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{ ******************************************************************
Eigenvalues and eigenvectors of a real upper Hessenberg matrix
****************************************************************** }
unit uhqr2;
interface
uses
utypes, uminmax;
procedure Hqr2(H : PMatrix;
Lb, Ub, I_low, I_igh : Integer;
Lambda : PCompVector;
Z : PMatrix);
implementation
procedure Hqr2(H : PMatrix;
Lb, Ub, I_low, I_igh : Integer;
Lambda : PCompVector;
Z : PMatrix);
{ ------------------------------------------------------------------
This function is a translation of the EISPACK subroutine hqr2
This procedure finds the eigenvalues and eigenvectors of a real
upper Hessenberg matrix by the QR method.
On input:
H contains the upper Hessenberg matrix.
Lb, Ub are the lowest and highest indices
of the elements of H
I_low and I_igh are integers determined by the balancing subroutine
Balance. If Balance has not been used, set I_low=Lb, I_igh=Ub
Z contains the transformation matrix produced by Eltran after the
reduction by Elmhes, or by Ortran after the reduction by Orthes, if
performed. If the eigenvectors of the Hessenberg matrix are desired,
Z must contain the identity matrix.
On output:
H has been destroyed.
Wr and Wi contain the real and imaginary parts, respectively, of
the eigenvalues. The eigenvalues are unordered except that complex
conjugate pairs of values appear consecutively with the eigenvalue
having the positive imaginary part first.
Z contains the real and imaginary parts of the eigenvectors. If the
i-th eigenvalue is real, the i-th column of Z contains its eigenvector.
If the i-th eigenvalue is complex with positive imaginary part, the i-th
and (i+1)-th columns of Z contain the real and imaginary parts of its
eigenvector. The eigenvectors are unnormalized. If an error exit is made,
none of the eigenvectors has been found.
The function returns an error code:
zero for normal return,
-j if the limit of 30*N iterations is exhausted
while the j-th eigenvalue is being sought
(N being the size of the matrix). The eigenvalues
should be correct for indices j+1,...,Ub.
------------------------------------------------------------------
Note: This is a crude translation. Many of the original goto's
have been kept !
------------------------------------------------------------------ }
procedure Cdiv(Ar, Ai, Br, Bi : Float; var Cr, Ci : Float);
{ Complex division, (Cr,Ci) = (Ar,Ai)/(Br,Bi) }
var
S, Ars, Ais, Brs, Bis : Float;
begin
S := Abs(Br) + Abs(Bi);
Ars := Ar / S;
Ais := Ai / S;
Brs := Br / S;
Bis := Bi / S;
S := Sqr(Brs) + Sqr(Bis);
Cr := (Ars * Brs + Ais * Bis) / S;
Ci := (Ais * Brs - Ars * Bis) / S;
end;
var
I, J, K, L, M, N, En, Na, Itn, Its, Mp2, Enm2 : Integer;
P, Q, R, S, T, W, X, Y, Ra, Sa, Vi, Vr, Zz, Norm, Tst1, Tst2 : Float;
NotLas : Boolean;
label
60, 70, 100, 130, 150, 170, 225, 260, 270, 280, 320, 330, 340,
600, 630, 635, 640, 680, 700, 710, 770, 780, 790, 795, 800;
begin
{ Store roots isolated by Balance and compute matrix norm }
K := Lb;
Norm := 0.0;
for I := Lb to Ub do
begin
for J := K to Ub do
Norm := Norm + Abs(H^[I]^[J]);
K := I;
if (I < I_low) or (I > I_igh) then
begin
Lambda^[I].X := H^[I]^[I];
Lambda^[I].Y := 0.0;
end;
end;
N := Ub - Lb + 1;
Itn := 30 * N;
En := I_igh;
T := 0.0;
60: { Search for next eigenvalues }
if En < I_low then goto 340;
Its := 0;
Na := En - 1;
Enm2 := Na - 1;
70: { Look for single small sub-diagonal element }
for L := En downto I_low do
begin
if L = I_low then goto 100;
S := Abs(H^[L - 1]^[L - 1]) + Abs(H^[L]^[L]);
if S = 0.0 then S := Norm;
Tst1 := S;
Tst2 := Tst1 + Abs(H^[L]^[L - 1]);
if Tst2 = Tst1 then goto 100;
end;
100: { Form shift }
X := H^[En]^[En];
if L = En then goto 270;
Y := H^[Na]^[Na];
W := H^[En]^[Na] * H^[Na]^[En];
if L = Na then goto 280;
if Itn = 0 then
begin
{ Set error -- all eigenvalues have not
converged after 30*N iterations }
SetErrCode(- En);
Exit;
end;
if (Its <> 10) and (Its <> 20) then goto 130;
{ Form exceptional shift }
T := T + X;
for I := I_low to En do
H^[I]^[I] := H^[I]^[I] - X;
S := Abs(H^[En]^[Na]) + Abs(H^[Na]^[Enm2]);
X := 0.75 * S;
Y := X;
W := - 0.4375 * S * S;
130:
Its := Its + 1;
Itn := Itn - 1;
{ Look for two consecutive small sub-diagonal elements }
for M := Enm2 downto L do
begin
Zz := H^[M]^[M];
R := X - Zz;
S := Y - Zz;
P := (R * S - W) / H^[M + 1]^[M] + H^[M]^[M + 1];
Q := H^[M + 1]^[M + 1] - Zz - R - S;
R := H^[M + 2]^[M + 1];
S := Abs(P) + Abs(Q) + Abs(R);
P := P / S;
Q := Q / S;
R := R / S;
if M = L then goto 150;
Tst1 := Abs(P) * (Abs(H^[M - 1]^[M - 1]) + Abs(Zz) + Abs(H^[M + 1]^[M + 1]));
Tst2 := Tst1 + Abs(H^[M]^[M - 1]) * (Abs(Q) + Abs(R));
if Tst2 = Tst1 then goto 150;
end;
150:
Mp2 := M + 2;
for I := Mp2 to En do
begin
H^[I]^[I - 2] := 0.0;
if I <> Mp2 then H^[I]^[I - 3] := 0.0;
end;
{ Double QR step involving rows L to En and columns M to En }
for K := M to Na do
begin
NotLas := (K <> Na);
if (K = M) then goto 170;
P := H^[K]^[K - 1];
Q := H^[K + 1]^[K - 1];
R := 0.0;
if NotLas then R := H^[K + 2]^[K - 1];
X := Abs(P) + Abs(Q) + Abs(R);
if X = 0.0 then goto 260;
P := P / X;
Q := Q / X;
R := R / X;
170: S := DSgn(Sqrt(P * P + Q * Q + R * R), P);
if K <> M then
H^[K]^[K - 1] := - S * X
else if L <> M then
H^[K]^[K - 1] := - H^[K]^[K - 1];
P := P + S;
X := P / S;
Y := Q / S;
Zz := R / S;
Q := Q / P;
R := R / P;
if NotLas then goto 225;
{ Row modification }
for J := K to Ub do
begin
P := H^[K]^[J] + Q * H^[K + 1]^[J];
H^[K]^[J] := H^[K]^[J] - P * X;
H^[K + 1]^[J] := H^[K + 1]^[J] - P * Y;
end;
J := Imin(En, K + 3);
{ Column modification }
for I := Lb to J do
begin
P := X * H^[I]^[K] + Y * H^[I]^[K + 1];
H^[I]^[K] := H^[I]^[K] - P;
H^[I]^[K + 1] := H^[I]^[K + 1] - P * Q;
end;
{ Accumulate transformations }
for I := I_low to I_igh do
begin
P := X * Z^[I]^[K] + Y * Z^[I]^[K + 1];
Z^[I]^[K] := Z^[I]^[K] - P;
Z^[I]^[K + 1] := Z^[I]^[K + 1] - P * Q;
end;
goto 260;
225:
{ Row modification }
for J := K to Ub do
begin
P := H^[K]^[J] + Q * H^[K + 1]^[J] + R * H^[K + 2]^[J];
H^[K]^[J] := H^[K]^[J] - P * X;
H^[K + 1]^[J] := H^[K + 1]^[J] - P * Y;
H^[K + 2]^[J] := H^[K + 2]^[J] - P * Zz;
end;
J := Imin(En, K + 3);
{ Column modification }
for I := Lb to J do
begin
P := X * H^[I]^[K] + Y * H^[I]^[K + 1] + Zz * H^[I]^[K + 2];
H^[I]^[K] := H^[I]^[K] - P;
H^[I]^[K + 1] := H^[I]^[K + 1] - P * Q;
H^[I]^[K + 2] := H^[I]^[K + 2] - P * R;
end;
{ Accumulate transformations }
for I := I_low to I_igh do
begin
P := X * Z^[I]^[K] + Y * Z^[I]^[K + 1] + Zz * Z^[I]^[K + 2];
Z^[I]^[K] := Z^[I]^[K] - P;
Z^[I]^[K + 1] := Z^[I]^[K + 1] - P * Q;
Z^[I]^[K + 2] := Z^[I]^[K + 2] - P * R;
end;
260: end;
goto 70;
270: { One root found }
H^[En]^[En] := X + T;
Lambda^[En].X := H^[En]^[En];
Lambda^[En].Y := 0.0;
En := Na;
goto 60;
280: { Two roots found }
P := 0.5 * (Y - X);
Q := P * P + W;
Zz := Sqrt(Abs(Q));
H^[En]^[En] := X + T;
X := H^[En]^[En];
H^[Na]^[Na] := Y + T;
if Q < 0.0 then goto 320;
{ Real pair }
Zz := P + DSgn(Zz, P);
Lambda^[Na].X := X + Zz;
Lambda^[En].X := Lambda^[Na].X;
if Zz <> 0.0 then Lambda^[En].X := X - W / Zz;
Lambda^[Na].Y := 0.0;
Lambda^[En].Y := 0.0;
X := H^[En]^[Na];
S := Abs(X) + Abs(Zz);
P := X / S;
Q := Zz / S;
R := Sqrt(P * P + Q * Q);
P := P / R;
Q := Q / R;
{ Row modification }
for J := Na to Ub do
begin
Zz := H^[Na]^[J];
H^[Na]^[J] := Q * Zz + P * H^[En]^[J];
H^[En]^[J] := Q * H^[En]^[J] - P * Zz;
end;
{ Column modification }
for I := Lb to En do
begin
Zz := H^[I]^[Na];
H^[I]^[Na] := Q * Zz + P * H^[I]^[En];
H^[I]^[En] := Q * H^[I]^[En] - P * Zz;
end;
{ Accumulate transformations }
for I := I_low to I_igh do
begin
Zz := Z^[I]^[Na];
Z^[I]^[Na] := Q * Zz + P * Z^[I]^[En];
Z^[I]^[En] := Q * Z^[I]^[En] - P * Zz;
end;
goto 330;
320: { Complex pair }
Lambda^[Na].X := X + P;
Lambda^[En].X := Lambda^[Na].X;
Lambda^[Na].Y := Zz;
Lambda^[En].Y := - Zz;
330:
En := Enm2;
goto 60;
340:
if Norm = 0.0 then Exit;
{ All roots found. Backsubstitute to find
vectors of upper triangular form }
for En := Ub downto Lb do
begin
P := Lambda^[En].X;
Q := Lambda^[En].Y;
Na := En - 1;
if Q < 0.0 then
goto 710
else if Q = 0.0 then
goto 600
else
goto 800;
600: { Real vector }
M := En;
H^[En]^[En] := 1.0;
if Na < Lb then goto 800;
for I := Na downto Lb do
begin
W := H^[I]^[I] - P;
R := 0.0;
for J := M to En do
R := R + H^[I]^[J] * H^[J]^[En];
if Lambda^[I].Y >= 0.0 then goto 630;
Zz := W;
S := R;
goto 700;
630: M := I;
if Lambda^[I].Y <> 0.0 then goto 640;
T := W;
if T <> 0.0 then goto 635;
Tst1 := Norm;
T := Tst1;
repeat
T := 0.01 * T;
Tst2 := Norm + T;
until Tst2 <= Tst1;
635: H^[I]^[En] := - R / T;
goto 680;
640: { Solve real equations }
X := H^[I]^[I + 1];
Y := H^[I + 1]^[I];
Q := Sqr(Lambda^[I].X - P) + Sqr(Lambda^[I].Y);
T := (X * S - Zz * R) / Q;
H^[I]^[En] := T;
if Abs(X) > Abs(Zz) then
H^[I + 1]^[En] := (- R - W * T) / X
else
H^[I + 1]^[En] := (- S - Y * T) / Zz;
680: { Overflow control }
T := Abs(H^[I]^[En]);
if T = 0.0 then goto 700;
Tst1 := T;
Tst2 := Tst1 + 1.0 / Tst1;
if Tst2 > Tst1 then goto 700;
for J := I to En do
H^[J]^[En] := H^[J]^[En] / T;
700: end;
{ End real vector }
goto 800;
{ Complex vector }
710: M := Na;
{ Last vector component chosen imaginary so that
eigenvector matrix is triangular }
if Abs(H^[En]^[Na]) > Abs(H^[Na]^[En]) then
begin
H^[Na]^[Na] := Q / H^[En]^[Na];
H^[Na]^[En] := - (H^[En]^[En] - P) / H^[En]^[Na];
end
else
Cdiv(0.0, - H^[Na]^[En], H^[Na]^[Na] - P, Q, H^[Na]^[Na], H^[Na]^[En]);
H^[En]^[Na] := 0.0;
H^[En]^[En] := 1.0;
Enm2 := Na - 1;
if Enm2 < Lb then goto 800;
for I := Enm2 downto Lb do
begin
W := H^[I]^[I] - P;
Ra := 0.0;
Sa := 0.0;
for J := M to En do
begin
Ra := Ra + H^[I]^[J] * H^[J]^[Na];
Sa := Sa + H^[I]^[J] * H^[J]^[En];
end;
if Lambda^[I].Y >= 0.0 then goto 770;
Zz := W;
R := Ra;
S := Sa;
goto 795;
770: M := I;
if Lambda^[I].Y <> 0.0 then goto 780;
Cdiv(- Ra, - Sa, W, Q, H^[I]^[Na], H^[I]^[En]);
goto 790;
{ Solve complex equations }
780: X := H^[I]^[I + 1];
Y := H^[I + 1]^[I];
Vr := Sqr(Lambda^[I].X - P) + Sqr(Lambda^[I].Y) - Sqr(Q);
Vi := (Lambda^[I].X - P) * 2.0 * Q;
if (Vr = 0.0) and (Vi = 0.0) then
begin
Tst1 := Norm * (Abs(W) + Abs(Q) + Abs(X) + Abs(Y) + Abs(Zz));
Vr := Tst1;
repeat
Vr := 0.01 * Vr;
Tst2 := Tst1 + Vr;
until Tst2 <= Tst1;
end;
Cdiv(X * R - Zz * Ra + Q * Sa, X * S - Zz * Sa - Q * Ra, Vr, Vi, H^[I]^[Na], H^[I]^[En]);
if Abs(X) > Abs(Zz) + Abs(Q) then
begin
H^[I + 1]^[Na] := (- Ra - W * H^[I]^[Na] + Q * H^[I]^[En]) / X;
H^[I + 1]^[En] := (- Sa - W * H^[I]^[En] - Q * H^[I]^[Na]) / X;
end
else
Cdiv(- R - Y * H^[I]^[Na], - S - Y * H^[I]^[En], Zz, Q, H^[I + 1]^[Na], H^[I + 1]^[En]);
790: { Overflow control }
T := FMax(Abs(H^[I]^[Na]), Abs(H^[I]^[En]));
if T = 0.0 then goto 795;
Tst1 := T;
Tst2 := Tst1 + 1.0 / Tst1;
if Tst2 > Tst1 then goto 795;
for J := I to En do
begin
H^[J]^[Na] := H^[J]^[Na] / T;
H^[J]^[En] := H^[J]^[En] / T;
end;
795: end;
{ End complex vector }
800: end;
{ End back substitution.
Vectors of isolated roots }
for I := Lb to Ub do
if (I < I_low) or (I > I_igh) then
for J := I to Ub do
Z^[I]^[J] := H^[I]^[J];
{ Multiply by transformation matrix to give
vectors of original full matrix. }
for J := Ub downto I_low do
begin
M := Imin(J, I_igh);
for I := I_low to I_igh do
begin
Zz := 0.0;
for K := I_low to M do
Zz := Zz + Z^[I]^[K] * H^[K]^[J];
Z^[I]^[J] := Zz;
end;
end;
SetErrCode(0);
end;
end.
|