# Overview¶

This is document for the sample program which uses K $$\omega$$ library in the ISSP Math Library; this program computes the Green’s function with $$K\omega$$. For the details of K $$\omega$$ library, See “$$K\omega$$ manual” in this package.

## Calculation in this program¶

This program compute the Green’s function

\begin{align} G_{i}(z) = \langle i | (z-{\hat H})^{-1}| i \rangle \equiv {\boldsymbol \varphi}_i^{*} \cdot (z-{\hat H})^{-1} {\boldsymbol \varphi}_i, \end{align}

where $$| i \rangle$$ is a wavefunction, $${\cal H}$$ is the Hamiltonian, and $$z$$ is a complex frequency.

$${\cal H}$$ in the above equation is obtained by either the

following two ways:

• Input $${\cal H}$$ as a file with the MatrixMarket format

• Construct $${\cal H}$$ as a Hamiltonian of the Heisenberg model in this program.

In the computation of the Green’s function, we use either the following two method according to the type of $${\hat H}$$ (a real- or a complex- number).

• $${\hat H}$$ of real numbers : Shifted Bi-Conjugate Gradient(BiCG) method

• $${\hat H}$$ of complex numbers : Shifted Conjugate Orthogonal Conjugate Gradient(COCG) method