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{ ******************************************************************
Singular value decomposition
****************************************************************** }
unit usvd;
interface
uses
utypes, uminmax, utrigo;
procedure SV_Decomp(A : PMatrix;
Lb, Ub1, Ub2 : Integer;
S : PVector;
V : PMatrix);
{ ------------------------------------------------------------------
Singular value decomposition. Factors the matrix A (n x m, with
n >= m) as a product U * S * V' where U is a (n x m) column-
orthogonal matrix, S a (m x m) diagonal matrix with elements >= 0
(the singular values) and V a (m x m) orthogonal matrix. This
routine is used in conjunction with SV_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 U
S = vector of singular values
V = orthogonal matrix
------------------------------------------------------------------
Possible results :
MatOk : No error
MatNonConv : Non-convergence
MatErrDim : Non-compatible dimensions (n < m)
------------------------------------------------------------------
NB : This procedure destroys the original matrix A
------------------------------------------------------------------ }
procedure SV_SetZero(S : PVector;
Lb, Ub : Integer;
Tol : Float);
{ ------------------------------------------------------------------
Sets the singular values to zero if they are lower than a
specified threshold.
------------------------------------------------------------------
Input parameters : S = vector of singular values
Tol = relative tolerance
Threshold value will be Tol * Max(S)
Lb = index of first vector element
Ub = index of last vector element
------------------------------------------------------------------
Output parameter : S = modified singular values
------------------------------------------------------------------ }
procedure SV_Solve(U : PMatrix;
S : PVector;
V : PMatrix;
B : PVector;
Lb, Ub1, Ub2 : Integer;
X : PVector);
{ ------------------------------------------------------------------
Solves a system of equations by singular value decomposition,
after the matrix has been transformed by SV_Decomp, and the lowest
singular values have been set to zero by SV_SetZero.
------------------------------------------------------------------
Input parameters : U, S, V = vector and matrices
from SV_Decomp
B = constant vector
Lb, Ub1, Ub2 = as in SV_Decomp
------------------------------------------------------------------
Output parameter : X = solution vector
= V * Diag(1/s(i)) * U' * B, for s(i) <> 0
------------------------------------------------------------------ }
procedure SV_Approx(U : PMatrix;
S : PVector;
V : PMatrix;
Lb, Ub1, Ub2 : Integer;
A : PMatrix);
{ ------------------------------------------------------------------
Approximates a matrix A by the product USV', after the lowest
singular values have been set to zero by SV_SetZero.
------------------------------------------------------------------
Input parameters : U, S, V = vector and matrices
from SV_Decomp
Lb, Ub1, Ub2 = as in SV_Decomp
------------------------------------------------------------------
Output parameter : A = approximated matrix
------------------------------------------------------------------ }
implementation
procedure SV_Decomp(A : PMatrix;
Lb, Ub1, Ub2 : Integer;
S : PVector;
V : PMatrix);
{ ----------------------------------------------------------------------
This procedure is a translation of the EISPACK subroutine SVD
This procedure determines the singular value decomposition A = U.S.V'
of a real M by N rectangular matrix. Householder bidiagonalization and
a variant of the QR algorithm are used.
----------------------------------------------------------------------
This is a crude translation. Many of the original goto's
have been kept!
---------------------------------------------------------------------- }
var
I, J, K, L, I1, K1, L1, Mn, Its : Integer;
C, F, G, H, T, X, Y, Z, Tst1, Tst2, Scale : Float;
R : PVector;
label
190, 210, 270, 290, 360, 390, 430, 460,
475, 490, 520, 540, 565, 580, 650, 700;
begin
if Ub2 > Ub1 then
begin
SetErrCode(MatErrDim);
Exit
end;
DimVector(R, Ub2);
Scale := 0.0;
G := 0.0;
X := 0.0;
{ Householder reduction to bidiagonal form }
for I := Lb to Ub2 do
begin
L := I + 1;
R^[I] := Scale * G;
G := 0.0;
T := 0.0;
Scale := 0.0;
if I > Ub1 then goto 210;
for K := I to Ub1 do
Scale := Scale + Abs(A^[K]^[I]);
if Scale = 0.0 then goto 210;
for K := I to Ub1 do
begin
A^[K]^[I] := A^[K]^[I] / Scale;
T := T + Sqr(A^[K]^[I]);
end;
F := A^[I]^[I];
G := - DSgn(Sqrt(T), F);
H := F * G - T;
A^[I]^[I] := F - G;
if I = Ub2 then goto 190;
for J := L to Ub2 do
begin
T := 0.0;
for K := I to Ub1 do
T := T + A^[K]^[I] * A^[K]^[J];
F := T / H;
for K := I to Ub1 do
A^[K]^[J] := A^[K]^[J] + F * A^[K]^[I];
end;
190: for K := I to Ub1 do
A^[K]^[I] := Scale * A^[K]^[I];
210: S^[I] := Scale * G;
G := 0.0;
T := 0.0;
Scale := 0.0;
if (I > Ub1) or (I = Ub2) then goto 290;
for K := L to Ub2 do
Scale := Scale + Abs(A^[I]^[K]);
if Scale = 0.0 then goto 290;
for K := L to Ub2 do
begin
A^[I]^[K] := A^[I]^[K] / Scale;
T := T + Sqr(A^[I]^[K]);
end;
F := A^[I]^[L];
G := - DSgn(Sqrt(T), F);
H := F * G - T;
A^[I]^[L] := F - G;
for K := L to Ub2 do
R^[K] := A^[I]^[K] / H;
if I = Ub1 then goto 270;
for J := L to Ub1 do
begin
T := 0.0;
for K := L to Ub2 do
T := T + A^[J]^[K] * A^[I]^[K];
for K := L to Ub2 do
A^[J]^[K] := A^[J]^[K] + T * R^[K];
end;
270: for K := L to Ub2 do
A^[I]^[K] := Scale * A^[I]^[K];
290: X := FMax(X, Abs(S^[I]) + Abs(R^[I]));
end;
{ Accumulation of right-hand transformations }
for I := Ub2 downto Lb do
begin
if I = Ub2 then goto 390;
if G = 0.0 then goto 360;
for J := L to Ub2 do
{ Double division avoids possible underflow }
V^[J]^[I] := (A^[I]^[J] / A^[I]^[L]) / G;
for J := L to Ub2 do
begin
T := 0.0;
for K := L to Ub2 do
T := T + A^[I]^[K] * V^[K]^[J];
for K := L to Ub2 do
V^[K]^[J] := V^[K]^[J] + T * V^[K]^[I];
end;
360: for J := L to Ub2 do
begin
V^[I]^[J] := 0.0;
V^[J]^[I] := 0.0;
end;
390: V^[I]^[I] := 1.0;
G := R^[I];
L := I;
end;
{ Accumulation of left-hand transformations }
Mn := IMin(Ub1, Ub2);
for I := Mn downto Lb do
begin
L := I + 1;
G := S^[I];
if I = Ub2 then goto 430;
for J := L to Ub2 do
A^[I]^[J] := 0.0;
430: if G = 0.0 then goto 475;
if I = Mn then goto 460;
for J := L to Ub2 do
begin
T := 0.0;
for K := L to Ub1 do
T := T + A^[K]^[I] * A^[K]^[J];
{ Double division avoids possible underflow }
F := (T / A^[I]^[I]) / G;
for K := I to Ub1 do
A^[K]^[J] := A^[K]^[J] + F * A^[K]^[I];
end;
460: for J := I to Ub1 do
A^[J]^[I] := A^[J]^[I] / G;
goto 490;
475: for J := I to Ub1 do
A^[J]^[I] := 0.0;
490: A^[I]^[I] := A^[I]^[I] + 1.0;
end;
{ Diagonalization of the bidiagonal form }
Tst1 := X;
for K := Ub2 downto Lb do
begin
K1 := K - 1;
Its := 0;
520: { Test for splitting }
for L := K downto Lb do
begin
L1 := L - 1;
Tst2 := Tst1 + Abs(R^[L]);
if Tst2 = Tst1 then goto 565;
{ R^[Lb] is always zero, so there is no exit
through the bottom of the loop }
Tst2 := Tst1 + Abs(S^[L1]);
if Tst2 = Tst1 then goto 540;
end;
540: { Cancellation of R^[L] if L greater than 1 }
C := 0.0;
T := 1.0;
for I := L to K do
begin
F := T * R^[I];
R^[I] := C * R^[I];
Tst2 := Tst1 + Abs(F);
if Tst2 = Tst1 then goto 565;
G := S^[I];
H := Pythag(F, G);
S^[I] := H;
C := G / H;
T := - F / H;
for J := Lb to Ub1 do
begin
Y := A^[J]^[L1];
Z := A^[J]^[I];
A^[J]^[L1] := Y * C + Z * T;
A^[J]^[I] := - Y * T + Z * C;
end;
end;
565: { Test for convergence }
Z := S^[K];
if L = K then goto 650;
if Its = 30 then
begin
SetErrCode(MatNonConv);
DelVector(R, Ub2);
Exit;
end;
{ Shift from bottom 2 by 2 minor }
Its := Its + 1;
X := S^[L];
Y := S^[K1];
G := R^[K1];
H := R^[K];
F := 0.5 * (((G + Z) / H) * ((G - Z) / Y) + Y / H - H / Y);
G := Pythag(F, 1.0);
F := X - (Z / X) * Z + (H / X) * (Y / (F + DSgn(G, F)) - H);
{ Next QR transformation }
C := 1.0;
T := 1.0;
for I1 := L to K1 do
begin
I := I1 + 1;
G := R^[I];
Y := S^[I];
H := T * G;
G := C * G;
Z := Pythag(F, H);
R^[I1] := Z;
C := F / Z;
T := H / Z;
F := X * C + G * T;
G := - X * T + G * C;
H := Y * T;
Y := Y * C;
for J := Lb to Ub2 do
begin
X := V^[J]^[I1];
Z := V^[J]^[I];
V^[J]^[I1] := X * C + Z * T;
V^[J]^[I] := - X * T + Z * C;
end;
Z := Pythag(F, H);
S^[I1] := Z;
{ Rotation can be arbitrary if Z is zero }
if Z = 0.0 then goto 580;
C := F / Z;
T := H / Z;
580: F := C * G + T * Y;
X := - T * G + C * Y;
for J := Lb to Ub1 do
begin
Y := A^[J]^[I1];
Z := A^[J]^[I];
A^[J]^[I1] := Y * C + Z * T;
A^[J]^[I] := - Y * T + Z * C;
end;
end;
R^[L] := 0.0;
R^[K] := F;
S^[K] := X;
goto 520;
650: { Convergence }
if Z >= 0.0 then goto 700;
{ S^[K] is made non-negative }
S^[K] := - Z;
for J := Lb to Ub2 do
V^[J]^[K] := - V^[J]^[K];
700: end;
DelVector(R, Ub2);
SetErrCode(MatOk);
end;
procedure SV_SetZero(S : PVector;
Lb, Ub : Integer;
Tol : Float);
var
Threshold : Float;
I : Integer;
begin
Threshold := S^[Lb];
for I := Lb + 1 to Ub do
if S^[I] > Threshold then Threshold := S^[I];
Threshold := Tol * Threshold;
for I := Lb to Ub do
if S^[I] < Threshold then S^[I] := 0.0;
end;
procedure SV_Solve(U : PMatrix;
S : PVector;
V : PMatrix;
B : PVector;
Lb, Ub1, Ub2 : Integer;
X : PVector);
var
I, J, K : Integer;
Sum : Float;
Tmp : PVector;
begin
DimVector(Tmp, Ub2);
for J := Lb to Ub2 do
begin
Sum := 0.0;
if S^[J] > 0.0 then
begin
for I := Lb to Ub1 do
Sum := Sum + U^[I]^[J] * B^[I];
Sum := Sum / S^[J];
end;
Tmp^[J] := Sum;
end;
for J := Lb to Ub2 do
begin
Sum := 0.0;
for K := Lb to Ub2 do
Sum := Sum + V^[J]^[K] * Tmp^[K];
X^[J] := Sum;
end;
DelVector(Tmp, Ub2);
end;
procedure SV_Approx(U : PMatrix;
S : PVector;
V : PMatrix;
Lb, Ub1, Ub2 : Integer;
A : PMatrix);
var
I, J, K : Integer;
begin
for I := Lb to Ub1 do
for J := Lb to Ub2 do
begin
A^[I]^[J] := 0.0;
for K := Lb to Ub2 do
if S^[K] > 0.0 then
A^[I]^[J] := A^[I]^[J] + U^[I]^[K] * V^[J]^[K];
end;
end;
end.
|
