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{ ******************************************************************
QR decomposition
Ref.: 'Matrix Computations' by Golub & Van Loan
Pascal implementation contributed by Mark Vaughan
****************************************************************** }
unit uqr;
interface
uses
utypes;
procedure QR_Decomp(A : PMatrix;
Lb, Ub1, Ub2 : Integer;
R : PMatrix);
{ ------------------------------------------------------------------
QR decomposition. Factors the matrix A (n x m, with n >= m) as a
product Q * R where Q is a (n x m) column-orthogonal matrix, and R
a (m x m) upper triangular matrix. This routine is used in
conjunction with QR_Solve to solve a system of equations.
------------------------------------------------------------------
Input parameters : A = matrix
Lb = index of first matrix element
Ub1 = index of last matrix element in 1st dim.
Ub2 = index of last matrix element in 2nd dim.
------------------------------------------------------------------
Output parameter : A = contains the elements of Q
R = upper triangular matrix
------------------------------------------------------------------
Possible results : MatOk
MatErrDim
MatSing
------------------------------------------------------------------
NB : This procedure destroys the original matrix A
------------------------------------------------------------------ }
procedure QR_Solve(Q, R : PMatrix;
B : PVector;
Lb, Ub1, Ub2 : Integer;
X : PVector);
{ ------------------------------------------------------------------
Solves a system of equations by the QR decomposition,
after the matrix has been transformed by QR_Decomp.
------------------------------------------------------------------
Input parameters : Q, R = matrices from QR_Decomp
B = constant vector
Lb, Ub1, Ub2 = as in QR_Decomp
------------------------------------------------------------------
Output parameter : X = solution vector
------------------------------------------------------------------ }
implementation
procedure QR_Decomp(A : PMatrix;
Lb, Ub1, Ub2 : Integer;
R : PMatrix);
var
I, J, K : Integer;
Sum : Float;
begin
if Ub2 > Ub1 then
begin
SetErrCode(MatErrDim);
Exit
end;
for K := Lb to Ub2 do
begin
{ Compute the "k"th diagonal entry in R }
Sum := 0.0;
for I := Lb to Ub1 do
Sum := Sum + Sqr(A^[I]^[K]);
if Sum = 0.0 then
begin
SetErrCode(MatSing);
Exit;
end;
R^[K]^[K] := Sqrt(Sum);
{ Divide the entries in the "k"th column of A by the computed "k"th }
{ diagonal element of R. this begins the process of overwriting A }
{ with Q . . . }
for I := Lb to Ub1 do
A^[I]^[K] := A^[I]^[K] / R^[K]^[K];
for J := (K + 1) to Ub2 do
begin
{ Complete the remainder of the row entries in R }
Sum := 0.0;
for I := Lb to Ub1 do
Sum := Sum + A^[I]^[K] * A^[I]^[J];
R^[K]^[J] := Sum;
{ Update the column entries of the Q/A matrix }
for I := Lb to Ub1 do
A^[I]^[J] := A^[I]^[J] - A^[I]^[K] * R^[K]^[J];
end;
end;
SetErrCode(MatOk);
end;
procedure QR_Solve(Q, R : PMatrix;
B : PVector;
Lb, Ub1, Ub2 : Integer;
X : PVector);
var
I, J : Integer;
Sum : Float;
begin
{ Form Q'B and store the result in X }
for J := Lb to Ub2 do
begin
X^[J] := 0.0;
for I := Lb to Ub1 do
X^[J] := X^[J] + Q^[I]^[J] * B^[I];
end;
{ Update X with the solution vector }
X^[Ub2] := X^[Ub2] / R^[Ub2]^[Ub2];
for I := (Ub2 - 1) downto Lb do
begin
Sum := 0.0;
for J := (I + 1) to Ub2 do
Sum := Sum + R^[I]^[J] * X^[J];
X^[I] := (X^[I] - Sum) / R^[I]^[I];
end;
end;
end.
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