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";
}
ReturnMatrix Inverter1(const CroutMatrix& X)
{
Matrix Y = X.i();
Y.Release();
return Y.ForReturn();
}
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";
}
