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sle.pas
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sle.pas
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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)
!! modifies randseed, might not exactly work as TP version!!!
Solve set of linear equations of the type Ax=b, for generic, and a
variety of special matrices.
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.
**********************************************************************}
{Solve set of linear equations of the type Ax=b, for generic, and a variety of
special matrices.
One (generic) function for overdetermined sets of this kind : slegls
overdetermined are sets that look like this: (I don't know if I
translated "overdetermined" right)
6 1 2 3 9
3 9 3 4 2
17 27 42 15 62
17 27 42 15 61
The two bottom rows look much alike, which introduces a big uncertainty in the
result, therefore these matrices need special treatment.
All procedures have similar procedure with a "L" appended to the name. We
didn't receive docs for those procedures. If you know what the difference is,
please mail us }
Unit sle;
interface
// gligli Single version with only slegls
type
ArbFloat = Single;
ArbInt = LONGINT;
const
highestfloatelement = High(ArbInt) div SizeOf(ArbFloat);
highestintelement = High(ArbInt) div SizeOf(ArbInt);
highestptrelement = High(ArbInt) div SizeOf(Pointer);
type
arfloat1 = array[1..highestfloatelement] of ArbFloat;
arint1 = array[1..highestintelement] of ArbInt;
ar2dr1 = array[1..highestptrelement] of ^arfloat1;
par2dr1 = ^ar2dr1;
{solve for overdetermined matrices, see unit comments}
Procedure slegls(Var a: ArbFloat; m, n, rwidtha: ArbInt; Var b, x: ArbFloat;
Var term: ArbInt);
implementation
Uses DSL,MDT;
{Here originally stood an exact copy of mdtgtr from unit mdt}
{Here originally stood an exact copy of dslgtr from unit DSL}
Procedure decomp(Var qr: ArbFloat; m, n, rwidthq: ArbInt; Var alpha: ArbFloat;
Var pivot, term: ArbInt);
Var i, j, jbar, k, ns, ii : ArbInt;
beta, sigma, alphak, qrkk, s : ArbFloat;
pqr, pal, y, sum : ^arfloat1;
piv : ^arint1;
Begin
term := 1;
pqr := @qr;
pal := @alpha;
piv := @pivot;
ns := n*sizeof(ArbFloat);
getmem(y, ns);
getmem(sum, ns);
For j:=1 To n Do
Begin
s := 0;
For i:=1 To m Do
s := s+sqr(pqr^[(i-1)*rwidthq+j]);
sum^[j] := s;
piv^[j] := j
End; {j}
For k:=1 To n Do
Begin
sigma := sum^[k];
jbar := k;
For j:=k+1 To n Do
If sigma < sum^[j] Then
Begin
sigma := sum^[j];
jbar := j
End;
If jbar <> k
Then
Begin
i := piv^[k];
piv^[k] := piv^[jbar];
piv^[jbar] := i;
sum^[jbar] := sum^[k];
sum^[k] := sigma;
For i:=1 To m Do
Begin
ii := (i-1)*rwidthq;
sigma := pqr^[ii+k];
pqr^[ii+k] := pqr^[ii+jbar];
pqr^[ii+jbar] := sigma
End; {i}
End; {column interchange}
sigma := 0;
For i:=k To m Do
sigma := sigma+sqr(pqr^[(i-1)*rwidthq+k]);
If sigma=0 Then
Begin
term := 2;
freemem(y, ns);
freemem(sum, ns);
exit
End;
qrkk := pqr^[(k-1)*rwidthq+k];
If qrkk < 0 Then
alphak := sqrt(sigma)
Else
alphak := -sqrt(sigma);
pal^[k] := alphak;
beta := 1/(sigma-qrkk*alphak);
pqr^[(k-1)*rwidthq+k] := qrkk-alphak;
For j:=k+1 To n Do
Begin
s := 0;
For i:=k To m Do
Begin
ii := (i-1)*rwidthq;
s := s+pqr^[ii+k]*pqr^[ii+j]
End; {i}
y^[j] := beta*s
End; {j}
For j:=k+1 To n Do
Begin
For i:=k To m Do
Begin
ii := (i-1)*rwidthq;
pqr^[ii+j] := pqr^[ii+j]-pqr^[ii+k]*y^[j]
End; {i}
sum^[j] := sum^[j]-sqr(pqr^[(k-1)*rwidthq+j])
End {j}
End; {k}
freemem(y, ns);
freemem(sum, ns);
End; {decomp}
Procedure decomp1(Var qr1; m, n: ArbInt; Var alpha1: ArbFloat;
Var pivot1, term: ArbInt);
Var i, j, jbar, k, ns : ArbInt;
beta, sigma, alphak, qrkk, s : ArbFloat;
qr : ar2dr1 absolute qr1;
alpha : arfloat1 absolute alpha1;
pivot : arint1 absolute pivot1;
y, sum : ^arfloat1;
Begin
term := 1;
ns := n*sizeof(ArbFloat);
getmem(y, ns);
getmem(sum, ns);
For j:=1 To n Do
Begin
s := 0;
For i:=1 To m Do
s := s+sqr(qr[i]^[j]);
sum^[j] := s;
pivot[j] := j
End; {j}
For k:=1 To n Do
Begin
sigma := sum^[k];
jbar := k;
For j:=k+1 To n Do
If sigma < sum^[j]
Then
Begin
sigma := sum^[j];
jbar := j
End;
If jbar <> k
Then
Begin
i := pivot[k];
pivot[k] := pivot[jbar];
pivot[jbar] := i;
sum^[jbar] := sum^[k];
sum^[k] := sigma;
For i:=1 To m Do
Begin
sigma := qr[i]^[k];
qr[i]^[k] := qr[i]^[jbar];
qr[i]^[jbar] := sigma
End; {i}
End; {column interchange}
sigma := 0;
For i:=k To m Do
sigma := sigma+sqr(qr[i]^[k]);
If sigma=0
Then
Begin
term := 2;
freemem(y, ns);
freemem(sum, ns);
exit
End;
qrkk := qr[k]^[k];
If qrkk < 0 Then alphak := sqrt(sigma)
Else alphak := -sqrt(sigma);
alpha[k] := alphak;
beta := 1/(sigma-qrkk*alphak);
qr[k]^[k] := qrkk-alphak;
For j:=k+1 To n Do
Begin
s := 0;
For i:=k To m Do
s := s+qr[i]^[k]*qr[i]^[j];
y^[j] := beta*s
End; {j}
For j:=k+1 To n Do
Begin
For i:=k To m Do
qr[i]^[j] := qr[i]^[j]-qr[i]^[k]*y^[j];
sum^[j] := sum^[j]-sqr(qr[k]^[j])
End {j}
End; {k}
freemem(y, ns);
freemem(sum, ns);
End; {decomp1}
Procedure solve(Var qr: ArbFloat; m, n, rwidthq: ArbInt; Var alpha: ArbFloat;
Var pivot: ArbInt; Var r, y: ArbFloat);
Var i, j, ii : ArbInt;
gamma, s : ArbFloat;
pqr, pal, pr, py, z : ^arfloat1;
piv : ^arint1;
Begin
pqr := @qr;
pal := @alpha;
piv := @pivot;
pr := @r;
py := @y;
getmem(z, n*sizeof(ArbFloat));
For j:=1 To n Do
Begin
gamma := 0;
For i:=j To m Do
gamma := gamma+pqr^[(i-1)*rwidthq+j]*pr^[i];
gamma := gamma/(pal^[j]*pqr^[(j-1)*rwidthq+j]);
For i:=j To m Do
pr^[i] := pr^[i]+gamma*pqr^[(i-1)*rwidthq+j]
End; {j}
z^[n] := pr^[n]/pal^[n];
For i:=n-1 Downto 1 Do
Begin
s := pr^[i];
ii := (i-1)*rwidthq;
For j:=i+1 To n Do
s := s-pqr^[ii+j]*z^[j];
z^[i] := s/pal^[i]
End; {i}
For i:=1 To n Do
py^[piv^[i]] := z^[i];
freemem(z, n*sizeof(ArbFloat));
End; {solve}
Procedure solve1(Var qr1; m, n: ArbInt; Var alpha1: ArbFloat;
Var pivot1: ArbInt; Var r1, y1: ArbFloat);
Var i, j : ArbInt;
gamma, s : ArbFloat;
qr : ar2dr1 absolute qr1;
alpha : arfloat1 absolute alpha1;
r : arfloat1 absolute r1;
y : arfloat1 absolute y1;
pivot : arint1 absolute pivot1;
z : ^arfloat1;
Begin
getmem(z, n*sizeof(ArbFloat));
For j:=1 To n Do
Begin
gamma := 0;
For i:=j To m Do
gamma := gamma+qr[i]^[j]*r[i];
gamma := gamma/(alpha[j]*qr[j]^[j]);
For i:=j To m Do
r[i] := r[i]+gamma*qr[i]^[j]
End; {j}
z^[n] := r[n]/alpha[n];
For i:=n-1 Downto 1 Do
Begin
s := r[i];
For j:=i+1 To n Do
s := s-qr[i]^[j]*z^[j];
z^[i] := s/alpha[i]
End; {i}
For i:=1 To n Do
y[pivot[i]] := z^[i];
freemem(z, n*sizeof(ArbFloat));
End; {solve1}
Procedure slegls(Var a: ArbFloat; m, n, rwidtha: ArbInt; Var b, x: ArbFloat;
Var term: ArbInt);
Var i, j, ns, ms, ii : ArbInt;
normy0, norme1, s : ArbFloat;
pa, pb, px, qr, alpha, e, y, r : ^arfloat1;
pivot : ^arint1;
Begin
If (n<1) Or (m<n)
Then
Begin
term := 3;
exit
End;
pa := @a;
pb := @b;
px := @x;
ns := n*sizeof(ArbFloat);
ms := m*sizeof(ArbFloat);
getmem(qr, m*ns);
getmem(alpha, ns);
getmem(e, ns);
getmem(y, ns);
getmem(r, m*sizeof(ArbFloat));
getmem(pivot, n*sizeof(ArbInt));
For i:=1 To m Do
move(pa^[(i-1)*rwidtha+1], qr^[(i-1)*n+1], ns);
decomp(qr^[1], m, n, n, alpha^[1], pivot^[1], term);
If term=2
Then
Begin
freemem(qr, m*ns);
freemem(alpha, ns);
freemem(e, ns);
freemem(y, ns);
freemem(r, m*sizeof(ArbFloat));
freemem(pivot, n*sizeof(ArbInt));
exit
End;
move(pb^[1], r^[1], ms);
solve(qr^[1], m, n, n, alpha^[1], pivot^[1], r^[1], y^[1]);
For i:=1 To m Do
Begin
s := pb^[i];
ii := (i-1)*rwidtha;
For j:=1 To n Do
s := s-pa^[ii+j]*y^[j];
r^[i] := s
End; {i}
solve(qr^[1], m, n, n, alpha^[1], pivot^[1], r^[1], e^[1]);
normy0 := 0;
norme1 := 0;
For i:=1 To n Do
Begin
normy0 := normy0+sqr(y^[i]);
norme1 := norme1+sqr(e^[i])
End; {i}
If norme1 > 0.0625*normy0
Then
Begin
term := 2;
freemem(qr, m*ns);
freemem(alpha, ns);
freemem(e, ns);
freemem(y, ns);
freemem(r, m*sizeof(ArbFloat));
freemem(pivot, n*sizeof(ArbInt));
exit
End;
For i:=1 To n Do
px^[i] := y^[i];
freemem(qr, m*ns);
freemem(alpha, ns);
freemem(e, ns);
freemem(y, ns);
freemem(r, m*sizeof(ArbFloat));
freemem(pivot, n*sizeof(ArbInt));
End; {slegls}
Begin
{$ifdef fixate_random}
randseed := 12345
{$endif}
End.