void QRZT(Matrix& X, LowerTriangularMatrix& L)
{
REPORT
Tracer et("QRZT(1)");
int n = X.Ncols(); int s = X.Nrows(); L.resize(s);
if (n == 0 || s == 0) { L = 0.0; return; }
Real* xi = X.Store(); int k;
for (int i=0; i<s; i++)
{
Real sum = 0.0;
Real* xi0=xi; k=n; while(k--) { sum += square(*xi++); }
sum = sqrt(sum);
if (sum == 0.0)
{
REPORT
k=n; while(k--) { *xi0++ = 0.0; }
for (int j=i; j<s; j++) L.element(j,i) = 0.0;
}
else
{
L.element(i,i) = sum;
Real* xj0=xi0; k=n; while(k--) { *xj0++ /= sum; }
for (int j=i+1; j<s; j++)
{
sum=0.0;
xi=xi0; Real* xj=xj0; k=n; while(k--) { sum += *xi++ * *xj++; }
xi=xi0; k=n; while(k--) { *xj0++ -= sum * *xi++; }
L.element(j,i) = sum;
}
}
}
}
void BetaShifted_HGVolatilityParam::initializeShiftMatrix(
LMMTenorStructure_PTR pLMMTenorStructure,
const LowerTriangularMatrix& betas,
const std::vector<double>& liborsInitValues)
{
assert(betas.size1() == betas.size2() && betas.size2() == liborsInitValues.size() && pLMMTenorStructure->get_nbLIBOR());
//unused first column, first row is automatically null since it is lower triangular matrix
LowerTriangularMatrix m(betas.size1(), betas.size2());
for(size_t k=0; k<m.size1();++k)
{
shift_matrix_(k,0) = 1.0e100;
}
for(size_t liborIndex=1; liborIndex<m.size1();++liborIndex)
{
for(size_t timeIndex=1;timeIndex<=liborIndex;++timeIndex)
{
double beta = betas(liborIndex,timeIndex);
double liborInitValue = liborsInitValues[liborIndex];
shift_matrix_(liborIndex,timeIndex) = 1+ (1-beta)/beta*liborInitValue;
}
}
}
inline
Matrix<C>::Matrix(const LowerTriangularMatrix<C>& M)
: entries_(M.row_dimension()*M.column_dimension()),
rowdim_(M.row_dimension()), coldim_(M.column_dimension())
{
for (size_type i(0); i < rowdim_; i++)
for (size_type j(0); j <= i; j++)
{
this->operator () (i, j) = M(i, j);
}
}
void MLE_D_FI::MakeCovariance()
{
if (Covariance.Nrows()==0)
{
LowerTriangularMatrix LTI = LT.i();
Covariance << LTI.t() * LTI;
SE << Covariance; // get diagonal
int n = Covariance.Nrows();
for (int i=1; i <= n; i++) SE(i) = sqrt(SE(i));
}
}
void Shifted_HGVolatilityParam::reset_g_matrix(const LowerTriangularMatrix& other_g)
{
assert(other_g.size1() == g_matrix_.size1() );
assert(other_g.size2() == g_matrix_.size2() );
for(size_t indexLibor=1; indexLibor<g_matrix_.size1();++indexLibor)
{
for(size_t indexTime=1; indexTime<=indexLibor; ++indexTime)
{
g_matrix_(indexLibor,indexTime) = other_g(indexLibor,indexTime);
}
}
}
void Print(const LowerTriangularMatrix& X)
{
++PCN;
cout << "\nMatrix type: " << X.Type().Value() << " (";
cout << X.Nrows() << ", ";
cout << X.Ncols() << ")\n\n";
if (X.IsZero()) { cout << "All elements are zero\n" << flush; return; }
int nr=X.Nrows();
for (int i=1; i<=nr; i++)
{
for (int j=1; j<=i; j++) cout << X(i,j) << "\t";
cout << "\n";
}
cout << flush; ++PCZ;
}
///
///
/// Function to calculate the transformation matrix X ( p = X.f )
/// according to the Whiten Crusher Model described in
/// the JKMRC monograph:
/// Napier-Munn et al.
/// "Mineral Comminution Circuits - Their Operation and Optimisation",
/// JKMRC monograph 1996,
/// p138ff
///
void WhitenCrusherTransformationMatrix(const LowerTriangularMatrix& B, const DiagonalMatrix& C, Matrix& X)
{
int matrixSize = B.Nrows() ;
IdentityMatrix I(matrixSize);
/// do the matrix calculations
X = (I - C) * (I - (B * C)).i() ;
}
void QRZT(Matrix& X, LowerTriangularMatrix& L)
{
REPORT
Tracer et("QZT(1)");
int n = X.Ncols(); int s = X.Nrows(); L.ReSize(s);
Real* xi = X.Store(); int k;
for (int i=0; i<s; i++)
{
Real sum = 0.0;
Real* xi0=xi; k=n; while(k--) { sum += square(*xi++); }
sum = sqrt(sum);
L.element(i,i) = sum;
if (sum==0.0) Throw(SingularException(L));
Real* xj0=xi0; k=n; while(k--) { *xj0++ /= sum; }
for (int j=i+1; j<s; j++)
{
sum=0.0;
xi=xi0; Real* xj=xj0; k=n; while(k--) { sum += *xi++ * *xj++; }
xi=xi0; k=n; while(k--) { *xj0++ -= sum * *xi++; }
L.element(j,i) = sum;
}
}
}
void test3(Real* y, Real* x1, Real* x2, int nobs, int npred)
{
cout << "\n\nTest 3 - Cholesky\n";
// traditional sum of squares and products method of calculation
// with subtraction of means - using Cholesky decomposition
Matrix X(nobs,npred);
X.Column(1) << x1; X.Column(2) << x2;
ColumnVector Y(nobs); Y << y;
ColumnVector Ones(nobs); Ones = 1.0;
RowVector M = Ones.t() * X / nobs;
Matrix XC(nobs,npred);
XC = X - Ones * M;
ColumnVector YC(nobs);
Real m = Sum(Y) / nobs; YC = Y - Ones * m;
SymmetricMatrix SSQ; SSQ << XC.t() * XC;
// Cholesky decomposition of SSQ
LowerTriangularMatrix L = Cholesky(SSQ);
// calculate estimate
ColumnVector A = L.t().i() * (L.i() * (XC.t() * YC));
// calculate estimate of constant term
Real a = m - (M * A).AsScalar();
// Get variances of estimates from diagonal elements of invoice of SSQ
DiagonalMatrix D; D << L.t().i() * L.i();
ColumnVector V = D.AsColumn();
Real v = 1.0/nobs + (L.i() * M.t()).SumSquare();
// Calculate fitted values and residuals
int npred1 = npred+1;
ColumnVector Fitted = X * A + a;
ColumnVector Residual = Y - Fitted;
Real ResVar = Residual.SumSquare() / (nobs-npred1);
// Get diagonals of Hat matrix (an expensive way of doing this)
Matrix X1(nobs,npred1); X1.Column(1)<<Ones; X1.Columns(2,npred1)<<X;
DiagonalMatrix Hat; Hat << X1 * (X1.t() * X1).i() * X1.t();
// print out answers
cout << "\nEstimates and their standard errors\n\n";
cout.setf(ios::fixed, ios::floatfield);
cout << setw(11) << setprecision(5) << a << " ";
cout << setw(11) << setprecision(5) << sqrt(v*ResVar) << endl;
ColumnVector SE(npred);
for (int i=1; i<=npred; i++) SE(i) = sqrt(V(i)*ResVar);
cout << setw(11) << setprecision(5) << (A | SE) << endl;
cout << "\nObservations, fitted value, residual value, hat value\n";
cout << setw(9) << setprecision(3) <<
(X | Y | Fitted | Residual | Hat.AsColumn());
cout << "\n\n";
}
void updateQRZT(Matrix& X, LowerTriangularMatrix& L)
{
REPORT
Tracer et("updateQRZT");
int n = X.Ncols(); int s = X.Nrows();
if (s != L.Nrows())
Throw(ProgramException("Incompatible dimensions",X,L));
if (n == 0 || s == 0) return;
Real* xi = X.Store(); int k;
for (int i=0; i<s; i++)
{
Real r = L.element(i,i);
Real sum = 0.0;
Real* xi0=xi; k=n; while(k--) { sum += square(*xi++); }
sum = sqrt(sum + square(r));
if (sum == 0.0)
{
REPORT
k=n; while(k--) { *xi0++ = 0.0; }
for (int j=i; j<s; j++) L.element(j,i) = 0.0;
}
else
{
Real frs = fabs(r) + sum;
Real a0 = sqrt(frs / sum); Real alpha = a0 / frs;
if (r <= 0) { REPORT L.element(i,i) = sum; alpha = -alpha; }
else { REPORT L.element(i,i) = -sum; }
Real* xj0=xi0; k=n; while(k--) { *xj0++ *= alpha; }
for (int j=i+1; j<s; j++)
{
sum = 0.0;
xi=xi0; Real* xj=xj0; k=n; while(k--) { sum += *xi++ * *xj++; }
sum += a0 * L.element(j,i);
xi=xi0; k=n; while(k--) { *xj0++ -= sum * *xi++; }
L.element(j,i) -= sum * a0;
}
}
}
}
void trymatd()
{
Tracer et("Thirteenth test of Matrix package");
Tracer::PrintTrace();
Matrix X(5,20);
int i,j;
for (j=1;j<=20;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=20; j++) X(i,j) = (long)X(i-1,j) * j % 1001;
SymmetricMatrix S; S << X * X.t();
Matrix SM = X * X.t() - S;
Print(SM);
LowerTriangularMatrix L = Cholesky(S);
Matrix Diff = L*L.t()-S; Clean(Diff, 0.000000001);
Print(Diff);
{
Tracer et1("Stage 1");
LowerTriangularMatrix L1(5);
Matrix Xt = X.t(); Matrix Xt2 = Xt;
QRZT(X,L1);
Diff = L - L1; Clean(Diff,0.000000001); Print(Diff);
UpperTriangularMatrix Ut(5);
QRZ(Xt,Ut);
Diff = L - Ut.t(); Clean(Diff,0.000000001); Print(Diff);
Matrix Y(3,20);
for (j=1;j<=20;j++) Y(1,j) = 22-j;
for (i=2;i<=3;i++) for (j=1;j<=20; j++)
Y(i,j) = (long)Y(i-1,j) * j % 101;
Matrix Yt = Y.t(); Matrix M,Mt; Matrix Y2=Y;
QRZT(X,Y,M); QRZ(Xt,Yt,Mt);
Diff = Xt - X.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Yt - Y.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Mt - M.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Y2 * Xt2 * S.i() - M * L.i();
Clean(Diff,0.000000001); Print(Diff);
}
ColumnVector C1(5);
{
Tracer et1("Stage 2");
X.ReSize(5,5);
for (j=1;j<=5;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=5; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
for (i=1;i<=5;i++) C1(i) = i*i;
CroutMatrix A = X;
ColumnVector C2 = A.i() * C1; C1 = X.i() * C1;
X = C1 - C2; Clean(X,0.000000001); Print(X);
}
{
Tracer et1("Stage 3");
X.ReSize(7,7);
for (j=1;j<=7;j++) X(1,j) = j+1;
for (i=2;i<=7;i++) for (j=1;j<=7; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
C1.ReSize(7);
for (i=1;i<=7;i++) C1(i) = i*i;
RowVector R1 = C1.t();
Diff = R1 * X.i() - ( X.t().i() * R1.t() ).t(); Clean(Diff,0.000000001);
Print(Diff);
}
{
Tracer et1("Stage 4");
X.ReSize(5,5);
for (j=1;j<=5;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=5; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
C1.ReSize(5);
for (i=1;i<=5;i++) C1(i) = i*i;
CroutMatrix A1 = X*X;
ColumnVector C2 = A1.i() * C1; C1 = X.i() * C1; C1 = X.i() * C1;
X = C1 - C2; Clean(X,0.000000001); Print(X);
}
{
Tracer et1("Stage 5");
int n = 40;
SymmetricBandMatrix B(n,2); B = 0.0;
for (i=1; i<=n; i++)
{
B(i,i) = 6;
if (i<=n-1) B(i,i+1) = -4;
if (i<=n-2) B(i,i+2) = 1;
}
B(1,1) = 5; B(n,n) = 5;
SymmetricMatrix A = B;
ColumnVector X(n);
X(1) = 429;
for (i=2;i<=n;i++) X(i) = (long)X(i-1) * 31 % 1001;
X = X / 100000L;
// the matrix B is rather ill-conditioned so the difficulty is getting
// good agreement (we have chosen X very small) may not be surprising;
// maximum element size in B.i() is around 1400
ColumnVector Y1 = A.i() * X;
LowerTriangularMatrix C1 = Cholesky(A);
ColumnVector Y2 = C1.t().i() * (C1.i() * X) - Y1;
Clean(Y2, 0.000000001); Print(Y2);
UpperTriangularMatrix CU = C1.t().i();
LowerTriangularMatrix CL = C1.i();
Y2 = CU * (CL * X) - Y1;
Clean(Y2, 0.000000001); Print(Y2);
Y2 = B.i() * X - Y1; Clean(Y2, 0.000000001); Print(Y2);
LowerBandMatrix C2 = Cholesky(B);
Matrix M = C2 - C1; Clean(M, 0.000000001); Print(M);
ColumnVector Y3 = C2.t().i() * (C2.i() * X) - Y1;
Clean(Y3, 0.000000001); Print(Y3);
CU = C1.t().i();
CL = C1.i();
Y3 = CU * (CL * X) - Y1;
Clean(Y3, 0.000000001); Print(Y3);
Y3 = B.i() * X - Y1; Clean(Y3, 0.000000001); Print(Y3);
SymmetricMatrix AI = A.i();
Y2 = AI*X - Y1; Clean(Y2, 0.000000001); Print(Y2);
SymmetricMatrix BI = B.i();
BandMatrix C = B; Matrix CI = C.i();
M = A.i() - CI; Clean(M, 0.000000001); Print(M);
M = B.i() - CI; Clean(M, 0.000000001); Print(M);
M = AI-BI; Clean(M, 0.000000001); Print(M);
M = AI-CI; Clean(M, 0.000000001); Print(M);
M = A; AI << M; M = AI-A; Clean(M, 0.000000001); Print(M);
C = B; BI << C; M = BI-B; Clean(M, 0.000000001); Print(M);
}
{
Tracer et1("Stage 5");
SymmetricMatrix A(4), B(4);
A << 5
<< 1 << 4
<< 2 << 1 << 6
<< 1 << 0 << 1 << 7;
B << 8
<< 1 << 5
<< 1 << 0 << 9
<< 2 << 1 << 0 << 6;
LowerTriangularMatrix AB = Cholesky(A) * Cholesky(B);
Matrix M = Cholesky(A) * B * Cholesky(A).t() - AB*AB.t();
Clean(M, 0.000000001); Print(M);
M = A * Cholesky(B); M = M * M.t() - A * B * A;
Clean(M, 0.000000001); Print(M);
}
{
Tracer et1("Stage 6");
int N=49;
int i;
SymmetricBandMatrix S(N,1);
Matrix B(N,N+1); B=0;
for (i=1;i<=N;i++) { S(i,i)=1; B(i,i)=1; B(i,i+1)=-1; }
for (i=1;i<N; i++) S(i,i+1)=-.5;
DiagonalMatrix D(N+1); D = 1;
B = B.t()*S.i()*B - (D-1.0/(N+1))*2.0;
Clean(B, 0.000000001); Print(B);
}
{
Tracer et1("Stage 7");
// Copying and moving CroutMatrix
Matrix A(7,7);
A.Row(1) << 3 << 2 << -1 << 4 << -3 << 5 << 9;
A.Row(2) << -8 << 7 << 2 << 0 << 7 << 0 << -1;
A.Row(3) << 2 << -2 << 3 << 1 << 9 << 0 << 3;
A.Row(4) << -1 << 5 << 2 << 2 << 5 << -1 << 2;
A.Row(5) << 4 << -4 << 1 << 9 << -8 << 7 << 5;
A.Row(6) << 1 << -2 << 5 << -1 << -2 << 5 << 1;
A.Row(7) << -6 << 3 << -1 << 8 << -1 << 2 << 2;
RowVector D(30); D = 0;
Real x = determinant(A);
CroutMatrix B = A;
D(1) = determinant(B) / x - 1;
Matrix C = A * Inverter1(B) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// Test copy constructor (in Inverter2 and ordinary copy)
CroutMatrix B1; B1 = B;
D(2) = determinant(B1) / x - 1;
C = A * Inverter2(B1) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// Do it again with release
B.release(); B1 = B;
D(2) = B.nrows(); D(3) = B.ncols(); D(4) = B.size();
D(5) = determinant(B1) / x - 1;
B1.release();
C = A * Inverter2(B1) - IdentityMatrix(7);
D(6) = B1.nrows(); D(7) = B1.ncols(); D(8) = B1.size();
Clean(C, 0.000000001); Print(C);
// see if we get an implicit invert
B1 = -A;
D(9) = determinant(B1) / x + 1; // odd number of rows - sign will change
C = -A * Inverter2(B1) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// check for_return
B = LU1(A); B1 = LU2(A); CroutMatrix B2 = LU3(A);
C = A * B.i() - IdentityMatrix(7); Clean(C, 0.000000001); Print(C);
D(10) = (B == B1 ? 0 : 1) + (B == B2 ? 0 : 1);
// check lengths
D(13) = B.size()-49;
// check release(2)
B1.release(2);
B2 = B1; D(15) = B == B2 ? 0 : 1;
CroutMatrix B3 = B1; D(16) = B == B3 ? 0 : 1;
D(17) = B1.size();
// some oddments
B1 = B; B1 = B1.i(); C = A - B1.i(); Clean(C, 0.000000001); Print(C);
B1 = B; B1.release(); B1 = B1; B2 = B1;
D(19) = B == B1 ? 0 : 1; D(20) = B == B2 ? 0 : 1;
B1.cleanup(); B2 = B1; D(21) = B1.size(); D(22) = B2.size();
GenericMatrix GM = B; C = A.i() - GM.i(); Clean(C, 0.000000001); Print(C);
B1 = GM; D(23) = B == B1 ? 0 : 1;
B1 = A * 0; B2 = B1; D(24) = B2.is_singular() ? 0 : 1;
// check release again - see if memory moves
const Real* d = B.const_data();
const int* i = B.const_data_indx();
B1 = B;
const Real* d1 = B1.const_data();
const int* i1 = B1.const_data_indx();
B1.release(); B2 = B1;
const Real* d2 = B2.const_data();
const int* i2 = B2.const_data_indx();
D(25) = (d != d1 ? 0 : 1) + (d1 == d2 ? 0 : 1)
+ (i != i1 ? 0 : 1) + (i1 == i2 ? 0 : 1);
Clean(D, 0.000000001); Print(D);
}
{
Tracer et1("Stage 8");
// Same for BandLUMatrix
BandMatrix A(7,3,2);
A.Row(1) << 3 << 2 << -1;
A.Row(2) << -8 << 7 << 2 << 0;
A.Row(3) << 2 << -2 << 3 << 1 << 9;
A.Row(4) << -1 << 5 << 2 << 2 << 5 << -1;
A.Row(5) << -4 << 1 << 9 << -8 << 7 << 5;
A.Row(6) << 5 << -1 << -2 << 5 << 1;
A.Row(7) << 8 << -1 << 2 << 2;
RowVector D(30); D = 0;
Real x = determinant(A);
BandLUMatrix B = A;
D(1) = determinant(B) / x - 1;
Matrix C = A * Inverter1(B) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// Test copy constructor (in Inverter2 and ordinary copy)
BandLUMatrix B1; B1 = B;
D(2) = determinant(B1) / x - 1;
C = A * Inverter2(B1) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// Do it again with release
B.release(); B1 = B;
D(2) = B.nrows(); D(3) = B.ncols(); D(4) = B.size();
D(5) = determinant(B1) / x - 1;
B1.release();
C = A * Inverter2(B1) - IdentityMatrix(7);
D(6) = B1.nrows(); D(7) = B1.ncols(); D(8) = B1.size();
Clean(C, 0.000000001); Print(C);
// see if we get an implicit invert
B1 = -A;
D(9) = determinant(B1) / x + 1; // odd number of rows - sign will change
C = -A * Inverter2(B1) - IdentityMatrix(7);
Clean(C, 0.000000001); Print(C);
// check for_return
B = LU1(A); B1 = LU2(A); BandLUMatrix B2 = LU3(A);
C = A * B.i() - IdentityMatrix(7); Clean(C, 0.000000001); Print(C);
D(10) = (B == B1 ? 0 : 1) + (B == B2 ? 0 : 1);
// check lengths
D(11) = B.bandwidth().lower()-3;
D(12) = B.bandwidth().upper()-2;
D(13) = B.size()-42;
D(14) = B.size2()-21;
// check release(2)
B1.release(2);
B2 = B1; D(15) = B == B2 ? 0 : 1;
BandLUMatrix B3 = B1; D(16) = B == B3 ? 0 : 1;
D(17) = B1.size();
// Compare with CroutMatrix
CroutMatrix CM = A;
C = CM.i() - B.i(); Clean(C, 0.000000001); Print(C);
D(18) = determinant(CM) / x - 1;
// some oddments
B1 = B; CM = B1.i(); C = A - CM.i(); Clean(C, 0.000000001); Print(C);
B1 = B; B1.release(); B1 = B1; B2 = B1;
D(19) = B == B1 ? 0 : 1; D(20) = B == B2 ? 0 : 1;
B1.cleanup(); B2 = B1; D(21) = B1.size(); D(22) = B2.size();
GenericMatrix GM = B; C = A.i() - GM.i(); Clean(C, 0.000000001); Print(C);
B1 = GM; D(23) = B == B1 ? 0 : 1;
B1 = A * 0; B2 = B1; D(24) = B2.is_singular() ? 0 : 1;
// check release again - see if memory moves
const Real* d = B.const_data(); const Real* dd = B.const_data();
const int* i = B.const_data_indx();
B1 = B;
const Real* d1 = B1.const_data(); const Real* dd1 = B1.const_data();
const int* i1 = B1.const_data_indx();
B1.release(); B2 = B1;
const Real* d2 = B2.const_data(); const Real* dd2 = B2.const_data();
const int* i2 = B2.const_data_indx();
D(25) = (d != d1 ? 0 : 1) + (d1 == d2 ? 0 : 1)
+ (dd != dd1 ? 0 : 1) + (dd1 == dd2 ? 0 : 1)
+ (i != i1 ? 0 : 1) + (i1 == i2 ? 0 : 1);
Clean(D, 0.000000001); Print(D);
}
{
Tracer et1("Stage 9");
// Modification of Cholesky decomposition
int i, j;
// Build test matrix
Matrix X(100, 10);
MultWithCarry mwc; // Uniform random number generator
for (i = 1; i <= 100; ++i) for (j = 1; j <= 10; ++j)
X(i, j) = 2.0 * (mwc.Next() - 0.5);
Matrix X1 = X; // save copy
// Form sums of squares and products matrix and Cholesky decompose
SymmetricMatrix A; A << X.t() * X;
UpperTriangularMatrix U1 = Cholesky(A).t();
// Do QR decomposition of X and check we get same triangular matrix
UpperTriangularMatrix U2;
QRZ(X, U2);
Matrix Diff = U1 - U2; Clean(Diff, 0.000000001); Print(Diff);
// Try adding new row to X and updating triangular matrix
RowVector NewRow(10);
for (j = 1; j <= 10; ++j) NewRow(j) = 2.0 * (mwc.Next() - 0.5);
UpdateCholesky(U2, NewRow);
X = X1 & NewRow; QRZ(X, U1);
Diff = U1 - U2; Clean(Diff, 0.000000001); Print(Diff);
// Try removing two rows and updating triangular matrix
DowndateCholesky(U2, X1.Row(20));
DowndateCholesky(U2, X1.Row(35));
X = X1.Rows(1,19) & X1.Rows(21,34) & X1.Rows(36,100) & NewRow; QRZ(X, U1);
Diff = U1 - U2; Clean(Diff, 0.000000001); Print(Diff);
// Circular shifts
CircularShift(X, 3,6);
CircularShift(X, 5,5);
CircularShift(X, 4,5);
CircularShift(X, 1,6);
CircularShift(X, 6,10);
}
{
Tracer et1("Stage 10");
// Try updating QRZ, QRZT decomposition
TestUpdateQRZ tuqrz1(10, 100, 50, 25); tuqrz1.DoTest();
tuqrz1.Reset(); tuqrz1.ClearRow(1); tuqrz1.DoTest();
tuqrz1.Reset(); tuqrz1.ClearRow(1); tuqrz1.ClearRow(2); tuqrz1.DoTest();
tuqrz1.Reset(); tuqrz1.ClearRow(5); tuqrz1.ClearRow(6); tuqrz1.DoTest();
tuqrz1.Reset(); tuqrz1.ClearRow(10); tuqrz1.DoTest();
TestUpdateQRZ tuqrz2(15, 100, 0, 0); tuqrz2.DoTest();
tuqrz2.Reset(); tuqrz2.ClearRow(1); tuqrz2.DoTest();
tuqrz2.Reset(); tuqrz2.ClearRow(1); tuqrz2.ClearRow(2); tuqrz2.DoTest();
tuqrz2.Reset(); tuqrz2.ClearRow(5); tuqrz2.ClearRow(6); tuqrz2.DoTest();
tuqrz2.Reset(); tuqrz2.ClearRow(15); tuqrz2.DoTest();
TestUpdateQRZ tuqrz3(5, 0, 10, 0); tuqrz3.DoTest();
}
// cout << "\nEnd of Thirteenth test\n";
}
void trymata()
{
// cout << "\nTenth test of Matrix package\n";
Tracer et("Tenth test of Matrix package");
Tracer::PrintTrace();
int i; int j;
UpperTriangularMatrix U(8);
for (i=1;i<=8;i++) for (j=i;j<=8;j++) U(i,j)=i+j*j+5;
Matrix X(8,6);
for (i=1;i<=8;i++) for (j=1;j<=6;j++) X(i,j)=i*j+1.0;
Matrix Y = U.i()*X; Matrix MU=U;
Y=Y-MU.i()*X; Clean(Y,0.00000001); Print(Y);
Y = U.t().i()*X; Y=Y-MU.t().i()*X; Clean(Y,0.00000001); Print(Y);
UpperTriangularMatrix UX(8);
for (i=1;i<=8;i++) for (j=i;j<=8;j++) UX(i,j)=i+j+1;
UX(4,4)=0; UX(4,5)=0;
UpperTriangularMatrix UY = U.i() * UX;
{ X=UX; MU=U; Y=UY-MU.i()*X; Clean(Y,0.000000001); Print(Y); }
LowerTriangularMatrix LY = U.t().i() * UX.t();
{ Y=LY-MU.i().t()*X.t(); Clean(Y,0.000000001); Print(Y); }
DiagonalMatrix D(8); for (i=1;i<=8;i++) D(i,i)=i+1;
{ X=D.i()*MU; }
{ UY=U; UY=D.i()*UY; Y=UY-X; Clean(Y,0.00000001); Print(Y); }
{ UY=D.i()*U; Y=UY-X; Clean(Y,0.00000001); Print(Y); }
// X=MU.t();
// LY=D.i()*U.t(); Y=D*LY-X; Clean(Y,0.00000001); Print(Y);
// LowerTriangularMatrix L=U.t();
// LY=D.i()*L; Y=D*LY-X; Clean(Y,0.00000001); Print(Y);
U.ReSize(8);
for (i=1;i<=8;i++) for (j=i;j<=8;j++) U(i,j)=i+j*j+5;
MU = U;
MU = U.i() - MU.i(); Clean(MU,0.00000001); Print(MU);
MU = U.t().i() - U.i().t(); Clean(MU,0.00000001); Print(MU);
// test LINEQ
{
ColumnVector X1(4), X2(4);
X1(1)=1; X1(2)=2; X1(3)=3; X1(4)=4;
X2(1)=1; X2(2)=10; X2(3)=100; X2(4)=1000;
Matrix A(4,4);
A(1,1)=1; A(1,2)=3; A(1,3)=0; A(1,4)=0;
A(2,1)=3; A(2,2)=2; A(2,3)=5; A(2,4)=0;
A(3,1)=0; A(3,2)=5; A(3,3)=4; A(3,4)=1;
A(4,1)=0; A(4,2)=0; A(4,3)=1; A(4,4)=3;
process(A,X1,X2);
BandMatrix B(4,1,1); B.Inject(A);
process(B,X1,X2);
UpperTriangularMatrix U(4);
U(1,1)=1; U(1,2)=2; U(1,3)=3; U(1,4)=4;
U(2,2)=8; U(2,3)=7; U(2,4)=6;
U(3,3)=2; U(3,4)=7;
U(4,4)=1;
process(U,X1,X2);
// check rowwise load
UpperTriangularMatrix U1(4);
U1.Row(1) << 1 << 2 << 3 << 4;
U1.Row(2) << 8 << 7 << 6;
U1.Row(3) << 2 << 7;
U1.Row(4) << 1;
U1 -= U;
Print(U1);
LowerTriangularMatrix L = U.t();
process(L,X1,X2);
}
// test inversion of poorly conditioned matrix
// a user complained this didn't work under OS9
{
Matrix M(4,4);
M << 8.613057e+00 << 8.693985e+00 << -2.322050e-01 << 0.000000e+00
<< 8.693985e+00 << 8.793868e+00 << -2.346310e-01 << 0.000000e+00
<< -2.322050e-01 << -2.346310e-01 << 6.264000e-03 << 0.000000e+00
<< 0.000000e+00 << 0.000000e+00 << 0.000000e+00 << 3.282806e+03 ;
Matrix MI = M.i();
DiagonalMatrix I(4); I = 1;
Matrix Diff = MI * M - I;
Clean(Diff,0.00000001); Print(Diff);
// Alternatively do Cholesky
SymmetricMatrix SM; SM << M;
LowerTriangularMatrix LT = Cholesky(SM).i();
MI = LT.t() * LT; Diff = MI * M - I;
Clean(Diff,0.00000001); Print(Diff);
}
// cout << "\nEnd of tenth test\n";
}
void trymatc()
{
// cout << "\nTwelfth test of Matrix package\n";
Tracer et("Twelfth test of Matrix package");
Tracer::PrintTrace();
DiagonalMatrix D(15); D=1.5;
Matrix A(15,15);
int i,j;
for (i=1;i<=15;i++) for (j=1;j<=15;j++) A(i,j)=i*i+j-150;
{ A = A + D; }
ColumnVector B(15);
for (i=1;i<=15;i++) B(i)=i+i*i-150.0;
{
Tracer et1("Stage 1");
ColumnVector B1=B;
B=(A*2.0).i() * B1;
Matrix X = A*B-B1/2.0;
Clean(X, 0.000000001); Print(X);
A.ReSize(3,5);
for (i=1; i<=3; i++) for (j=1; j<=5; j++) A(i,j) = i+100*j;
B = A.AsColumn()+10000;
RowVector R = (A+10000).AsColumn().t();
Print( RowVector(R-B.t()) );
}
{
Tracer et1("Stage 2");
B = A.AsColumn()+10000;
Matrix XR = (A+10000).AsMatrix(15,1).t();
Print( RowVector(XR-B.t()) );
}
{
Tracer et1("Stage 3");
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0+10000;
Matrix MR = (A+10000).AsColumn().t();
Print( RowVector(MR-B.t()) );
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0;
MR = A.AsColumn().t();
Print( RowVector(MR-B.t()) );
}
{
Tracer et1("Stage 4");
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0;
RowVector R = A.AsColumn().t();
Print( RowVector(R-B.t()) );
}
{
Tracer et1("Stage 5");
RowVector R = (A.AsColumn()-5000).t();
B = ((R.t()+10000) - A.AsColumn())-5000;
Print( RowVector(B.t()) );
}
{
Tracer et1("Stage 6");
B = A.AsColumn(); ColumnVector B1 = (A+10000).AsColumn() - 10000;
Print(ColumnVector(B1-B));
}
{
Tracer et1("Stage 7");
Matrix X = B.AsMatrix(3,5); Print(Matrix(X-A));
for (i=1; i<=3; i++) for (j=1; j<=5; j++) B(5*(i-1)+j) -= i+100*j;
Print(B);
}
{
Tracer et1("Stage 8");
A.ReSize(7,7); D.ReSize(7);
for (i=1; i<=7; i++) for (j=1; j<=7; j++) A(i,j) = i*j*j;
for (i=1; i<=7; i++) D(i,i) = i;
UpperTriangularMatrix U; U << A;
Matrix X = A; for (i=1; i<=7; i++) X(i,i) = i;
A.Inject(D); Print(Matrix(X-A));
X = U; U.Inject(D); A = U; for (i=1; i<=7; i++) X(i,i) = i;
Print(Matrix(X-A));
}
{
Tracer et1("Stage 9");
A.ReSize(7,5);
for (i=1; i<=7; i++) for (j=1; j<=5; j++) A(i,j) = i+100*j;
Matrix Y = A; Y = Y - ((const Matrix&)A); Print(Y);
Matrix X = A; // X.Release();
Y = A; Y = ((const Matrix&)X) - A; Print(Y); Y = 0.0;
Y = ((const Matrix&)X) - ((const Matrix&)A); Print(Y);
}
{
Tracer et1("Stage 10");
// some tests on submatrices
UpperTriangularMatrix U(20);
for (i=1; i<=20; i++) for (j=i; j<=20; j++) U(i,j)=100 * i + j;
UpperTriangularMatrix V = U.SymSubMatrix(1,5);
UpperTriangularMatrix U1 = U;
U1.SubMatrix(4,8,5,9) /= 2;
U1.SubMatrix(4,8,5,9) += 388 * V;
U1.SubMatrix(4,8,5,9) *= 2;
U1.SubMatrix(4,8,5,9) += V;
U1 -= U; UpperTriangularMatrix U2 = U1;
U1 << U1.SubMatrix(4,8,5,9);
U2.SubMatrix(4,8,5,9) -= U1; Print(U2);
U1 -= (777*V); Print(U1);
U1 = U; U1.SubMatrix(4,8,5,9) -= U.SymSubMatrix(1,5);
U1 -= U; U2 = U1; U1 << U1.SubMatrix(4,8,5,9);
U2.SubMatrix(4,8,5,9) -= U1; Print(U2);
U1 += V; Print(U1);
U1 = U;
U1.SubMatrix(3,10,15,19) += 29;
U1 -= U;
Matrix X = U1.SubMatrix(3,10,15,19); X -= 29; Print(X);
U1.SubMatrix(3,10,15,19) *= 0; Print(U1);
LowerTriangularMatrix L = U.t();
LowerTriangularMatrix M = L.SymSubMatrix(1,5);
LowerTriangularMatrix L1 = L;
L1.SubMatrix(5,9,4,8) /= 2;
L1.SubMatrix(5,9,4,8) += 388 * M;
L1.SubMatrix(5,9,4,8) *= 2;
L1.SubMatrix(5,9,4,8) += M;
L1 -= L; LowerTriangularMatrix L2 = L1;
L1 << L1.SubMatrix(5,9,4,8);
L2.SubMatrix(5,9,4,8) -= L1; Print(L2);
L1 -= (777*M); Print(L1);
L1 = L; L1.SubMatrix(5,9,4,8) -= L.SymSubMatrix(1,5);
L1 -= L; L2 =L1; L1 << L1.SubMatrix(5,9,4,8);
L2.SubMatrix(5,9,4,8) -= L1; Print(L2);
L1 += M; Print(L1);
L1 = L;
L1.SubMatrix(15,19,3,10) -= 29;
L1 -= L;
X = L1.SubMatrix(15,19,3,10); X += 29; Print(X);
L1.SubMatrix(15,19,3,10) *= 0; Print(L1);
}
{
Tracer et1("Stage 11");
// more tests on submatrices
Matrix M(20,30);
for (i=1; i<=20; i++) for (j=1; j<=30; j++) M(i,j)=100 * i + j;
Matrix M1 = M;
for (j=1; j<=30; j++)
{ ColumnVector CV = 3 * M1.Column(j); M.Column(j) += CV; }
for (i=1; i<=20; i++)
{ RowVector RV = 5 * M1.Row(i); M.Row(i) -= RV; }
M += M1; Print(M);
}
// cout << "\nEnd of twelfth test\n";
}
void trymatc()
{
// cout << "\nTwelfth test of Matrix package\n";
Tracer et("Twelfth test of Matrix package");
Tracer::PrintTrace();
DiagonalMatrix D(15); D=1.5;
Matrix A(15,15);
int i,j;
for (i=1;i<=15;i++) for (j=1;j<=15;j++) A(i,j)=i*i+j-150;
{ A = A + D; }
ColumnVector B(15);
for (i=1;i<=15;i++) B(i)=i+i*i-150.0;
{
Tracer et1("Stage 1");
ColumnVector B1=B;
B=(A*2.0).i() * B1;
Matrix X = A*B-B1/2.0;
Clean(X, 0.000000001); Print(X);
A.ReSize(3,5);
for (i=1; i<=3; i++) for (j=1; j<=5; j++) A(i,j) = i+100*j;
B = A.AsColumn()+10000;
RowVector R = (A+10000).AsColumn().t();
Print( RowVector(R-B.t()) );
}
{
Tracer et1("Stage 2");
B = A.AsColumn()+10000;
Matrix XR = (A+10000).AsMatrix(15,1).t();
Print( RowVector(XR-B.t()) );
}
{
Tracer et1("Stage 3");
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0+10000;
Matrix MR = (A+10000).AsColumn().t();
Print( RowVector(MR-B.t()) );
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0;
MR = A.AsColumn().t();
Print( RowVector(MR-B.t()) );
}
{
Tracer et1("Stage 4");
B = (A.AsMatrix(15,1)+A.AsColumn())/2.0;
RowVector R = A.AsColumn().t();
Print( RowVector(R-B.t()) );
}
{
Tracer et1("Stage 5");
RowVector R = (A.AsColumn()-5000).t();
B = ((R.t()+10000) - A.AsColumn())-5000;
Print( RowVector(B.t()) );
}
{
Tracer et1("Stage 6");
B = A.AsColumn(); ColumnVector B1 = (A+10000).AsColumn() - 10000;
Print(ColumnVector(B1-B));
}
{
Tracer et1("Stage 7");
Matrix X = B.AsMatrix(3,5); Print(Matrix(X-A));
for (i=1; i<=3; i++) for (j=1; j<=5; j++) B(5*(i-1)+j) -= i+100*j;
Print(B);
}
{
Tracer et1("Stage 8");
A.ReSize(7,7); D.ReSize(7);
for (i=1; i<=7; i++) for (j=1; j<=7; j++) A(i,j) = i*j*j;
for (i=1; i<=7; i++) D(i,i) = i;
UpperTriangularMatrix U; U << A;
Matrix X = A; for (i=1; i<=7; i++) X(i,i) = i;
A.Inject(D); Print(Matrix(X-A));
X = U; U.Inject(D); A = U; for (i=1; i<=7; i++) X(i,i) = i;
Print(Matrix(X-A));
}
{
Tracer et1("Stage 9");
A.ReSize(7,5);
for (i=1; i<=7; i++) for (j=1; j<=5; j++) A(i,j) = i+100*j;
Matrix Y = A; Y = Y - ((const Matrix&)A); Print(Y);
Matrix X = A;
Y = A; Y = ((const Matrix&)X) - A; Print(Y); Y = 0.0;
Y = ((const Matrix&)X) - ((const Matrix&)A); Print(Y);
}
{
Tracer et1("Stage 10");
// some tests on submatrices
UpperTriangularMatrix U(20);
for (i=1; i<=20; i++) for (j=i; j<=20; j++) U(i,j)=100 * i + j;
UpperTriangularMatrix V = U.SymSubMatrix(1,5);
UpperTriangularMatrix U1 = U;
U1.SubMatrix(4,8,5,9) /= 2;
U1.SubMatrix(4,8,5,9) += 388 * V;
U1.SubMatrix(4,8,5,9) *= 2;
U1.SubMatrix(4,8,5,9) += V;
U1 -= U; UpperTriangularMatrix U2 = U1;
U1 << U1.SubMatrix(4,8,5,9);
U2.SubMatrix(4,8,5,9) -= U1; Print(U2);
U1 -= (777*V); Print(U1);
U1 = U; U1.SubMatrix(4,8,5,9) -= U.SymSubMatrix(1,5);
U1 -= U; U2 = U1; U1 << U1.SubMatrix(4,8,5,9);
U2.SubMatrix(4,8,5,9) -= U1; Print(U2);
U1 += V; Print(U1);
U1 = U;
U1.SubMatrix(3,10,15,19) += 29;
U1 -= U;
Matrix X = U1.SubMatrix(3,10,15,19); X -= 29; Print(X);
U1.SubMatrix(3,10,15,19) *= 0; Print(U1);
LowerTriangularMatrix L = U.t();
LowerTriangularMatrix M = L.SymSubMatrix(1,5);
LowerTriangularMatrix L1 = L;
L1.SubMatrix(5,9,4,8) /= 2;
L1.SubMatrix(5,9,4,8) += 388 * M;
L1.SubMatrix(5,9,4,8) *= 2;
L1.SubMatrix(5,9,4,8) += M;
L1 -= L; LowerTriangularMatrix L2 = L1;
L1 << L1.SubMatrix(5,9,4,8);
L2.SubMatrix(5,9,4,8) -= L1; Print(L2);
L1 -= (777*M); Print(L1);
L1 = L; L1.SubMatrix(5,9,4,8) -= L.SymSubMatrix(1,5);
L1 -= L; L2 =L1; L1 << L1.SubMatrix(5,9,4,8);
L2.SubMatrix(5,9,4,8) -= L1; Print(L2);
L1 += M; Print(L1);
L1 = L;
L1.SubMatrix(15,19,3,10) -= 29;
L1 -= L;
X = L1.SubMatrix(15,19,3,10); X += 29; Print(X);
L1.SubMatrix(15,19,3,10) *= 0; Print(L1);
}
{
Tracer et1("Stage 11");
// more tests on submatrices
Matrix M(20,30);
for (i=1; i<=20; i++) for (j=1; j<=30; j++) M(i,j)=100 * i + j;
Matrix M1 = M;
for (j=1; j<=30; j++)
{ ColumnVector CV = 3 * M1.Column(j); M.Column(j) += CV; }
for (i=1; i<=20; i++)
{ RowVector RV = 5 * M1.Row(i); M.Row(i) -= RV; }
M += M1; Print(M);
}
{
Tracer et1("Stage 12");
// more tests on Release
Matrix M(20,30);
for (i=1; i<=20; i++) for (j=1; j<=30; j++) M(i,j)=100 * i + j;
Matrix M1 = M;
M.Release();
Matrix M2 = M;
Matrix X = M; Print(X);
X = M1 - M2; Print(X);
#ifndef DONT_DO_NRIC
nricMatrix N = M1;
nricMatrix N1 = N;
N.Release();
nricMatrix N2 = N;
nricMatrix Y = N; Print(Y);
Y = N1 - N2; Print(Y);
N = M1 / 2; N1 = N * 2; N.Release(); N2 = N * 2; Y = N; Print(N);
Y = (N1 - M1) | (N2 - M1); Print(Y);
#endif
}
{
Tracer et("Stage 13");
// test sum of squares of rows or columns
MultWithCarry mwc;
DiagonalMatrix DM; Matrix X;
// rectangular matrix
Matrix A(20, 15);
FillWithValues(mwc, A);
// sum of squares of rows
DM << A * A.t();
ColumnVector CV = A.sum_square_rows();
X = CV - DM.AsColumn(); Clean(X, 0.000000001); Print(X);
DM << A.t() * A;
RowVector RV = A.sum_square_columns();
X = RV - DM.AsRow(); Clean(X, 0.000000001); Print(X);
X = RV - A.t().sum_square_rows().t(); Clean(X, 0.000000001); Print(X);
X = CV - A.t().sum_square_columns().t(); Clean(X, 0.000000001); Print(X);
// UpperTriangularMatrix
A.ReSize(17,17); FillWithValues(mwc, A);
UpperTriangularMatrix UT; UT << A;
Matrix A1 = UT;
X = UT.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = UT.sum_square_columns() - A1.sum_square_columns(); Print(X);
// LowerTriangularMatrix
LowerTriangularMatrix LT; LT << A;
A1 = LT;
X = LT.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = LT.sum_square_columns() - A1.sum_square_columns(); Print(X);
// SymmetricMatrix
SymmetricMatrix SM; SM << A;
A1 = SM;
X = SM.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = SM.sum_square_columns() - A1.sum_square_columns(); Print(X);
// DiagonalMatrix
DM << A;
A1 = DM;
X = DM.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = DM.sum_square_columns() - A1.sum_square_columns(); Print(X);
// BandMatrix
BandMatrix BM(17, 3, 5); BM.Inject(A);
A1 = BM;
X = BM.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = BM.sum_square_columns() - A1.sum_square_columns(); Print(X);
// SymmetricBandMatrix
SymmetricBandMatrix SBM(17, 4); SBM.Inject(A);
A1 = SBM;
X = SBM.sum_square_rows() - A1.sum_square_rows(); Print(X);
X = SBM.sum_square_columns() - A1.sum_square_columns(); Print(X);
// IdentityMatrix
IdentityMatrix IM(29);
X = IM.sum_square_rows() - 1; Print(X);
X = IM.sum_square_columns() - 1; Print(X);
// Matrix with zero rows
A1.ReSize(0,10);
X.ReSize(1,10); X = 0; X -= A1.sum_square_columns(); Print(X);
X.ReSize(0,1); X -= A1.sum_square_rows(); Print(X);
// Matrix with zero columns
A1.ReSize(10,0);
X.ReSize(10,1); X = 0; X -= A1.sum_square_rows(); Print(X);
X.ReSize(1,0); X -= A1.sum_square_columns(); Print(X);
}
{
Tracer et("Stage 14");
// test extend orthonormal
MultWithCarry mwc;
Matrix A(20,5); FillWithValues(mwc, A);
// Orthonormalise
UpperTriangularMatrix R;
Matrix A_old = A;
QRZ(A,R);
// Check decomposition
Matrix X = A * R - A_old; Clean(X, 0.000000001); Print(X);
// Check orthogonality
X = A.t() * A - IdentityMatrix(5);
Clean(X, 0.000000001); Print(X);
// Try orthonality extend
SquareMatrix A1(20);
A1.Columns(1,5) = A;
extend_orthonormal(A1,5);
// check columns unchanged
X = A - A1.Columns(1,5); Print(X);
// Check orthogonality
X = A1.t() * A1 - IdentityMatrix(20);
Clean(X, 0.000000001); Print(X);
X = A1 * A1.t() - IdentityMatrix(20);
Clean(X, 0.000000001); Print(X);
// Test with smaller number of columns
Matrix A2(20,15);
A2.Columns(1,5) = A;
extend_orthonormal(A2,5);
// check columns unchanged
X = A - A2.Columns(1,5); Print(X);
// Check orthogonality
X = A2.t() * A2 - IdentityMatrix(15);
Clean(X, 0.000000001); Print(X);
// check it works with no columns to start with
A2.ReSize(100,100);
extend_orthonormal(A2,0);
// Check orthogonality
X = A2.t() * A2 - IdentityMatrix(100);
Clean(X, 0.000000001); Print(X);
X = A2 * A2.t() - IdentityMatrix(100);
Clean(X, 0.000000001); Print(X);
}
// cout << "\nEnd of twelfth test\n";
}
void trymatd()
{
Tracer et("Thirteenth test of Matrix package");
Tracer::PrintTrace();
Matrix X(5,20);
int i,j;
for (j=1;j<=20;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=20; j++) X(i,j) = (long)X(i-1,j) * j % 1001;
SymmetricMatrix S; S << X * X.t();
Matrix SM = X * X.t() - S;
Print(SM);
LowerTriangularMatrix L = Cholesky(S);
Matrix Diff = L*L.t()-S; Clean(Diff, 0.000000001);
Print(Diff);
{
Tracer et1("Stage 1");
LowerTriangularMatrix L1(5);
Matrix Xt = X.t(); Matrix Xt2 = Xt;
QRZT(X,L1);
Diff = L - L1; Clean(Diff,0.000000001); Print(Diff);
UpperTriangularMatrix Ut(5);
QRZ(Xt,Ut);
Diff = L - Ut.t(); Clean(Diff,0.000000001); Print(Diff);
Matrix Y(3,20);
for (j=1;j<=20;j++) Y(1,j) = 22-j;
for (i=2;i<=3;i++) for (j=1;j<=20; j++)
Y(i,j) = (long)Y(i-1,j) * j % 101;
Matrix Yt = Y.t(); Matrix M,Mt; Matrix Y2=Y;
QRZT(X,Y,M); QRZ(Xt,Yt,Mt);
Diff = Xt - X.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Yt - Y.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Mt - M.t(); Clean(Diff,0.000000001); Print(Diff);
Diff = Y2 * Xt2 * S.i() - M * L.i();
Clean(Diff,0.000000001); Print(Diff);
}
ColumnVector C1(5);
{
Tracer et1("Stage 2");
X.ReSize(5,5);
for (j=1;j<=5;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=5; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
for (i=1;i<=5;i++) C1(i) = i*i;
CroutMatrix A = X;
ColumnVector C2 = A.i() * C1; C1 = X.i() * C1;
X = C1 - C2; Clean(X,0.000000001); Print(X);
}
{
Tracer et1("Stage 3");
X.ReSize(7,7);
for (j=1;j<=7;j++) X(1,j) = j+1;
for (i=2;i<=7;i++) for (j=1;j<=7; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
C1.ReSize(7);
for (i=1;i<=7;i++) C1(i) = i*i;
RowVector R1 = C1.t();
Diff = R1 * X.i() - ( X.t().i() * R1.t() ).t(); Clean(Diff,0.000000001);
Print(Diff);
}
{
Tracer et1("Stage 4");
X.ReSize(5,5);
for (j=1;j<=5;j++) X(1,j) = j+1;
for (i=2;i<=5;i++) for (j=1;j<=5; j++)
X(i,j) = (long)X(i-1,j) * j % 1001;
C1.ReSize(5);
for (i=1;i<=5;i++) C1(i) = i*i;
CroutMatrix A1 = X*X;
ColumnVector C2 = A1.i() * C1; C1 = X.i() * C1; C1 = X.i() * C1;
X = C1 - C2; Clean(X,0.000000001); Print(X);
}
{
Tracer et1("Stage 5");
int n = 40;
SymmetricBandMatrix B(n,2); B = 0.0;
for (i=1; i<=n; i++)
{
B(i,i) = 6;
if (i<=n-1) B(i,i+1) = -4;
if (i<=n-2) B(i,i+2) = 1;
}
B(1,1) = 5; B(n,n) = 5;
SymmetricMatrix A = B;
ColumnVector X(n);
X(1) = 429;
for (i=2;i<=n;i++) X(i) = (long)X(i-1) * 31 % 1001;
X = X / 100000L;
// the matrix B is rather ill-conditioned so the difficulty is getting
// good agreement (we have chosen X very small) may not be surprising;
// maximum element size in B.i() is around 1400
ColumnVector Y1 = A.i() * X;
LowerTriangularMatrix C1 = Cholesky(A);
ColumnVector Y2 = C1.t().i() * (C1.i() * X) - Y1;
Clean(Y2, 0.000000001); Print(Y2);
UpperTriangularMatrix CU = C1.t().i();
LowerTriangularMatrix CL = C1.i();
Y2 = CU * (CL * X) - Y1;
Clean(Y2, 0.000000001); Print(Y2);
Y2 = B.i() * X - Y1; Clean(Y2, 0.000000001); Print(Y2);
LowerBandMatrix C2 = Cholesky(B);
Matrix M = C2 - C1; Clean(M, 0.000000001); Print(M);
ColumnVector Y3 = C2.t().i() * (C2.i() * X) - Y1;
Clean(Y3, 0.000000001); Print(Y3);
CU = C1.t().i();
CL = C1.i();
Y3 = CU * (CL * X) - Y1;
Clean(Y3, 0.000000001); Print(Y3);
Y3 = B.i() * X - Y1; Clean(Y3, 0.000000001); Print(Y3);
SymmetricMatrix AI = A.i();
Y2 = AI*X - Y1; Clean(Y2, 0.000000001); Print(Y2);
SymmetricMatrix BI = B.i();
BandMatrix C = B; Matrix CI = C.i();
M = A.i() - CI; Clean(M, 0.000000001); Print(M);
M = B.i() - CI; Clean(M, 0.000000001); Print(M);
M = AI-BI; Clean(M, 0.000000001); Print(M);
M = AI-CI; Clean(M, 0.000000001); Print(M);
M = A; AI << M; M = AI-A; Clean(M, 0.000000001); Print(M);
C = B; BI << C; M = BI-B; Clean(M, 0.000000001); Print(M);
}
{
Tracer et1("Stage 5");
SymmetricMatrix A(4), B(4);
A << 5
<< 1 << 4
<< 2 << 1 << 6
<< 1 << 0 << 1 << 7;
B << 8
<< 1 << 5
<< 1 << 0 << 9
<< 2 << 1 << 0 << 6;
LowerTriangularMatrix AB = Cholesky(A) * Cholesky(B);
Matrix M = Cholesky(A) * B * Cholesky(A).t() - AB*AB.t();
Clean(M, 0.000000001); Print(M);
M = A * Cholesky(B); M = M * M.t() - A * B * A;
Clean(M, 0.000000001); Print(M);
}
{
Tracer et1("Stage 6");
int N=49;
int i;
SymmetricBandMatrix S(N,1);
Matrix B(N,N+1); B=0;
for (i=1;i<=N;i++) { S(i,i)=1; B(i,i)=1; B(i,i+1)=-1; }
for (i=1;i<N; i++) S(i,i+1)=-.5;
DiagonalMatrix D(N+1); D = 1;
B = B.t()*S.i()*B - (D-1.0/(N+1))*2.0;
Clean(B, 0.000000001); Print(B);
}
{
Tracer et1("Stage 7");
// See if you can pass a CroutMatrix to a function
Matrix A(4,4);
A.Row(1) << 3 << 2 << -1 << 4;
A.Row(2) << -8 << 7 << 2 << 0;
A.Row(3) << 2 << -2 << 3 << 1;
A.Row(4) << -1 << 5 << 2 << 2;
CroutMatrix B = A;
Matrix C = A * Inverter(B) - IdentityMatrix(4);
Clean(C, 0.000000001); Print(C);
}
// cout << "\nEnd of Thirteenth test\n";
}
