10.2. Lanczos solver¶

In Cytnx, an eigenvalue problem can be solved for a custom linear operator defined using the LinOp class by using the Lanczos iterative solver.

For this, you can pass a LinOp or any of its child classes to linalg.Lanczos_ER:

Lanczos_ER(Hop, k=1, is_V=true, maxiter=10000, CvgCrit=1.0e-14, is_row=false, Tin=Tensor(), max_krydim=4)¶

Perform Lanczos for Hermitian/symmetric matrices or linear function.

Parameters:

Hop (LinOp) – the Linear Operator defined by LinOp class or its inheritance.
k (uint64) – the number of lowest k eigenvalues
is_V (bool) – if set to true, the eigenvectors will be returned
maxiter (uint64) – the maximum number of iteration steps for each k
CvgCrit (double) – the convergence criterion of the energy
is_row (bool) – whether the returned eigenvectors should be in row-major form
Tin (Tensor) – the initial vector, should be a Tensor with rank-1
max_krydim (uint32) – the maximum Krylov subspace dimension for each iteration

Returns:

[eigvals (Tensor), eigvecs (Tensor)(option)]

Return type:

vector<Tensor> (C++ API)/list of Tensor(python API)

For example, we consider a simple example where we wrap a (4x4) matrix inside a custom operator. We can easily generalize the matvec to be any custom sparse structure.

In Python:

class MyOp(cytnx.LinOp):
  def __init__(self):
    cytnx.LinOp.__init__(self,"mv",4)

  def matvec(self,v):
    A = cytnx.arange(16).reshape(4,4)
    A += A.permute(1,0)
    return cytnx.linalg.Dot(A,v)


op = MyOp()

v0 = cytnx.arange(4) # trial state
ev = cytnx.linalg.Lanczos(op, k=1, Tin=v0, method="ER", max_krydim = 2)

print(ev[0]) #eigenval
print(ev[1]) #eigenvec

In C++:

using namespace cytnx;
class MyOp : public LinOp {
 public:
  MyOp() : LinOp("mv", 6) {
    // Create Hermitian Matrix
    A = arange(36).reshape(6, 6);
    A += A.permute(1, 0);
  }

 private:
  Tensor A;
  Tensor matvec(const Tensor &v) override { return linalg::Dot(A, v); }
};

auto op = MyOp();

auto v0 = arange(6);  // trial state
auto ev = linalg::Lanczos_ER(&op, 1, true, 10000, 1.0e-14, false, v0);

cout << ev[0] << endl;  // eigenval
cout << ev[1] << endl;  // eigenvec

Output >>

Total elem: 1
type  : Double (Float64)
cytnx device: CPU
Shape : (1)
[-7.41657e+00 ]




Total elem: 4
type  : Double (Float64)
cytnx device: CPU
Shape : (4)
[-7.94119e-01 -3.69644e-01 5.48299e-02 4.79304e-01 ]

Note

The ER stand for explicitly restarted. The Lanczos method used is based on this reference, which can reproduce degeneracies correctly.
The Lanczos solver only works for symmetric/Hermitian operators.
In cases where the operator dimensionality is small, the max_krydim parameter can be reduced for correct convergence.