static void Main()
{
var ma = new matrix("1 2;5 6");
ma.print();
var ma2 = ma * ma.T;
(ma2).print();
WriteLine($"{ma[0].norm()}");
WriteLine($"{ma[1].norm()}");
WriteLine($"{ma[0,0]} {ma[0,1]}");
WriteLine($"{ma[1,0]} {ma[1,1]}");
matrix v1 = (ma).cols(0, 0);
(v1).print();
vector v2 = ma[1];
(v2).print();
matrix v3 = (ma2).cols(0, 0);
(v3).print();
matrix v4 = (ma2).rows(1, 1);
(v4).print();
ma[1, 1] = 22;
ma.print();
}
public static void Main()
{
int n = 3;
matrix A = new matrix(n, n);
matrix V = new matrix(n, n);
vector e = new vector(n);
var rand = new Random();
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
A[i, j] = rand.NextDouble() * 10;
A[j, i] = A[i, j];
}
}
A.print("Symmetric matrix A:");
matrix Ao = new matrix(n, n);
Ao = A.copy();
int count = jacobi.cycsweep(A, e, V);
WriteLine($"Number of sweeps: {count}");
V.print("Eigenvectors");
A.print("new A");
e.print("Eigenvalues");
(V.T * Ao * V).print("V^T*A*V diagonal, test");
}
static void Main()
{
WriteLine("Exam problem 21: \nTwo-sided Jacobi algorithm for Singular Value Decomposition (SVD) \n");;
int n = 7;
WriteLine($"Generating Random {n}x{n} matrix");
matrix A = randomMatrix(n, n), U = new matrix(n, n), V = new matrix(n, n);
matrix D = A.copy();
A.print("Matrix A generated:");
WriteLine("Making composition...");
int sweeps = twoSidedJacobiSVD(D, U, V);
WriteLine($"Procedure done!\nNo. of sweeps: {sweeps}.\nNow A --> U*D*V^T, where");
D.print("D: ");
U.print("U: ");
V.print("V: ");
matrix R = (U * D) * V.transpose();
R.print("Test that UDV^T = A");
A.print("A, for reference");
matrix UUt = U * U.transpose(), VVt = V * V.transpose();
UUt.print("Test U*U^T = 1");
VVt.print("Test V*V^T = 1");
}
public static void Main()
{
WriteLine();
WriteLine("Create a matrix and diagonalize it via the cyclic, value-by-value and" +
" classic method.");
var rand = new Random();
int n = 5;
// Create a random matrix
matrix A = new matrix(n, n);
for (int i = 0; i < n; i++)
{
A[i, i] = 2 - 4 * rand.NextDouble();
for (int j = i + 1; j < n; j++)
{
A[i, j] = 2 - 3.32 * rand.NextDouble();
A[j, i] = A[i, j];
}
}
// Print the random matrix.
WriteLine("\nSetting up a random symmetric matrix:");
A.print();
WriteLine();
// Diagonalize it via the three different methods
matrix Acopy = A.copy();
matrix AcopyII = A.copy();
matrix V = new matrix(n, n);
vector e = new vector(n);
vector eRot = new vector(n);
vector eClassic = new vector(n);
int sweeps = jacobi.cycle(A, e, V);
int entries = (n * n - n) / 2;
int rotations = jacobi.findEigenvalue(Acopy, eRot, V, n, true);
int rotationsClassic = jacobi.classic(AcopyII, eClassic, V);
WriteLine("The cyclic method used {0} operations", sweeps * entries);
WriteLine("The value-by-value method used {0} operations", rotations);
WriteLine("The classic method used {0} operations", rotationsClassic);
WriteLine("\nAfter cyclic diagonalization the matrix looks like:");
A.print();
WriteLine("\nAfter value-by-value diagonalization the matrix looks like:");
Acopy.print();
WriteLine("\nAfter classic Jacobi diagonalization the matrix looks like:");
AcopyII.print();
Write("\n\n");
WriteLine("The found eigenvalues are:");
WriteLine("E(cyclic)\t\tE(val_by_val)\t\tE(classic)");
for (int j = 0; j < n; j++)
{
WriteLine("{0}\t{1}\t{2}", e[j], eRot[j], eClassic[j]);
}
}
static void Main()
{
int n = 6;
int m = 4;
// Problem A1:
WriteLine($"Problem A1: QR-factorization on a random {n}x{m} matrix A: \n");
matrix A = generatematrix(n, m);
// Printing our random matrix A
A.print("Matrix A =");
QRdecompositionGS decomp = new QRdecompositionGS(A);
// Printing the orthogonal matrix Q
matrix Q = decomp.Q;
Q.print("\nDecomposed component Q = ");
// Printing the upper triangle matrix R
matrix R = decomp.R;
R.print("\nDecomposed component R = ");
// Checking that Q^T*Q is equal to the identity
(Q.transpose() * Q).print("\nQ^(T)*Q = ");
// Checking that QR=A by substracting A from QR to yield the null matrix
(Q * R - A).print("\n Q*R-A = ");
n = 3;
// Problem A2:
WriteLine($"\nProblem A2: QR-factorization on a random {n}x{n} matrix A solving Ax=b: \n");
vector b = generatevector(n);
A = generatematrix(n, n);
decomp = new QRdecompositionGS(A);
b.print("Vector b = ");
A.print("\nSquare matrix A = ");
vector x = decomp.solve(b);
// Printing the solution to Ax=b
x.print("\nVector x = ");
WriteLine("\nChecking that x is indeed the solution: ");
(A * x - b).print("\nA*x-b = ");
// Problem B:
A = generatematrix(n, n);
decomp = new QRdecompositionGS(A);
WriteLine($"\nProblem B: Matrix inverse by Gram-Schmidt QR factorization \n");
matrix B = decomp.inverse();
B.print("A^(-1) = ");
(A * B).print("\n A*A^(-1) = ");
// Problem C:
}
class main { public static void Main()
{
int n = RandomInt(3, 5);
matrix A = new matrix(n, n);
for (int i = 0; i < n; i = i + 1)
{
for (int j = i; j < n; j = j + 1)
{
if (i == j)
{
A[i, j] = RandomDouble(5, 20);
}
else
{
A[i, j] = RandomDouble(5, 20); A[j, i] = A[i, j];
}
}
}
A.print("Randomly generated symmetric square matrix A=\n");
matrix A_2 = A.copy(); matrix V = new matrix(n, n); vector e = new vector(n);
// Diagonalization
int sweeps = JacobiDiag(A, e, V);
A.print("\nD=\n");
matrix A1 = V.T * A_2 * V;
V.print("\nV=\n");
A1.print("\nV(^T)AV=\n");
e.print("\nObtained eigenvalues of A:\n\n");
double sum = 0;
for (int i = 0; i < n; i = i + 1)
{
if (Round(A1[i, i]) == Round(e[i]))
{
sum = sum + 1;
}
}
if (sum == n)
{
Out.WriteLine($"\nDiagonalization succesfull: Completed in {sweeps} iterations");
}
else
{
Out.WriteLine("\nDiagonalization failed");
}
}
static void probA1()
{
Write("Problem A1:\n");
var rand = new System.Random();
int m = 2 + rand.Next(6);
int n = m + rand.Next(6);
Write("First I create a random tall matrix with random dimmensions\n");
matrix A = makeRandomMatrix(n, m);
A.print("Random matrix A:");
qr_gs decomposer = new qr_gs(A);
Write("\nThen I decompose it into Q and R using the Gramm Schmitt method trough my qr_gs class:\n");
matrix Q = decomposer.Q;
matrix R = decomposer.R;
Q.print("Decomposition Q = ");
R.print("Decomposition R = ");
Write("\nCheck that Q^(T)Q is equal to the identity matrix:\n");
(Q.transpose() * Q).print("Q^(T)Q = ");
Write("\nCheck that Q*R = A, or equivalently that Q*R-A is a matrix of all zeros:\n");
(Q * R - A).print("Q*R-A");
}
static void probA2()
{
Write("\nProblem A2:\n");
// Need to make the tests for A2
var rand = new System.Random();
// Remember, square matrix:
Write("\nMake a random square matrix, A, with random dimmensions, and a random vector, b, of the same dimmension:\n");
int n = 2 + rand.Next(10);
matrix A = makeRandomMatrix(n, n);
vector b = makeRandomVector(n);
A.print("A = ");
b.print("b = ");
qr_gs decomposer = new qr_gs(A);
Write("\nSolve the system A*x = b for the vector x:\n");
//decomposer.Q.print("Decomposition Q: ");
//decomposer.R.print("Decomposition R: ");
//(decomposer.Q.transpose()*b).print("Q^(T)*b = ");
vector x = decomposer.solve(b);
x.print("Solution: x = ");
Write("\nCheck that the vector x satisfies A*x=b:\n");
(A * x - b).print("A*x-b = ");
}
static void Main()
{
/// Test on random matrix ///
WriteLine("/////////////// Part A - Test on random matrix ///////////////");
int n = 5;
var rand = new Random(1);
matrix A = new matrix(n, n);
vector e = new vector(n);
matrix V = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
A[i, j] = rand.NextDouble();
A[j, i] = A[i, j];
}
}
A.print($"random {n}x{n} matrix"); WriteLine();
matrix B = A.copy();
int sweeps = jacobi.cyclic(B, e, V);
WriteLine($"Number of sweeps in jacobi={sweeps}"); WriteLine();
matrix D = (V.T * A * V);
(D).print("Should be a diagonal matrix V.T*B*V="); WriteLine();
e.print("Eigenvalues should equal the diagonal elements above");
vector D_diagonal = new vector(n);
for (int i = 0; i < n; i++)
{
D_diagonal[i] = D[i, i];
}
if (D_diagonal.approx(e))
{
WriteLine("Eigenvalues agree \tTest passed");
}
else
{
WriteLine("Test failed");
}
WriteLine();
matrix A2 = (V * D * V.T);
(A2).print("Check that V*D*V.T=A");
if (A2.approx(A))
{
WriteLine("V*D*V.T = A \tTest passed");
}
else
{
WriteLine("V*D*V.T != A \tTest failed");
}
}
static void B2()
{
WriteLine("\n Problem B2\n");
int n = 8;
matrix A = rand_sym_mat(n);
matrix B = A.copy();
WriteLine("Random symmetric matrix A with size {0}", n);
WriteLine("Perform eigenvalue by eigenvalue calculation for first 4 values");
int evalnum = 4;
bool val_by_val = true;
jcbi_cycl test_vbv = new jcbi_cycl(B, val_by_val, evalnum);
Write("Found eigenvalues = ");
for (int i = 0; i < evalnum; i++)
{
Write(" {0:g3} ", test_vbv.Eigvals[i]);
}
Write("\n");
((test_vbv.V).T * A * test_vbv.V - test_vbv.D).print("V.T*A*V - D = ", "{0,10:f6}");
WriteLine("\n Total rotations used = {0}", test_vbv.Rotations);
B.print("\n Matrix copy used, after value by value operations = ", "{0,10:f6}");
matrix C = A.copy();
WriteLine("\nSimilar calculation with cyclic sweeps");
jcbi_cycl test_cycl = new jcbi_cycl(C);
test_cycl.Eigvals.print("Found eigenvalues = ");
(test_cycl.V.T * A * test_cycl.V - test_cycl.D).print("V.T*A*V - D = ", "{0,10:f6}");
WriteLine("\n Total rotations used = {0}", test_cycl.Rotations);
C.print("\n Matrix copy used, after cyclic sweeps = ", "{0,10:f6}");
}
static void Main()
{
var rnd = new Random();
var A = new matrix(3, 3);
for (int i = 0; i < A.size1; i++)
{
for (int j = 0; j < A.size2; j++)
{
A[i, j] = 10 * (rnd.NextDouble() - 0.5);
}
}
var Q = A.copy();
var R = new matrix(Q.size2, Q.size2);
WriteLine("\n__________________________________________________________________________________________________________");
WriteLine("Question B:\nMatrix inverse by Gram-Schmidt QR factorization");
WriteLine("Random matrix");
A.print("A = ");
qr_gs.qr_gs_decomp(Q, R);
WriteLine("Factorized into");
Q.print("Q = ");
R.print("R = ");
var B = qr_gs.qr_gs_inverse(Q, R);
WriteLine("Inverse of A:");
B.print("A^-1=B=");
WriteLine("Product should give identity");
(A * B).print("I=AB=");
WriteLine("__________________________________________________________________________________________________________\n");
}
public static void Main(string[] args)
{
int n = 5; //size of real symmetric matrix
matrix A = new matrix(n, n);
var rnd = new Random(1);
for (int i = 0; i < n; i++) /* Construction of a random vector */
{
for (int j = 0; j < n; j++)
{
A[i, j] = (rnd.NextDouble() * 10);
A[j, i] = A[i, j]; /* We make sure that the matrix is symmetric */
}
}
A.print("Random symmetric matrix, A:");
matrix V = new matrix(n, n);
vector e = new vector(n);
matrix B = A.copy(); /* We make a copy of A */
int sweeps = jacobi.cyclic(A, e, V); /* We use the jacobi matlib for the implementation of the cyclic sweeps: public static int cyclic(matrix A, vector e, matrix V=null){...} */
WriteLine($"\nNumber of sweeps: {sweeps}");
matrix D = (V.T * B * V);
// D.print("\nEigenvalue-decomposition, V^T*A*V (should be diagonal):");
D.printfloat("\nEigenvalue-decomposition, V^T*A*V (should be diagonal):"); /* using my print2 to get 0's instead of values with e.g. "e-16"*/
e.print("\nEigenvalues:\n");
}
public static void Main()
{
// First part of A
int n = 5; // Size of quadratic matrix M.
var rnd = new Random(1);
Console.WriteLine("Part A1");
matrix M = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
M[i, j] = 4 * rnd.NextDouble() - 1.5;
M[j, i] = M[i, j];
}
}
matrix M2 = M.copy();
M.print("We choose an arbitrary matrix M = ");
vector D = new vector(n);
matrix V = new matrix(n, n);
int sweeps = Jacobi_diagonalization.cyclic_sweep(M, V, D);
matrix temp2 = V.T * M2 * V;
Console.WriteLine($"Total number of sweeps done is = {sweeps}");
temp2.print("V.T*M*V = ");
D.print("Eigenvalues = ");
}
static void Main()
{
WriteLine("My last 2 digits in my student number is 22 so 22 mod 22 = 0, which is then the exam project I have done.");
WriteLine("In this project I use the Lanczos algorithm to produce the Matrixes V and T where A = V T V^{T}");
WriteLine("The real symmetric matrix we choose as en example is");
//Define symmetric matrix
var A = new matrix("1 2 3;2 4 5;3 5 8");
A.print("A = ");
WriteLine("We then apply the Lanczos algortihm to get the V and T matrices.");
//Call V and T matrix from Lanczos algorithm
matrix T = Lanczos.execute(A, 3, 0);
matrix V = Lanczos.execute(A, 3, 1);
//Print them
T.print("T = ");
V.print("V = ");
//Recreate A to check that T and V are correct
matrix A2 = V * T * V.transpose();
A2.print("Were A should be V T V^T. We calculate this to check, V T V^T = ");
WriteLine("Which gives exactley A again");
}
static void probC()
{
Write("Problem C:\n");
var rand = new System.Random();
int n = 5 + rand.Next(3);
matrix A = makeRandomMatrix(n, n);
A.print("Make a random square matrix of random size: A = ");
qr_givens decomp_givens = new qr_givens(A);
qr_gs decomp_gs = new qr_gs(A);
Write("\nDo the Givens decomposition and store the restult in the matrix G, which cotains the elements of the component R in the upper triangular part, and the angles for the Givens rotations in the relevant sub-diagonal entries. Compare with the R matrix found from the Gramm-Schmitt method:");
decomp_givens.G.print("Givens G: ");
decomp_gs.R.print("Gram-Schmitt R: ");
Write("The upper triangular parts are the same.\n");
vector b = makeRandomVector(n);
b.print("\nMake a random vector b with the same size as A: b = ");
Write("\nCompare using the Givens rotations to apply Q^(T) to b, and doing the explicit matrix multiplication with the Q matrix found by the Gramm-Schmitt method:\n");
decomp_givens.applyQT(b).print("Givens: Q^(T)*b = ");
(decomp_gs.Q.transpose() * b).print("GS: Q^(T)*b = ");
vector x = decomp_givens.solve(b);
x.print("\nSolution to A*x=b using Givens: ");
(A * x - b).print("\nCheck solution satisfies A*x = b: A*x-b = ");
Write("\nFind the inverse of A, and check that A^(-1)*A is the identity matrix:\n");
decomp_givens.inverse().print("A^(-1)");
(decomp_givens.inverse() * A).print("A^(-1)*A = ");
}
static void Main()
{
var ma = new matrix("1 2;5 6");
ma.print();
(ma * ma.T).print();
}
// Comparison of time and rotations needed to calculate the lowest
// eigenvalue for the cyclic and single value method.
static void probB5()
{
Write("\n\nProblem B.5:\n");
int n = 7;
matrix A1 = matrixHelp.makeRandSymMatrix(n);
matrix A2 = A1.copy();
Write("Finding the largest eigenvalue of the matrix:\n");
A1.print();
matrix V1 = new matrix(n, n);
matrix V2 = new matrix(n, n);
vector e1 = new vector(n);
vector e2 = new vector(n);
int rot1 = jacobi.cyclic(A1, e1, V1);
int rot2 = jacobi.values(A2, e2, 1, V2, true);
Write("\n");
Write($"Largest found by cyclic jacobi: {e1[n-1]}\n");
Write($"Largest found by single row jacobi: {e2[0]}\n");
Write("\nThe single row method was changed by changing the calculation of the rotation angle theta to: 0.5*Atan2(-Apq, App-Aqq)\n");
}
public static int Main()
{
// int n=5;
int m = 4;
var rand = new Random(1);
// Assignment B
matrix B = new matrix(m, m);
for (int i = 0; i < m; i++)
{
for (int j = 0; j < m; j++)
{
B[i, j] = 10 * (rand.NextDouble());
}
}
WriteLine("Assignment B: ");
B.print($"Square matrix A with size {m}x{m}, to do assigment B:");
var qr_B = new qrdecompositionGS(B);
var B_inv = qr_B.inverse();
B_inv.print("The invers matrix of A, B=:");
var bbinv = B * B_inv;
bbinv.print("A*B=I");
return(0);
}
public static void Main()
{
Write("Part 1:\n");
Write("Demonstrating that the Lanczos iterations produces the right V and T matrix:\n");
int n = 6;
matrix A = matrixHelp.makeRandSymMatrix(6);
Write($"Here for n = {6}, m = n\n");
string fmt = ":F4";
A.print("The randomly generated symmetric matrix: A = ", fmt);
int m = n;
matrix V = new matrix(n, m);
matrix T = new matrix(m, m);
lanczos.iterations(A, V, T);
T.print("The found tridiagonal matrix T:", fmt);
V.print("The found orthogonal matrix V:", fmt);
Write("\nIf V is orthogonal, then V*V^T = I:\n");
(V * V.T).print("V*V^T = ", fmt);
Write("\nAs n = m here, so V*T*V^T should give A:\n");
(V * T * V.T - A).print("V*T*V^T - A = ", fmt);
}
static void Main()
{
int n = 6;
WriteLine($"Generate symmetric {n}x{n} matrix A");
matrix A = randomMatrix(n, n);
restoreUpperTriang(A);
A.print("A: ");
matrix V = new matrix(n, n);
vector e = new vector(n);
int sweeps;
// Cyclic
sweeps = Jacobi.cyclic(A, V, e);
WriteLine($"Cyclic Jacobi done in {sweeps} sweeps");
e.print("Cyclic eigenvalues: ");
// Lowest k
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
int k = n / 3;
vector v = Jacobi.lowestK(A, V, e, k);
v.print($"Lowest {k} values:");
A.print($"Test that {k} rows are cleared");
// Highest k
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
k = n / 3;
v = Jacobi.highestK(A, V, e, k);
v.print($"Highest {k} values:");
A.print($"Test that {k} rows are cleared");
restoreUpperTriang(A);
V = new matrix(n, n);
e = new vector(n);
int rots = Jacobi.classic(A, V, e);
e.print($"Classical done! rotations: {rots}");
A.print($"Test that all rows are cleared");
}
class main { public static void Main()
{
int n = RandomInt(4, 8);
matrix A = new matrix(n, n);
for (int i = 0; i < n; i = i + 1)
{
for (int j = i; j < n; j = j + 1)
{
if (i == j)
{
A[i, j] = RandomDouble(5, 20);
}
else
{
A[i, j] = RandomDouble(5, 20); A[j, i] = A[i, j];
}
}
}
A.print("Randomly generated symmetric square matrix A=\n");
matrix A_2 = A.copy(); matrix V = new matrix(n, n); vector e = new vector(n);
// Diagonalization
int c = 1; // Number of lowest eigenvalues
int sweeps = JacobiDiag(A, e, V, c);
A.print("\nD=\n");
matrix A1 = V.T * A_2 * V;
V.print("\nV=\n");
A1.print("\nV(^T)AV=\n");
//A.print("\nD=\n");
if (Round(A[0, 0]) == Round(e[0]))
{
Out.WriteLine($"\nElimination of first row succesfull: Completed in {sweeps} iterations");
}
else
{
Out.WriteLine("\nElimination failed");
}
WriteLine("A[p,p] and A[q,q] are analytically given and defined through recursion of their entries in the pre transformation matrix.");
WriteLine("Assumning an even weighing from s^2 and c^2 tt is evident that A'[p,p]<A'[q,q] by their definitions within the algorithm.");
WriteLine("\nAll comparisons are shown in the figure for a static matrix, along with the computation time of the sweep method for randomly generated matrices.\n");
WriteLine("\n To obtain the largest eigenvalue first instead of the lowest we use the analytical expression given for A'[p,p]=c^2A[p,p]-2scA[p,q]+s^2A[q,q] again. Since A'[p,p] for our given theta is minimal and is a composition of sin and cos functions and thus, have the periodicity of cos and sin functions we must conclude that a phaseshift of pi/2 will maximize the expression to yield the largest eigenvalue first and still diagonalize the matrix.");
}
static void Main()
{
var rand = new Random();
// We can pull out new random numbers between 0 and 1 with rand.NextDouble() and
// stuff it into a matrix
int n = 4;
matrix A = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
A[i, j] = 2 - 4 * rand.NextDouble();
}
}
WriteLine("Set up a random matrix, and try to find the inverse matrix via QR" +
" decomposition.\n");
WriteLine("Printing random matrix A:");
A.print();
// Factorize the matrix into two matrices Q and R, and calculate the inverse matrix B
var qrGS = new qrDecompositionGS(A);
matrix B = qrGS.inverse();
matrix Q = qrGS.Q;
matrix R = qrGS.R;
WriteLine("\nPrinting matrix R:");
R.print();
WriteLine("\nPrinting matrix Q:");
Q.print();
WriteLine("\nPrinting matrix B (inverse to A):");
B.print();
// Let's check if B is actually the inverse by calculating A*B which then should
// give the identity matrix
matrix AB = A * B;
WriteLine("\nPrinting matrix A*B:");
AB.print();
matrix I = new matrix(n, n);
I.set_identity();
bool approx = AB.approx(I);
if (approx == true)
{
WriteLine("A*B is approximately equal to the identity matrix.");
}
else
{
WriteLine("A*B is not approximately equal to the identity matrix.");
}
}
static void Main(string[] args)
{
int n = 5;
matrix D = GenRandSymMatrix(n, n);
matrix A = D.copy();
A.print($"Symmetric matrix A:");
(vector e, matrix V, int n1) = cyclic(D);
for (int i = 0; i < n; i++)
{
for (int j = i + 1; j < n; j++)
{
D[j, i] = D[i, j]; // complete the matrix
}
}
matrix VTAV = V.transpose() * A * V;
e.print($"Eigenvalues e:");
V.print($"Eigenvectors V:");
A.print($"D matrix: ");
VTAV.print($"Matrix V^TAV:");
if (D.approx(VTAV))
{
WriteLine("V^TAV = D, test passed.");
}
else
{
WriteLine("V^TAV != D, test failed.");
}
matrix B = A.copy();
(vector e2, _, int n2) = lowest_eigens(A, 3);
(vector e4, _, int n4) = highest_eigens(A, 3);
e.print($"\nEigenvalues from cyclic method found after {n1, 3} rotations: ");
e2.print($"3 lowest eigenvalues from row-by-row method found after {n2, 3} rotations: ");
e4.print($"3 highest eigenvalues from row-by-row method found after {n4, 3} rotations: ");
(vector e3, _, int n3) = classic(B);
e3.print($"Eigenvalues from the classic method found after {n3, 3} rotations: ");
WriteLine();
}
static void Main()
{
// Testing that the jacobiAlgorithm works:
int n = 5;
vector e = new vector(n);
matrix V = new matrix(n, n);
matrix A = generateRandommatrix(n, n);
matrix B = A.copy();
int sweeps = jacobi.jacobi_cyclic(A, e, V);
WriteLine("---------Problem A1----------");
WriteLine("\nTesting that the Jacobi algorithm works as intended:");
A.print("\nRandom square matrix A:");
WriteLine("\nEigenvalue decomposition of A=V*D*V^T:");
(V.transpose() * B * V).print("\nD = V^T*A*V = ");
e.print("\nListed eigenvalues:\t");
WriteLine($"\nNumber of sweeps required to diagonalize A: {sweeps}");
// Particle in a box
n = 50;
matrix H = Hamiltonian.boxHamilton(n);
B = H.copy();
V = new matrix(n, n);
e = new vector(n);
sweeps = jacobi.jacobi_cyclic(H, e, V);
WriteLine("\n---------Problem A2----------");
WriteLine("\nNow solving for a particle in a box.");
WriteLine($"\nThe dimensions of the soon to be diagonalized Hamiltonian is {n}x{n}");
WriteLine("\nComparison of the calculated eigenvalues and the exact:\n");
WriteLine("n Calculated Exact:");
for (int k = 0; k < n / 3; k++)
{
double exact = PI * PI * (k + 1) * (k + 1);
double calculated = e[k];
WriteLine($"{k} {calculated:f4}\t {exact:f4}");
}
// plotting time
StreamWriter solutions = new StreamWriter("SolutionsA.txt");
for (int i = 0; i < n; i++)
{
solutions.Write($"{i*(1.0/n)}");
for (int k = 0; k < 5; k++)
{
solutions.Write($" {V[i,k]}");
}
solutions.Write("\n");
}
solutions.Close();
}
}//randomMatrix
static void randomTest(int n)
{
matrix a = randomSymmetricMatrix(n);
a.print("A = ");
diagJacobi test = new diagJacobi(a, eigVec: true);
Write($"Number of sweeps: {test.sweeps}\n");
(((test.v).transpose()) * a * (test.v)).print("V^T*A*V = ");
(test.l).print("eigenvalues = ");
} //randomTest
}//randomMatrix
static void randomTest(int n)
{
Write("==== Test of diagonalizatrion on random matrix A: ====\n");
matrix a = randomSymmetricMatrix(n);
a.print("A = ");
diagJacobi test = new diagJacobi(a, eigVec: true);
Write("Performing Jacobi diagonalization:\n");
Write($"Number of sweeps: {test.sweeps}\n");
(((test.v).transpose()) * a * (test.v)).print("V^T * A * V = ");
(test.l).print("eigenvalues = ");
}//randomTest
public static void Main()
{
Write("==== B ====\n");
Write("Testing inverse by Gram-Schmidt in random matrix A:\n");
matrix a = rndMat.randomMatrix(5, 5);
a.print("A = ");
var qr = new qrDecompositionGS(a);
Write("Inverse is found to be:\n");
(qr.inverse()).print("A^(-1) = ");
Write("Checking that this is indeed the desired inverse matrix by calculating I=A^-1 * A:\n");
(qr.inverse() * a).print("A^(-1)*A = ");
} //Main
public static void Main()
{
Write("==== A.1 ====\n");
Write("Testing on random tall matrix A:\n");
matrix a = rndMat.randomMatrix(5, 3);
a.print("A =");
qrDecompositionGS qr = new qrDecompositionGS(a);
Write("Q and R matrices found from QR-factorization:\n");
(qr.q).print("Q = ");;
(qr.r).print("R = ");;
Write("Testing that Q is orthogonal:\n");
(qr.q.transpose() * qr.q).print("Q^T*Q = ");
Write("Calculating the difference between A and QR, which should be 0:\n");
(a - qr.q * qr.r).print("A-QR = ");
Write("\n==== A.2 ====\n");
Write("Testing solver on random matrix A:\n");
a = rndMat.randomMatrix(5, 5);
a.print("A = ");
qr = new qrDecompositionGS(a);
Write("and random vector b:\n");
vector b = rndMat.randomVector(5);
b.print("b = ");
Write("Solution to Ax=b is found to be:\n");
vector x = qr.solve(b);
x.print("x = ");
Write("Checking that x is a solution by calculating 0=Ax-b:\n");
vector diff = a * x - b;
diff.print("A*x-b = ");
} //Main
static void Main()
{
var rnd = new Random(1);
int n = 4;
var A = new matrix(n, n);
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
A[i, j] = 2 * (rnd.NextDouble() - 0.5);
}
}
A.print("Matrix inverse.\nRandom square matrix A:");
var qra = new gramschmidt(A);
var B = qra.inverse();
B.print("\nThe inverse of A, A^-1:");
var Id = new matrix(n, n);
for (int i = 0; i < n; i++)
{
Id[i, i] = 1;
}
var C = A * B;
C.print("\ncheck: A*A^-1:");
if (Id.approx(A * B))
{
Write("\nA*A^-1 = Id, test passed\n");
}
else
{
Write("\nA*A^-1 != Id, test failed\n");
}
var D = B * A;
D.print("\ncheck: A^-1*A=");
if (Id.approx(B * A))
{
Write("\nA^-1*A = I, test passed\n");
}
else
{
Write("\nA^-1*A != I, test failed\n");
}
}
public static void Main(string[] args)
{
int n = 5, max_print = 8;
if (args.Length > 0)
{
n = int.Parse(args[0]);
}
Console.WriteLine("n={0}", n);
var rnd = new Random(1);
var a = new matrix(n, n);
vector e = new vector(n);
matrix v = new matrix(n, n);
if (n > max_print)
{
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
a[i, j] = 2 * (rnd.NextDouble() - 0.59);
}
}
int r = jacobi.cyclic(a, e, v);
Console.WriteLine("sweeps={0} e[n-1]={1}", r, e[n - 1]);
return;
}
else
{
for (int i = 0; i < n; i++)
{
for (int j = i; j < n; j++)
{
a[i, j] = rnd.NextDouble(); a[j, i] = a[i, j];
}
}
Console.WriteLine("Eigenvalue Decomposition A=V*D*V^T");
a.print("Random matrix A:");
matrix b = a.copy();
int sweeps = jacobi.cyclic(a, e, v);
System.Console.WriteLine("Number of sweeps={0}", sweeps);
var p = (v.T * b * v);
p.print("V^T*A*V (should be diagonal):");
e.print("Eigenvalues (should equal the diagonal elements above):\n");
}
}
