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Eigensystems.pas
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Eigensystems.pas
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// ###################################################################
// #### This file is part of the mathematics library project, and is
// #### offered under the licence agreement described on
// #### http://www.mrsoft.org/
// ####
// #### Copyright:(c) 2011, Michael R. . All rights reserved.
// ####
// #### Unless required by applicable law or agreed to in writing, software
// #### distributed under the License is distributed on an "AS IS" BASIS,
// #### WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// #### See the License for the specific language governing permissions and
// #### limitations under the License.
// ###################################################################
unit Eigensystems;
// ############################################
// #### Functions to extract eigenvalues and eigenvectors
// ############################################
interface
{$IFDEF FPC} {$MODESWITCH ADVANCEDRECORDS} {$ENDIF}
uses MatrixConst;
// ############################################
// #### functions for nonsymmetric matrices:
// executes the functions below in order to get the result.
function MatrixUnsymEigVecInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt; Eivec : PDouble; const LineWidthEivec : NativeInt) : TEigenvalueConvergence;
function MatrixUnsymEigVec(const A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt; Eivec : PDouble; const LineWidthEivec : NativeInt) : TEigenvalueConvergence;
function MatrixUnsymEigInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt; balance : boolean) : TEigenvalueConvergence;
function MatrixUnsymEig(const A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt; balance : boolean) : TEigenvalueConvergence;
// Given a Matrix A[0..width-1][0..Width-1], this routine replaces it by a balanced matrix
// with identical eigenvalues. A symmetric matrix is already balanced and is unaffected by this procedure.
procedure MatrixBalanceInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; Scale : PDouble; const LineWidthScale : NativeInt);
procedure MatrixBalanceBackInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; Scale : PDouble; const LineWidthScale : NativeInt);
// Reduction of Matrix A to hessenberg form by the elimination method. The real, nonsymmetric matrix
// A is replaced by an upper Hessenberg matrix with identical eigenvalues. Recommended, but not
// required, i sthat this routine be preceded by MatrixBalance. Non Hessenberg elements (which should be zero)
// are filled with random values and not replaced by zero elements.
procedure MatrixHessenbergPermInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; perm : PInteger; const LineWidthPerm : NativeInt);
procedure MatrixHessenbergPerm(dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; width : NativeInt; perm : PInteger; const LineWidthPerm : NativeInt);
procedure MatrixHessenbergInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt);
procedure MatrixHessenberg(dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; width : NativeInt);
// copies the hessenberg matrix to dest using the permutation vector from the hessenberg transformation. Initializes the destination matrix for the eigenvector
// finding routine for unsymmetric matrices
procedure MatrixInitEivecHess(hess : PDouble; const LineWidthHess : NativeInt; width : NativeInt; dest : PDouble; const LineWidthDest : NativeInt; perm : PInteger; const LineWidthPerm : NativeInt);
// Finds all eigenvalues of an upper Hessenberg matrix A. On input a can be exactly as output from MatrixHessenberg, on output
// A is destroyed. The real and imaginary parts of the eigenvalues are returned in wr[0..width-1] and wi[0..width-1], respectively
function MatrixEigHessenbergInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt) : TEigenvalueConvergence;
// the following function does not destroy A on output.
function MatrixEigHessenberg(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt) : TEigenvalueConvergence;
// finds all eigenvalues and eigenvectors in an upper hessenberg matrix A. On input A can be exactly as output from MatrixHessenbergPerm, on
// output A is destroyed. For complex eigenvalues (n, n+1) only one eigenvector is stored whereas the real part is stored in vector n and
// the imaginary part is stored in n+1. Note the function does not seem to correctly work with matrices with rank lower than width!
function MatrixEigVecHessenbergInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt; Eivec : PDouble; const LineWidthEivec : NativeInt) : TEigenvalueConvergence;
procedure MatrixNormEivecInPlace(Eivec : PDouble; const LineWidthEivec : NativeInt; width : NativeInt; WI : PDouble; const LineWidthWI : NativeInt);
// ############################################
// #### functions for symmetric matrices:
// Householder reduction of a real, symmetric matrix A[0..width-1][0..width-1]. On output
// A is replaced by the orthogonal matrix Q effecting the transformation. D[0..width-1] returns
// the diagonal elements of the tridiagonal matrix, and e[0..width - 1] of the off-diagonal elements with e[0] = 0.
procedure MatrixTridiagonalHouseInPlace(A : PDouble; const LineWidthA : NativeInt; const width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt);
procedure MatrixTridiagonalHouse(dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; const width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt);
function MatrixTridiagonalQLImplicitInPlace(Z : PDouble; const LineWidthZ : NativeInt; width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt) : TEigenvalueConvergence;
function MatrixEigTridiagonalMatrixInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; EigVals : PDouble; const LineWidthEigVals : NativeInt) : TEigenvalueConvergence;
function MatrixEigTridiagonalMatrix(Dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; width : NativeInt; EigVals : PDouble; const LineWidthEigVals : NativeInt) : TEigenvalueConvergence;
// ###########################################
// #### Reduction of a real general matrix A to upper Hessenberg form H by
// an orthogonal similarity transformation Q' * A * Q = H
// original DGEHRD
// A: On entry, the N-by-N general matrix to be reduced.
// On exit, the upper triangle and the first subdiagonal of A
// are overwritten with the upper Hessenberg matrix H, and the
// elements below the first subdiagonal, with the array TAU,
// represent the orthogonal matrix Q as a product of elementary
// reflectors.
procedure MatrixHessenberg2InPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; tau : PDouble; hessBlockSize : integer);
procedure ThrMtxHessenberg2InPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; tau : PDouble; hessBlockSize : integer);
// generate a real orthogonal matrix Q from the element reflectors (matrixhessenberg2inplace) to
// satisfy Q' * A * Q = H (or Q * H * Q' = A)
// the input matrix A is overwritten by the orhogonal matrix Q
procedure MatrixQFromHessenbergDecomp(A : PDouble; const LineWidthA : NativeInt; width : NativeInt;
tau : PDouble; BlockSize : NativeInt; work : PDouble; progress : TLinEquProgress = nil); overload;
procedure MatrixQFromHessenbergDecomp(A : PDouble; const LineWidthA : NativeInt; width : NativeInt;
tau : PDouble; progress : TLinEquProgress = nil); overload;
// Symmetric eigenvalue problem.
// the function first tries to reduce the input matrix to a tridiagonal one and
// then tries to find the eigenvalues and optionaly eigenvectors
// on entry A is an nxn matrix and W is an nx1 vector.
// on exit W contains the eigenvalues and if wanted A the eigenvectors
function MatrixEigUpperSymmetricMatrixInPlace2(A : PDouble; LineWidthA : NativeInt; N : NativeInt; W : PConstDoubleArr; EigValOnly : boolean; blockSize : integer; progress : TLinEquProgress = nil ) : TEigenvalueConvergence;
function ThrMtxEigUpperSymmetricMatrixInPlace2(A : PDouble; LineWidthA : NativeInt; N : NativeInt; W : PConstDoubleArr; EigValOnly : boolean; blockSize : integer; progress : TLinEquProgress = nil ) : TEigenvalueConvergence;
function UperSymEigMemorySize( N, blockSize : NativeInt; EigValOnly : boolean; Threaded : Boolean; var blkMultSize : integer ) : NativeInt;
function InternalEigValFromSymMatrix( D, E : PConstDoubleArr; N : NativeInt) : TEigenvalueConvergence;
function InternalEigValEigVecFromSymTridiagMatrix( D, E : PConstDoubleArr; Z : PDouble;
LineWidthZ : NativeInt; N : NativeInt; Work : PConstDoubleArr ) : TEigenvalueConvergence; overload;
// on entry A is a symmetric matrix (upper part will be used)
// on exit the diagonal and first superdiagonal of A are overwritten by the corresponding elements of
// the tridigonal matrix T, and the elements above the first superdiagonal, with the array TAU, represent the
// orthogonal matrix Q as a product of elementary reflectors.
// E and D conain the diagonal and off diagonal elements of the tridiagonal matrix T
// Tau (may be nil) contain the scalar factors of the elementary reflectors
// nb the blocksize used for block updates.
procedure MatrixUpperSymToTridiagInPlace( A : PDouble; LineWidthA : NativeInt; N : NativeInt;
D, E, Tau : PConstDoubleArr; nb : NativeInt );
implementation
uses Math, MathUtilFunc, MatrixASMStubSwitch, HouseholderReflectors, types,
BlockSizeSetup, ThreadedMatrixOperations, MtxThreadPool,
LinAlgQR,
SysUtils, MatrixRotations, Classes;
// #########################################################################
// #### Eigenvalue calculation - NR (aka Eispack)
// #########################################################################
function MatrixEigTridiagonalMatrixInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; EigVals : PDouble; const LineWidthEigVals : NativeInt) : TEigenvalueConvergence;
var E : Array of double;
begin
SetLength(E, width);
MatrixTridiagonalHouseInPlace(A, LineWidthA, width, EigVals, LineWidthEigVals, @E[0], sizeof(double));
Result := MatrixTridiagonalQLImplicitInPlace(A, LineWidthA, width, EigVals, LineWidthEigVals, @E[0], sizeof(double));
end;
function MatrixEigTridiagonalMatrix(Dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; width : NativeInt; EigVals : PDouble; const LineWidthEigVals : NativeInt) : TEigenvalueConvergence;
var E : Array of double;
begin
SetLength(E, width);
MatrixTridiagonalHouse(dest, LineWidthDest, A, LineWidthA, width, EigVals, LineWidthEigVals, @E[0], sizeof(double));
Result := MatrixTridiagonalQLImplicitInPlace(dest, LineWidthDest, width, EigVals, LineWidthEigVals, @E[0], sizeof(double));
end;
procedure MatrixTridiagonalHouse(dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; const width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt);
var pDest : PDouble;
i : NativeInt;
begin
Assert(width > 0, 'Dimension Error');
Assert(LineWidthA >= width*sizeof(double), 'Dimension error');
Assert(LineWidthD >= sizeof(double), 'Dimension error');
Assert(LineWidthE >= sizeof(double), 'Dimension error');
pDest := dest;
// copy data -> now we can perform an inline LU decomposition
for i := 0 to width - 1 do
begin
Move(A^, pDest^, sizeof(double)*width);
inc(PByte(A), LineWidthA);
inc(PByte(pDest), LineWidthDest);
end;
MatrixTridiagonalHouseInPlace(dest, LineWidthDest, width, D, LineWidthD, E, LineWidthE);
end;
procedure MatrixTridiagonalHouseInPlace(A : PDouble; const LineWidthA : NativeInt; const width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt);
var i, j : NativeInt;
l, k : NativeInt;
scale, hh, h, g, f : double;
pA, pAi, pAj, pAk : PDouble;
pEi, pEj, pEk : PDouble;
pD : PDouble;
begin
Assert(width > 0, 'Dimension Error');
Assert(LineWidthA >= width*sizeof(double), 'Dimension error');
Assert(LineWidthD >= sizeof(double), 'Dimension error');
Assert(LineWidthE >= sizeof(double), 'Dimension error');
pAi := A;
inc(PByte(pAi), (width-1)*LineWidthA);
pEi := E;
inc(PByte(pEi), (width-1)*LineWidthE);
for i := width - 1 downto 1 do
begin
l := i - 1;
h := 0;
scale := 0;
if l > 0 then
begin
pA := pAi;
for k := 0 to l do
begin
scale := scale + abs(pA^);
inc(pA);
end;
if scale = 0 then
begin
// skip transformation
dec(pA);
pEi^ := pA^;
end
else
begin
pA := pAi;
for k := 0 to l do
begin
pA^ := pA^/scale; // use scaled a's for transformation
h := h + sqr(pA^);
inc(pA);
end;
dec(pA);
f := pA^;
if f >= 0
then
g := -sqrt(h)
else
g := sqrt(h);
pEi^ := scale*g;
h := h - f*g;
pA^ := f - g; // store u in the ith row of a
f := 0;
pAj := A;
pEj := E;
for j := 0 to l do
begin
pA := pAj;
inc(pA, i);
pAk := pAi;
inc(pAk, j);
// next statement can be omitted if eigenvectors not wanted
pA^ := pAk^/h; // store u/H in ith column of a.
g := 0;
// Form an element of A*u in g
pAk := pAj;
pA := pAi;
for k := 0 to j do
begin
g := g + pA^*pAk^;
inc(pAk);
inc(pA);
end;
pAk := pAj;
pA := pAi;
inc(pA, j + 1);
inc(PByte(pAk), LineWidthA);
inc(pAk, j);
for k := j + 1 to l do
begin
g := g + pA^*pAk^;
inc(PByte(pAk), LineWidthA);
inc(pA);
end;
pEj^ := g/h;
pA := pAi;
inc(pA, j);
f := f + pEj^*pA^;
// next line
inc(PByte(pEj), LineWidthE);
inc(PByte(pAj), LineWidthA);
end;
hh := f/(h + h);
pAj := A;
pEj := E;
for j := 0 to l do
begin
pA := pAi;
inc(pA, j);
f := pA^;
pEj^ := pEj^ - hh*f;
g := pEj^;
pAk := pAj;
pEk := E;
pA := pAi;
for k := 0 to j do
begin
pAk^ := pAk^ - (f*pEk^ + g*pA^);
inc(pA);
inc(pAk);
inc(PByte(pEk), LineWidthE);
end;
inc(PByte(pAj), LineWidthA);
inc(PByte(pEj), LineWidthE);
end;
end;
end
else
begin
pA := pAi;
inc(pA, l);
pEi^ := pA^;
end;
pD := D;
inc(PByte(pD), i*LineWidthD);
pD^ := h;
dec(PByte(pAi), LineWidthA);
dec(PByte(pEi), LineWidthE);
end;
// next statement can be omitted if eigenvectors not wanted
D^ := 0;
E^ := 0;
// contents of this loop can be omitted if eigenvectors not wanted except for statement d[i] = a[i][i]
// begin accumulation of transformation matrices.
pAi := A;
pD := D;
for i := 0 to width - 1 do
begin
l := i - 1;
if (pD^ <> 0) then
begin
for j := 0 to l do
begin
g := 0;
pA := pAi;
pAk := A;
inc(pAk, j);
for k := 0 to l do
begin
g := g + pA^*pAk^;
inc(pA);
inc(PByte(pAk), LineWidthA);
end;
pA := A;
inc(pA, j);
pAk := A;
inc(pAk, i);
for k := 0 to l do
begin
pA^ := pA^ - g*pAk^;
inc(PByte(pA), LineWidthA);
inc(PByte(pAk), LineWidthA);
end;
end;
end;
pA := pAi;
inc(pA, i);
pD^ := pA^;
pA^ := 1;
pAj := A;
inc(pAj, i);
pA := pAi;
for j := 0 to l do
begin
pAj^ := 0;
pA^ := 0;
inc(PByte(pAj), LineWidthA);
inc(pA);
end;
inc(PByte(pAi), LineWidthA);
inc(PByte(pD), LineWidthD);
end;
end;
function MatrixTridiagonalQLImplicitInPlace(Z : PDouble; const LineWidthZ : NativeInt; width : NativeInt;
D : PDouble; const LineWidthD : NativeInt; E : PDouble; const LineWidthE : NativeInt) : TEigenvalueConvergence;
var m, l, iter, i, k : NativeInt;
s, r, p, g, f, dd, c, b : double;
pE, pEl, pEi : PDouble;
pD, pDl : PDouble;
pZ, pZi : PDouble;
const cMaxTridiagIter = 30;
begin
pE := E;
pEi := E;
inc(PByte(pEi), LineWidthE);
Result := qlNoConverge;
for i := 1 to width - 1 do
begin
pE^ := pEi^;
inc(PByte(pE), LineWidthE);
inc(PByte(pEi), LineWidthE);
end;
pE := E;
inc(PByte(pE), (width - 1)*LineWidthE);
pE^ := 0;
pDl := D;
pEl := E;
for l := 0 to width - 1 do
begin
iter := 0;
repeat
pD := pDl;
pE := pEl;
m := l;
while m < width - 1 do
begin
dd := abs(pD^);
inc(PByte(pD), LineWidthD);
dd := dd + abs(pD^);
if pE^ + dd = dd then
break;
inc(PByte(pE), LineWidthE);
inc(m);
end;
if m <> l then
begin
inc(iter);
if iter = cMaxTridiagIter then
exit;
g := (PDouble(NativeUint(pDl) + NativeUint(LineWidthD))^ - pDl^)/(2*pEl^);
r := pythag(g, 1);
pD := D;
inc(PByte(pD), m*LineWidthD);
g := pD^ - pDl^ + pEl^/(g + sign(r, g));
s := 1;
c := 1;
p := 0;
pE := E;
inc(PByte(pE), (m - 1)*LineWidthE);
for i := m - 1 downto l do
begin
f := s*pE^;
b := c*pE^;
r := pythag(f, g);
inc(PByte(pE), LineWidthE);
pE^ := r;
if r = 0 then
begin
pD^ := pD^ - p;
pE := E;
inc(PByte(pE), m*LineWidthE);
pE^ := 0;
break;
end;
s := f/r;
c := g/r;
g := pD^ - p;
dec(PByte(pD), LineWidthD);
r := (pD^ - g)*s + 2*c*b;
p := s*r;
inc(PByte(pD), LineWidthD);
pD^ := g + p;
g := c*r - b;
// next loop can be omitted if eigenvectors not wanted
pZ := Z;
pZi := Z;
inc(pZ, i);
inc(pZi, i+1);
for k := 0 to width - 1 do
begin
f := pZi^;
pZi^ := s*pZ^ + c*f;
pZ^ := c*pZ^ - s*f;
inc(PByte(pZi), LineWidthZ);
inc(PByte(pZ), LineWidthZ);
end;
dec(PByte(pD), LineWidthD);
dec(PByte(pE), 2*LineWidthE);
end;
if (r = 0) and (i >= l) then
continue;
pDl^ := pDl^ - p;
pEl^ := g;
pE := E;
inc(PByte(pE), m*LineWidthE);
pE^ := 0;
end;
until m = l;
inc(PByte(pDl), LineWidthD);
inc(PByte(pEl), LineWidthE);
end;
Result := qlOk;
end;
procedure MatrixBalanceBackInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; Scale : PDouble; const LineWidthScale : NativeInt);
var i : NativeInt;
begin
for i := 0 to width - 1 do
begin
if Scale^ <> 1 then
begin
// Apply similarity transformation
MatrixScaleAndAdd( GenPtr(A, 0, i, LineWidthA), LineWidthA, width, 1, 0, Scale^ );
MatrixScaleAndAdd( GenPtr(A, i, 0, LineWidthA), LineWidthA, 1, width, 0, 1/Scale^ );
end;
inc(PByte(Scale), LineWidthScale);
end;
end;
procedure MatrixBalanceInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; Scale : PDouble; const LineWidthScale : NativeInt);
var j, i : NativeInt;
last : boolean;
s, r, g, f, c, sqrdx : double;
pAi : PConstDoubleArr;
pAj, pScale : PDouble;
const RADIX = 2.0;
begin
assert(lineWidthA >= width*sizeof(double), 'Dimension error');
assert(LineWidthScale >= sizeof(double), 'Dimension error');
sqrdx := sqr(RADIX);
last := False;
MatrixInit(scale, LineWidthScale, 1, width, 1);
while last = False do
begin
last := True;
pScale := Scale;
for i := 0 to width - 1 do
begin
r := 0;
c := 0;
pAi := GenPtrArr(A, 0, i, LineWidthA);
pAj := GenPtr(A, i, 0, LineWidthA);
for j := 0 to width - 1 do
begin
if i <> j then
begin
c := c + abs(pAj^);
r := r + abs(pAi^[j]);
end;
inc(PByte(pAj), LineWidthA);
end;
if (c <> 0) and (r <> 0) then
begin
g := r/RADIX;
f := 1;
s := c + r;
// find the integer power of the machine radix that comes closest to balancing the matrix.
while c < g do
begin
f := f*RADIX;
c := c*sqrdx;
end;
g := r*RADIX;
while c > g do
begin
f := f/RADIX;
c := c/sqrdx;
end;
if (c + r)/f < 0.95*s then
begin
pScale^ := pScale^*f;
last := False;
g := 1/f;
// Apply similarity transformation
MatrixScaleAndAdd( GenPtr(A, 0, i, LineWidthA), LineWidthA, width, 1, 0, g );
MatrixScaleAndAdd( GenPtr(A, i, 0, LineWidthA), LineWidthA, 1, width, 0, f );
end;
end;
inc(PByte(pScale), LineWidthScale);
end;
end;
end;
procedure MatrixHessenbergPerm(dest : PDouble; const LineWidthDest : NativeInt; A : PDouble; const LineWidthA : NativeInt; width : NativeInt; perm : PInteger; const LineWidthPerm : NativeInt);
begin
assert(LineWidthA >= width*sizeof(double), 'Dimension Error');
assert(LineWidthDest >= width*sizeof(double), 'Dimension Error');
MatrixCopy(Dest, LineWidthDest, A, LineWidthA, width, width);
MatrixHessenbergPermInPlace(dest, LineWidthDest, width, perm, LineWidthPerm);
end;
procedure MatrixHessenbergPermInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; perm : PInteger; const LineWidthPerm : NativeInt);
var m, j, i : NativeInt;
x, y : double;
pAm, pAj, pAi : PDouble;
pA : PDouble;
pPerm : PInteger;
begin
assert(LineWidthA >= width*sizeof(double), 'Dimension Error');
pAm := A;
pPerm := Perm;
if Assigned(pPerm) then
begin
pPerm^ := 0;
inc(PByte(pPerm), LineWidthPerm);
end;
m := 1;
while m < width - 1 do
begin
inc(PByte(pAm), LineWidthA);
x := 0;
i := m;
// find the pivot
pAj := pAm;
inc(pAj, m - 1);
for j := m to width - 1 do
begin
if abs(pAj^) > abs(x) then
begin
x := pAj^;
i := j;
end;
inc(PByte(pAj), LineWidthA);
end;
if Assigned(pPerm) then
begin
pPerm^ := i;
inc(PByte(pPerm), LineWidthPerm);
end;
// interchange rows and columns
if i <> m then
begin
pAi := A;
inc(pAi, m - 1);
inc(PByte(pAi), i*LineWidthA);
pAj := pAm;
inc(pAj, m - 1);
for j := m - 1 to width - 1 do
begin
DoubleSwap(pAi^, pAj^);
inc(pAj);
inc(pAi);
end;
pAi := A;
inc(pAi, i);
pAj := A;
inc(pAj, m);
for j := 0 to width - 1 do
begin
DoubleSwap(pAi^, pAj^);
inc(PByte(pAi), LineWidthA);
inc(PByte(pAj), LineWidthA);
end;
end;
// carry out the elimination
if x <> 0 then
begin
pA := pAm;
inc(PByte(pA), LineWidthA);
inc(pA, m - 1);
i := m + 1;
while i < width do
begin
y := pA^;
if y <> 0 then
begin
y := y/x;
pA^ := y;
pAi := A;
inc(pAi, m);
inc(PByte(pAi), i*LineWidthA);
pAj := pAm;
inc(pAj, m);
for j := m to width - 1 do
begin
pAi^ := pAi^ - y*pAj^;
inc(pAj);
inc(pAi);
end;
pAi := A;
inc(pAi, i);
pAj := A;
inc(pAj, m);
for j := 0 to width - 1 do
begin
pAj^ := pAj^ + y*pAi^;
inc(PByte(pAi), LineWidthA);
inc(PByte(pAj), LineWidthA);
end;
end;
inc(PByte(pA), LineWidthA);
inc(i);
end;
end;
inc(m);
end;
end;
function MatrixEigHessenberg(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt) : TEigenvalueConvergence;
var pDest : PDouble;
dest : Array of double;
i : NativeInt;
begin
Assert(width > 0, 'Dimension Error');
Assert(LineWidthA >= width*sizeof(double), 'Dimension error');
setLength(dest, width*width);
pDest := @dest[0];
// copy data
for i := 0 to width - 1 do
begin
Move(A^, pDest^, sizeof(double)*width);
inc(PByte(A), LineWidthA);
inc(PByte(pDest), width*sizeof(double));
end;
Result := MatrixEigHessenbergInPlace(@dest[0], width*sizeof(double), width, WR, LineWidthWR, WI, LineWidthWI);
end;
function MatrixEigHessenbergInPlace(A : PDouble; const LineWidthA : NativeInt; width : NativeInt; WR : PDouble;
const LineWidthWR : NativeInt; WI : PDouble; const LineWidthWI : NativeInt) : TEigenvalueConvergence;
var nn, m, l, k : NativeInt;
j, its, i, mmin : NativeInt;
x, y, z : double;
u, v, w : double;
r, s, t : double;
q, p, anorm : double;
pAi, pAj, pAl, pA, pAm, pAk : PDouble;
pWr : PDouble;
pWi : PDouble;
const cMaxEigHessenbergIter = 30;
begin
Assert(width > 0, 'Dimension Error');
Assert(LineWidthA >= width*sizeof(double), 'Dimension error');
Assert(LineWidthWR >= sizeof(double), 'Dimension error');
Assert(LineWidthWI >= sizeof(double), 'Dimension error');
r := 0;
p := 0;
q := 0;
Result := qlNoConverge;
anorm := abs(A^);
pAi := A;
inc(PByte(pAi), LineWidthA);
// compute matrix norm for possible use in locating single small subdiagonal element
for i := 1 to width - 1 do
begin
pAj := pAi;
inc(pAj, i - 1);
for j := i - 1 to width - 1 do
begin
anorm := anorm + abs(pAj^);
inc(pAj);
end;
inc(PByte(pAi), LineWidthA);
end;
nn := width - 1;
t := 0;
// gets changed only by an exceptional shift.
while nn >= 0 do
begin
// begin search of the next eigenvalue
its := 0;
repeat
// begin iteration: look for single small subdiagonal element.
pAl := A;
inc(pAl, nn);
inc(PByte(pAl), nn*LineWidthA);
l := nn;
while l >= 1 do
begin
s := abs(pAl^);
dec(pAl);
dec(PByte(pAl), LineWidthA);
s := s + abs(pAl^);
if s = 0 then
s := anorm;
inc(PByte(pAl), LineWidthA);
if abs(pAl^) + s = s then
break;
dec(PByte(pAl), LineWidthA);
dec(l);
end;
pA := A;
inc(pA, nn);
inc(PByte(pA), nn*LineWidthA);
x := pA^;
// one root found
if l = nn then
begin
pWr := WR;
inc(PByte(pWr), nn*LineWidthWR);
pWr^ := x + t;
pWi := WI;
inc(PByte(pWi), nn*LineWidthWI);
pWi^ := 0;
dec(nn);
end
else
begin
dec(pA);
dec(PByte(pA), LineWidthA);
y := pA^;
inc(pA);
w := pA^;
dec(pA);
inc(PByte(pA), LineWidthA);
w := w*pA^;
// two roots found ...
if l = nn - 1 then
begin
p := 0.5*(y - x);
q := sqr(p) + w;
z := sqrt(abs(q));
x := x + t;
pWr := WR;
inc(PByte(pWr), (nn-1)*LineWidthWR);
pWi := WI;
inc(PByte(pWi), (nn-1)*LineWidthWI);
// ... a real pair
if q >= 0 then
begin
z := p + sign(z, p);
pWr^ := x + z;
inc(PByte(pWr), LineWidthWR);
pWr^ := x + z;
if z <> 0 then
pWr^ := x - w/z;
pWi^ := 0;
inc(PByte(pWi), LineWidthWI);
pWi^ := 0;
end
else
begin
// ...a complex pair
pWr^ := x + p;
inc(PByte(pWr), LineWidthWR);
pWr^ := x + p;
pWi^ := -z;
inc(PByte(pWi), LineWidthWI);
pWi^ := z;
end;
dec(nn, 2);
end
else
begin
// no roots found. Continue iteration
if its = cMaxEigHessenbergIter then
exit;
// form exceptional shift
if (its = 10) or (its = 20) then
begin
t := t + x;
pAi := A;
for i := 0 to width - 1 do
begin
pAi^ := pAi^ - x;
inc(pAi);
inc(PByte(pAi), LineWidthA);
end;
pA := A;
inc(pA, nn - 1);
inc(PByte(pA), nn*LineWidthA);
s := abs(pA^);
dec(pA);
dec(PByte(pA), LineWidthA);
s := s + abs(pA^);
x := 0.75*s;
y := x;
w := -0.4375*sqr(s);
end;
inc(its);
// form shift and the look for 2 consecutive small subdiagonal elements
pAm := A;
inc(pAm, nn - 2);
inc(PByte(pAm), (nn - 2)*LineWidthA);
m := nn - 2;
while m >= l do
begin
z := pAm^;
r := x - z;
s := y - z;
pAi := pAm;
inc(pAi);
pAl := pAm;
inc(PByte(pAl), LineWidthA);
p := (r*s - w)/pAl^ + pAi^;
inc(pAl);
q := pAl^ - z - r - s;
inc(PByte(pAl), LineWidthA);
r := pAl^;
// scale to prevent under- or overflow
s := abs(p) + abs(q) + abs(r);
p := p/s;
q := q/s;
r := r/s;
if m = l then
break;
pAi := pAm;
dec(pAi);
u := abs(pAi^)*(abs(q) + abs(r));
dec(PByte(pAi), LineWidthA);
dec(PByte(pAl), LineWidthA);
v := abs(p)*(abs(pAi^) + abs(z) + abs(pAl^));
if u + v = v then
break;
dec(pAm);
dec(PByte(pAm), LineWidthA);
dec(m);
end;
pAi := A;
inc(pAi, m);
inc(PByte(pAi), (m + 2)*LineWidthA);
for i := m + 2 to nn do
begin
pAi^ := 0;
if i <> m + 2 then
begin
dec(pAi);