbFloat;
var term: ArbInt);
var
xyc : r3Ar absolute XYC1;
l, b, d, u, c : ^arfloat1;
h2, h3, s2, s3: ArbFloat;
i : ArbInt; { Dimensions of set lin eqs to solve}
begin
term:=3;
if n < 2 then exit;
for i:=2 to n do if xyc[i-1].x>=xyc[i].x then exit;
term:=1;
getmem(u, n*SizeOf(ArbFloat));
getmem(l, n*Sizeof(ArbFloat));
getmem(d, n*SizeOf(ArbFloat));
getmem(c, n*SizeOf(ArbFloat));
getmem(b, n*SizeOf(ArbFloat));
h3:=xyc[2].x-xyc[1].x;
s3:=(xyc[2].y-xyc[1].y)/h3;
d^[1] := h3/3; u^[1] := h3/6; b^[1] := -dy1+s3;
for i:=2 to n-1 do
begin
h2:=h3; h3:=xyc[i+1].x-xyc[i].x;
s2:=s3; s3:=(xyc[i+1].y-xyc[i].y)/h3;
l^[i]:=h2/6;
d^[i]:=(h2+h3)/3;
u^[i]:=h3/6;
b^[i]:=s3-s2
end;
d^[n] := h3/3; l^[n] := h3/6; b^[n] := dyn-s3;
sledtr(n, l^[2], d^[1], u^[1], b^[1], c^[1], term);
for i:=1 to n do xyc[i].w := c^[i];
Freemem(b, n*SizeOf(ArbFloat));
Freemem(c, n*SizeOf(ArbFloat));
Freemem(d, n*SizeOf(ArbFloat));
Freemem(l, n*Sizeof(ArbFloat));
Freemem(u, n*SizeOf(ArbFloat));
end; {spl1cmpi}
procedure spl1peri(n: ArbInt; var xyc1: ArbFloat; var term: ArbInt);
var
xyc : r3Ar absolute XYC1;
l, b, d, u, c, k : ^arfloat1;
k2, kn1, dy1, cn,
h2, h3, s2, s3: ArbFloat;
i, m : ArbInt; { Dimensions of set lin eqs to solve}
begin
term:=3;
if n < 2 then exit;
if xyc[1].y<>xyc[n].y then exit;
for i:=2 to n do if xyc[i-1].x>=xyc[i].x then exit;
term:=1;
m := n-2;
xyc[1].w := 0; xyc[n].w := 0; { c1=cn=0 }
if m=0 then exit;
if m=1 then
begin
h2:=xyc[2].x-xyc[1].x;
s2:=(xyc[2].y-xyc[1].y)/h2;
h3:=xyc[3].x-xyc[2].x;
s3:=(xyc[3].y-xyc[2].y)/h3;
xyc[2].w := 6*(s3-s2)/(h2+h3);
xyc[3].w := -xyc[2].w;
xyc[1].w := xyc[3].w;
exit
end;
getmem(u, n*SizeOf(ArbFloat));
getmem(l, n*Sizeof(ArbFloat));
getmem(k, n*SizeOf(ArbFloat));
getmem(d, n*SizeOf(ArbFloat));
getmem(c, n*SizeOf(ArbFloat));
getmem(b, n*SizeOf(ArbFloat));
h3:=xyc[2].x-xyc[1].x;
s3:=(xyc[2].y-xyc[1].y)/h3;
k2 := h3/6; dy1 := s3;
for i:=2 to n-1 do
begin
h2:=h3; h3:=xyc[i+1].x-xyc[i].x;
s2:=s3; s3:=(xyc[i+1].y-xyc[i].y)/h3;
l^[i]:=h2/6;
d^[i]:=(h2+h3)/3;
u^[i]:=h3/6;
b^[i]:=s3-s2;
k^[i]:=0
end;
kn1 := h3/6; k^[2] := k2; k^[n-1] := kn1;
sledtr(m, l^[3], d^[2], u^[2], k^[2], k^[2], term);
sledtr(m, l^[3], d^[2], u^[2], b^[2], c^[2], term);
cn := (dy1-s3-k2*c^[2]-kn1*c^[n-1])/(2*(k2+kn1)-k2*k^[2]-kn1*k^[n-1]);
for i:=2 to n-1 do xyc[i].w := c^[i] - cn*k^[i];
xyc[1].w := cn; xyc[n].w := cn;
Freemem(b, n*SizeOf(ArbFloat));
Freemem(c, n*SizeOf(ArbFloat));
Freemem(d, n*SizeOf(ArbFloat));
Freemem(l, n*Sizeof(ArbFloat));
Freemem(k, n*SizeOf(ArbFloat));
Freemem(u, n*SizeOf(ArbFloat));
end; {spl1peri}
function spl1pprv(n: ArbInt; var xyc1: ArbFloat; t: ArbFloat; var term: ArbInt): ArbFloat;
var
xyc : r3Ar absolute XYC1;
i, j, m : ArbInt;
d, d3, h, dy : ArbFloat;
begin { Assumption : x[i]<x[i+1] i=1..n-1 }
spl1pprv := NaN;
term:=3; if n<2 then exit;
if (t<xyc[1].x) or (t>xyc[n].x) then exit;
term:=1;
i:=1; j:=n;
while j <> i+1 do
begin
m:=(i+j) div 2;
if t>=xyc[m].x then i:=m else j:=m
end; { x[i]<= t <=x[i+1] }
h := xyc[i+1].x-xyc[i].x;
d := t-xyc[i].x;
d3 :=(xyc[i+1].w-xyc[i].w)/h;
dy :=(xyc[i+1].y-xyc[i].y)/h-h*(2*xyc[i].w+xyc[i+1].w)/6;
spl1pprv:= xyc[i].y+d*(dy+d*(xyc[i].w/2+d*d3/6))
end; {spl1pprv}
procedure spl1nalf(n: ArbInt; var xyw1: ArbFloat; lambda:ArbFloat;
var xac1, residu: ArbFloat; var term: ArbInt);
var
xyw : r3Ar absolute xyw1;
xac : r3Ar absolute xac1;
i, j, ncd : ArbInt;
ca, crow : ArbFloat;
h, qty : ^arfloat1;
ch : ^arfloat0;
qtdq : ^arfloat1;
begin
term := 3; { testing input}
if n<2 then exit;
for i:=2 to n do if xyw[i-1].x>=xyw[i].x then exit;
for i:=1 to n do if xyw[i].w<=0 then exit;
if lambda<0 then exit;
term := 1;
Move(xyw, xac, n*SizeOf(r_3));
if n=2 then begin xac[1].w := 0; xac[2].w := 0; exit end;
Getmem(ch, (n+2)*SizeOf(ArbFloat)); FillChar(ch^, (n+2)*SizeOf(ArbFloat), 0);
Getmem(h, n*SizeOf(ArbFloat));
Getmem(qty, n*SizeOf(ArbFloat));
ncd := n-3; if ncd>2 then ncd := 2;
Getmem(qtdq, ((n-2)*(ncd+1)-(ncd*(ncd+1)) div 2)*SizeOf(ArbFloat));
for i:=2 to n do h^[i] := 1/(xyw[i].x-xyw[i-1].x); h^[1] := 0;
for i:=1 to n-2
do qty^[i] := (h^[i+1]*xyw[i].y -
(h^[i+1]+h^[i+2])*xyw[i+1].y +
h^[i+2]*xyw[i+2].y);
j := 1; i := 1;
qtdq^[j] := sqr(h^[i+1])/xyw[i].w +
sqr(h^[i+1]+h^[i+2])/xyw[i+1].w +
sqr(h^[i+2])/xyw[i+2].w +
lambda*(1/h^[i+1]+1/h^[i+2])/3;
Inc(j);
if ncd>0 then
begin i := 2;
qtdq^[j] := -h^[i+1]*(h^[i]+h^[i+1])/xyw[i].w
-h^[i+1]*(h^[i+1]+h^[i+2])/xyw[i+1].w +
lambda/h^[i+1]/6;
Inc(j);
qtdq^[j] := sqr(h^[i+1])/xyw[i].w +
sqr(h^[i+1]+h^[i+2])/xyw[i+1].w +
sqr(h^[i+2])/xyw[i+2].w +
lambda*(1/h^[i+1]+1/h^[i+2])/3;
Inc(j)
end;
for i:=3 to n-2
do begin
qtdq^[j] := h^[i]*h^[i+1]/xyw[i].w;
Inc(j);
qtdq^[j] := -h^[i+1]*(h^[i]+h^[i+1])/xyw[i].w
-h^[i+1]*(h^[i+1]+h^[i+2])/xyw[i+1].w +
lambda/h^[i+1]/6;
Inc(j);
qtdq^[j] := sqr(h^[i+1])/xyw[i].w +
sqr(h^[i+1]+h^[i+2])/xyw[i+1].w +
sqr(h^[i+2])/xyw[i+2].w +
lambda*(1/h^[i+1]+1/h^[i+2])/3;
Inc(j)
end;
{ Solving for c/lambda }
Slegpb(n-2, ncd, qtdq^[1], qty^[1], ch^[2], ca, term);
if term=1 then
begin
residu := 0;
for i:=1 to n do
begin
crow := (h^[i]*ch^[i-1] - (h^[i]+h^[i+1])*ch^[i]+h^[i+1]*ch^[i+1])
/xyw[i].w;
xac[i].y := xyw[i].y - crow;
residu := residu + sqr(crow)*xyw[i].w
end;
xac[1].w := 0;
for i:=2 to n-1 do xac[i].w := lambda*ch^[i];
xac[n].w := 0;
end;
Freemem(qtdq, ((n-2)*(ncd+1)-(ncd*(ncd+1)) div 2)*SizeOf(ArbFloat));
Freemem(qty, n*SizeOf(ArbFloat));
Freemem(h, n*SizeOf(ArbFloat));
Freemem(ch, (n+2)*SizeOf(ArbFloat));
end;
procedure spl2nalf(n: ArbInt; var xyzw1: ArbFloat; lambda:ArbFloat;
var xyg0, residu: ArbFloat; var term: ArbInt);
type R3 = array[1..3] of ArbFloat;
R33= array[1..3] of R3;
Rn3= array[1..$ffe0 div SizeOf(R3)] of R3;
var b,e21t,ht :^Rn3;
pfac :par2dr1;
e22 :R33;
i,j,l,i1,i2,n3 :ArbInt;
s,s1,px,py,hr,ca,
x,absdet,x1,x2,
absdetmax :ArbFloat;
vr :R4x;
wr :R2;
w,u :R3;
a_alfa_d :R4xAr absolute xyzw1;
a_gamma :nsp2rec absolute xyg0;
gamma :^arfloat1;
function e(var x,y:R2):ArbFloat;
const c1:ArbFloat=1/(16*pi);
var s:ArbFloat;
begin s:=sqr(x[1]-y[1]) +sqr(x[2]-y[2]);
if s=0 then e:=0 else e:=c1*s*ln(s)
end {e};
procedure pfxpfy(var a,b,c:R2;var f:r3; var pfx,pfy:ArbFloat);
var det:ArbFloat;
begin det:=(b[1]-a[1])*(c[2]-a[2]) - (b[2]-a[2])*(c[1]-a[1]);
pfx:=((f[2]-f[1])*(c[2]-a[2]) - (f[3]-f[1])*(b[2]-a[2]))/det;
pfy:=(-(f[2]-f[1])*(c[1]-a[1]) + (f[3]-f[1])*(b[1]-a[1]))/det
end {pfxpfy};
procedure pxpy(var a,b,c:R2; var px,py:ArbFloat);
var det : ArbFloat;
begin det:=(b[1]-a[1])*(c[2]-a[2]) - (b[2]-a[2])*(c[1]-a[1]);
px:=(b[2]-c[2])/det; py:=(c[1]-b[1])/det
end {pxpy};
function p(var x,a:R2; var px,py:ArbFloat):ArbFloat;
begin p:=1 + (x[1]-a[1])*px +(x[2]-a[2])*py end {p};
procedure slegpdlown(n: ArbInt; var a1; var bx1: ArbFloat;
var term: ArbInt);
var i, j, k, kmin1 : ArbInt;
h, lkk : ArbFloat;
a : ar2dr1 absolute a1;
x : arfloat1 absolute bx1;
begin
k:=0; term := 2;
while (k<n) do
begin
kmin1:=k; k:=k+1; lkk:=a[k]^[k];
for j:=1 to kmin1 do lkk:=lkk-sqr(a[k]^[j]);
if lkk<=0 then exit else
begin
a[k]^[k]:=sqrt(lkk); lkk:=a[k]^[k];
for i:=k+1 to n do
begin
h:=a[i]^[k];
for j:=1 to kmin1 do h:=h-a[k]^[j]*a[i]^[j];
a[i]^[k]:=h/lkk
end; {i}
h:=x[k];
for j:=1 to kmin1 do h:=h-a[k]^[j]*x[j];
x[k]:=h/lkk
end {lkk > 0}
end; {k}
for i:=n downto 1 do
begin
h:=x[i];
for j:=i+1 to n do h:=h-a[j]^[i]*x[j];
x[i]:=h/a[i]^[i];
end; {i}
term := 1
end;
begin
term := 3; if n<3 then exit;
n3 := n - 3;
i1:=1; x1:=a_alfa_d[1].xy[1]; i2:=1; x2:=x1;
for i:= 2 to n do
begin hr:=a_alfa_d[i].xy[1];
if hr < x1 then begin i1:=i; x1:=hr end else
if hr > x2 then begin i2:=i; x2:=hr end;
end;
vr:=a_alfa_d[n-2]; a_alfa_d[n-2]:=a_alfa_d[i1]; a_alfa_d[i1]:=vr;
vr:=a_alfa_d[n-1]; a_alfa_d[n-1]:=a_alfa_d[i2]; a_alfa_d[i2]:=vr;
for i:=1 to 2 do vr.xy[i]:=a_alfa_d[n-2].xy[i]-a_alfa_d[n-1].xy[i];
absdetmax:=-1; i1:=0;
for i:=1 to n do
begin for j:=1 to 2 do wr[j]:=a_alfa_d[i].xy[j]-a_alfa_d[n-2].xy[j];
if a_alfa_d[i].d<=0 then exit;
absdet:=abs(wr[1]*vr.xy[2]-wr[2]*vr.xy[1]);
if absdet > absdetmax then begin i1:=i; absdetmax:=absdet end;
end;
term := 4;
if absdetmax<=macheps*abs(x2-x1) then exit;
term := 1;
vr:=a_alfa_d[n]; a_alfa_d[n]:=a_alfa_d[i1]; a_alfa_d[i1]:=vr;
GetMem(e21t, n3*SizeOf(r3));
GetMem(b, n3*SizeOf(r3));
GetMem(gamma, n*SizeOf(ArbFloat));
pxpy(a_alfa_d[n-2].xy,a_alfa_d[n-1].xy,a_alfa_d[n].xy,px,py);
for i:=1 to n3 do b^[i][1]:=p(a_alfa_d[i].xy,a_alfa_d[n-2].xy,px,py);
pxpy(a_alfa_d[n-1].xy,a_alfa_d[n].xy,a_alfa_d[n-2].xy,px,py);
for i:=1 to n3 do b^[i][2]:=p(a_alfa_d[i].xy,a_alfa_d[n-1].xy,px,py);
pxpy(a_alfa_d[n].xy,a_alfa_d[n-2].xy,a_alfa_d[n-1].xy,px,py);
for i:=1 to n3 do b^[i][3]:=p(a_alfa_d[i].xy,a_alfa_d[n].xy,px,py);
e22[1,1]:=0; e22[2,2]:=0; e22[3,3]:=0;
e22[2,1]:=e(a_alfa_d[n-1].xy,a_alfa_d[n-2].xy); e22[1,2]:=e22[2,1];
e22[3,1]:=e(a_alfa_d[n].xy,a_alfa_d[n-2].xy); e22[1,3]:=e22[3,1];
e22[3,2]:=e(a_alfa_d[n].xy,a_alfa_d[n-1].xy); e22[2,3]:=e22[3,2];
for i:=1 to 3 do
for j:=1 to n3 do e21t^[j,i]:=e(a_alfa_d[n3+i].xy,a_alfa_d[j].xy);
GetMem(ht, n3*SizeOf(r3));
for i:=1 to 3 do
for j:=1 to n3 do
begin s:=0;
for l:= 1 to 3 do s:=s+e22[i,l]*b^[j][l]; ht^[j][i]:=s
end;
AllocateL2dr(n3,pfac);
for i:= 1 to n3 do
for j:= 1 to i do
begin if j=i then s1:=0 else s1:=e(a_alfa_d[i].xy,a_alfa_d[j].xy);
for l:= 1 to 3 do s1:=s1+b^[i][l]*(ht^[j][l]-e21t^[j][l])-e21t^[i][l]*b^[j][l];
if j=i then s:=1/a_alfa_d[i].d else s:=0;
for l:= 1 to 3 do s:=s+b^[i][l]*b^[j][l]/a_alfa_d[n3+l].d;
pfac^[i]^[j] := s1+s/lambda
end;
for i:= 1 to n3 do
gamma^[i]:=a_alfa_d[i].alfa-b^[i][1]*a_alfa_d[n-2].alfa-b^[i][2]*a_alfa_d[n-1].alfa-b^[i][3]*a_alfa_d[n].alfa;
slegpdlown(n3,pfac^[1],gamma^[1],term);
DeAllocateL2dr(n3,pfac);
FreeMem(ht, n3*SizeOf(r3));
if term=1 then
begin
for i:= 1 to 3 do
begin s:= 0;
for j:= 1 to n3 do
s:=s+b^[j][i]*gamma^[j]; w[i]:=s;
gamma^[n3+i]:=-w[i]
end;{w=btgamma}
for i:=1 to 3 do
begin s:=0;
for l:=1 to n3 do s:=s+e21t^[l][i]*gamma^[l];
s1:=0;
for l:=1 to 3 do s1:=s1+e22[i,l]*w[l];
u[i]:=a_alfa_d[n3+i].alfa+w[i]/(lambda*a_alfa_d[n3+i].d)+s1-s
end;
with a_gamma[0] do
pfxpfy(a_alfa_d[n-2].xy,a_alfa_d[n-1].xy,a_alfa_d[n].xy,u,xy[1],xy[2]);
residu:=0;for i:=1 to n3 do residu:=residu+sqr(gamma^[i])/a_alfa_d[i].d;
for i:= 1 to 3 do residu:=residu+sqr(w[i])/a_alfa_d[n3+i].d;
residu:=residu/sqr(lambda);
a_gamma[0].gamma := u[1];
for i:=1 to n do
begin
a_gamma[i].xy := a_alfa_d[i].xy;
a_gamma[i].gamma := gamma^[i]
end;
end;
FreeMem(gamma, n*SizeOf(ArbFloat));
FreeMem(b, n3*SizeOf(r3));
FreeMem(e21t, n3*SizeOf(r3))
end;
function spl2natv(n: ArbInt; var xyg0: ArbFloat; u, v: ArbFloat): ArbFloat;
const c1: ArbFloat=1/(16*pi);
var i : ArbInt;
s : ArbFloat;
a_gamma : nsp2rec absolute xyg0;
z : R2;
function e(var x,y:R2):ArbFloat;
var s:ArbFloat;
begin
s:=sqr(x[1]-y[1]) + sqr(x[2]-y[2]);
if s=0 then
e:= 0
else
e:=s*ln(s)
end {e};
function pf(var x,a:R2;fa,pfx,pfy:ArbFloat):ArbFloat;
begin
pf:=fa + (x[1]-a[1])*pfx + (x[2]-a[2])*pfy
end {pf};
begin
s:=0;
z[1] := u;
z[2] := v;
for i:=1 to n do
s:=s+a_gamma[i].gamma*e(z, a_gamma[i].xy);
with a_gamma[0] do
spl2natv :=s*c1+pf(z,a_gamma[n-2].xy, gamma, xy[1], xy[2])
end;
begin
end.