# Why lanczos algorithm is useful for finding the ground state energy?

1. Aug 18, 2009

### wdlang

i am now reading some materials on lanczos algorithm, one of the ten most important numerical algorithms in the 20th century

my puzzle is, why it is useful for finding out the ground state energy?

i can not see anything special about the ground state energy in the algorithm

2. Aug 21, 2009

### olgranpappy

it's useful for obtaining the largest eigenvalue of a giant matrix, e.g. So... consider the matrix

e^{-H}

which state has the largest eigenvalue?

3. Sep 5, 2009

### wsttiger

Lanczos is good for finding the lowest eigenvalues of a matrix ... hence, the ground state.

4. Sep 22, 2009

### maverick_starstrider

It can be shown that it converges quickest to extremal eigenvalues (in our case the minimal). But that isn't really WHY it's used. The reason why it is used is not really obvious unless you have some experience with other exact diagonalization methods. The Lanczos method preserves the sparcity of your Hamiltonian and thus greatly reduces the storage space required (which is very important when one actually wants to do one of these calculations). It also can be performed by keeping only 2-3 eigenvectors, which is also great for space. Finally, it lends itself to parellization.

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