From 6c25922942135be03d9ef1a7b8fa72e4db3b2784 Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 3 Oct 2024 15:27:57 +0000 Subject: [PATCH] build based on d8ffd5f --- dev/.documenter-siteinfo.json | 2 +- dev/api/index.html | 36 +- dev/index.html | 2 +- dev/usage/{28f7b5d4.svg => 86e74be1.svg} | 528 +++++++++++------------ dev/usage/index.html | 6 +- 5 files changed, 287 insertions(+), 287 deletions(-) rename dev/usage/{28f7b5d4.svg => 86e74be1.svg} (53%) diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 4990680..ac46fbb 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-03T15:20:47","documenter_version":"1.7.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.10.5","generation_timestamp":"2024-10-03T15:27:55","documenter_version":"1.7.0"}} \ No newline at end of file diff --git a/dev/api/index.html b/dev/api/index.html index 9317416..c3b092b 100644 --- a/dev/api/index.html +++ b/dev/api/index.html @@ -1,51 +1,51 @@ -API · SmoQyKPMCore.jl
GitHub

API

Note that for many methods there are two verions, one that relies on taking an instance of the KPMExpansion type as an argument, and a lower level one that does not.

SmoQyKPMCore.KPMExpansion — Type
mutable struct KPMExpansion{T<:AbstractFloat, Tfft<:FFTW.r2rFFTWPlan}

A type to represent the Chebyshev polynomial expansion used in the KPM algorithm.

Fields

  • M::Int: Order of the Chebyshev polynomial expansion.
  • bounds::NTuple{2,T}: Bounds on eigenspectrum in the KPM algorithm.
  • buf::Vector{T}: The first M elements of this vector of the Chebyshev expansion coefficients.
  • r2rplan::Tfft: Plan for performing DCT to efficiently calculate the Chebyshev expansion coefficients via Chebyshev-Gauss quadrature.
source
SmoQyKPMCore.KPMExpansion — Method
KPMExpansion(func::Function, bounds, M::Int, N::Int = 2*M)

Initialize an instance of the KPMExpansion type to approximate the univariate function func, called as func(x), with a order M Chebyshev polynomial expansion on the interval bounds[1] < x bounds[2]. Here, N ≥ M is the number of points at which func is evaluated on that specified interval, which are then used to calculate the expansion coeffiencents via Chebyshev-Gauss quadrature.

source
SmoQyKPMCore.kpm_update! — Function
kpm_update!(
+API · SmoQyKPMCore.jl
GitHub
API
Note that for many methods there are two verions, one that relies on taking an instance of the KPMExpansion type as an argument, and a lower level one that does not.
SmoQyKPMCore.KPMExpansion — Type
mutable struct KPMExpansion{T<:AbstractFloat, Tfft<:FFTW.r2rFFTWPlan}
A type to represent the Chebyshev polynomial expansion used in the KPM algorithm.
Fields
  • M::Int: Order of the Chebyshev polynomial expansion.
  • bounds::NTuple{2,T}: Bounds on eigenspectrum in the KPM algorithm.
  • buf::Vector{T}: The first M elements of this vector of the Chebyshev expansion coefficients.
  • r2rplan::Tfft: Plan for performing DCT to efficiently calculate the Chebyshev expansion coefficients via Chebyshev-Gauss quadrature.
source
SmoQyKPMCore.KPMExpansion — Method
KPMExpansion(func::Function, bounds, M::Int, N::Int = 2*M)
Initialize an instance of the KPMExpansion type to approximate the univariate function func, called as func(x), with a order M Chebyshev polynomial expansion on the interval bounds[1] < x bounds[2]. Here, N ≥ M is the number of points at which func is evaluated on that specified interval, which are then used to calculate the expansion coeffiencents via Chebyshev-Gauss quadrature.
source
SmoQyKPMCore.kpm_update! — Function
kpm_update!(
     kpm_expansion::KPMExpansion{T}, func::Function, bounds, M::Int, N::Int = 2*M
-) where {T<:AbstractFloat}
In-place update an instance of KPMExpansion to reflect new values for eigenspectrum bounds, expansion order M, and number of points at which the expanded function is evaluated when computing the expansion coefficients. This includes recomputing the expansion coefficients.
source
SmoQyKPMCore.kpm_update_bounds! — Function
kpm_update_bounds!(
+) where {T<:AbstractFloat}
In-place update an instance of KPMExpansion to reflect new values for eigenspectrum bounds, expansion order M, and number of points at which the expanded function is evaluated when computing the expansion coefficients. This includes recomputing the expansion coefficients.
source
SmoQyKPMCore.kpm_update_bounds! — Function
kpm_update_bounds!(
     kpm_expansion::KPMExpansion{T}, func::Function, bounds
-) where {T<:AbstractFloat}
In-place update an instance of KPMExpansion to reflect new values for eigenspectrum bounds, recomputing the expansion coefficients.
source
SmoQyKPMCore.kpm_update_order! — Function
kpm_update_order!(
     kpm_expansion::KPMExpansion, func::Function, M::Int, N::Int = 2*M
-)
In-place update the expansion order M for an instance of KPMExpansion, recomputing the expansion coefficients. It is also possible to udpate the number of point N the function func is evaluated at to calculate the expansion coefficients.
source
SmoQyKPMCore.kpm_coefs — Function
kpm_coefs(func::Function, bounds, M::Int, N::Int = 2*M)
Calculate and return the Chebyshev expansion coefficients. Refer to kpm_coefs! for more information.
source
SmoQyKPMCore.kpm_coefs! — Function
kpm_coefs!(
+)
In-place update the expansion order M for an instance of KPMExpansion, recomputing the expansion coefficients. It is also possible to udpate the number of point N the function func is evaluated at to calculate the expansion coefficients.
source
SmoQyKPMCore.kpm_coefs — Function
kpm_coefs(func::Function, bounds, M::Int, N::Int = 2*M)
Calculate and return the Chebyshev expansion coefficients. Refer to kpm_coefs! for more information.
source
SmoQyKPMCore.kpm_coefs! — Function
kpm_coefs!(
     coefs::AbstractVector{T}, func::Function, bounds,
     buf::Vector{T} = zeros(T, 2*length(coefs)),
     r2rplan = FFTW.plan_r2r!(zeros(T, 2*length(coefs)), FFTW.REDFT10)
-) where {T<:AbstractFloat}
Calculate and record the Chebyshev polynomial expansion coefficients to order M in the vector ceofs for the function func on the interval (bounds[1], bounds[2]). Let length(buf) be the number of evenly spaced points on the interval for which func is evaluated when performing Chebyshev-Gauss quadrature to compute the Chebyshev polynomial expansion coefficients.
source
SmoQyKPMCore.kpm_moments — Function
kpm_moments(
+) where {T<:AbstractFloat}
Calculate and record the Chebyshev polynomial expansion coefficients to order M in the vector ceofs for the function func on the interval (bounds[1], bounds[2]). Let length(buf) be the number of evenly spaced points on the interval for which func is evaluated when performing Chebyshev-Gauss quadrature to compute the Chebyshev polynomial expansion coefficients.
source
SmoQyKPMCore.kpm_moments — Function
kpm_moments(
     M::Int, A, bounds, R::T, tmp = zeros(eltype(R), size(R)..., 3)
-) where {T<:AbstractVecOrMat}
Calculate and return the first $M$ moments
\[\mu_m = \langle R | T_m(A) | R \rangle\]
if $R$ is a vector, and if $R$ is a matrix then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle R_n | T_m(A) | R_n \rangle,\]
where $| R_n \rangle$ is the n'th column of $R$.
source
kpm_moments(
+) where {T<:AbstractVecOrMat}
Calculate and return the first $M$ moments
\[\mu_m = \langle R | T_m(A) | R \rangle\]
if $R$ is a vector, and if $R$ is a matrix then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle R_n | T_m(A) | R_n \rangle,\]
where $| R_n \rangle$ is the n'th column of $R$.
source
kpm_moments(
     M::Int, A, bounds, U::T, V::T, tmp = zeros(eltype(V), size(V)..., 3)
-) where {T<:AbstractVecOrMat}
Calculate and return the first $M$ moments
\[\mu_m = \langle U | T_m(A) | V \rangle\]
if $U$ and $V$ are vector, and if they are matrices then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle U_n | T_m(A) | V_n \rangle,\]
where $| U_n \rangle$ and  $| V_n \rangle$ are are the n'th columns of each matrix.
source
SmoQyKPMCore.kpm_moments! — Function
kpm_moments!(
+) where {T<:AbstractVecOrMat}
Calculate and return the first $M$ moments
\[\mu_m = \langle U | T_m(A) | V \rangle\]
if $U$ and $V$ are vector, and if they are matrices then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle U_n | T_m(A) | V_n \rangle,\]
where $| U_n \rangle$ and  $| V_n \rangle$ are are the n'th columns of each matrix.
source
SmoQyKPMCore.kpm_moments! — Function
kpm_moments!(
     μ::AbstractVector, A, bounds, R::T, tmp = zeros(eltype(R), size(R)..., 3)
-) where {T<:AbstractVecOrMat}
Calculate the moments
\[\mu_m = \langle R | T_m(A) | R \rangle\]
if $R$ is a vector, and if $R$ is a matrix then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle R_n | T_m(A) | R_n \rangle,\]
where $| R_n \rangle$ is the n'th column of $R$.
source
kpm_moments!(
+) where {T<:AbstractVecOrMat}
Calculate the moments
\[\mu_m = \langle R | T_m(A) | R \rangle\]
if $R$ is a vector, and if $R$ is a matrix then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle R_n | T_m(A) | R_n \rangle,\]
where $| R_n \rangle$ is the n'th column of $R$.
source
kpm_moments!(
     μ::AbstractVector, A, bounds, U::T, V::T, tmp = zeros(eltype(V), size(V)..., 3)
-) where {T<:AbstractVecOrMat}
Calculate the moments
\[\mu_m = \langle U | T_m(A) | V \rangle\]
if $U$ and $V$ are vector, and if they are matrices then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle U_n | T_m(A) | V_n \rangle,\]
where $| U_n \rangle$ and  $| V_n \rangle$ are are the n'th columns of each matrix.
source
SmoQyKPMCore.kpm_density — Function
kpm_density(
+) where {T<:AbstractVecOrMat}
Calculate the moments
\[\mu_m = \langle U | T_m(A) | V \rangle\]
if $U$ and $V$ are vector, and if they are matrices then
\[\mu_m = \frac{1}{N} \sum_{n=1}^N \langle U_n | T_m(A) | V_n \rangle,\]
where $| U_n \rangle$ and  $| V_n \rangle$ are are the n'th columns of each matrix.
source
SmoQyKPMCore.kpm_density — Function
kpm_density(
     N::Int, μ::T, W
-) where {T<:AbstractVector}
Given the KPM moments $\mu$, evaluate the corresponding spectral density $\rho(\epsilon)$ at $N$ energy points $\epsilon$, where $W$ is the spectral bandwidth or the bounds of the spectrum. This function returns both $\rho(\epsilon)$ and $\epsilon$.
source
SmoQyKPMCore.kpm_density! — Function
kpm_density!(
+) where {T<:AbstractVector}
Given the KPM moments $\mu$, evaluate the corresponding spectral density $\rho(\epsilon)$ at $N$ energy points $\epsilon$, where $W$ is the spectral bandwidth or the bounds of the spectrum. This function returns both $\rho(\epsilon)$ and $\epsilon$.
source
SmoQyKPMCore.kpm_density! — Function
kpm_density!(
     ρ::AbstractVector{T},
     ϵ::AbstractVector{T},
     μ::AbstractVector{T},
     bounds,
     r2rplan = FFTW.plan_r2r!(ρ, FFTW.REDFT01)
-) where {T<:AbstractFloat}
Given the KPM moments $\mu$, calculate the spectral density $\rho(\epsilon)$ and corresponding energies $\epsilon$ where it is evaluated. Here bounds is bounds the eigenspecturm, such that the bandwidth is given by W = bounds[2] - bounds[1].
source
kpm_density!(
+) where {T<:AbstractFloat}
Given the KPM moments $\mu$, calculate the spectral density $\rho(\epsilon)$ and corresponding energies $\epsilon$ where it is evaluated. Here bounds is bounds the eigenspecturm, such that the bandwidth is given by W = bounds[2] - bounds[1].
source
kpm_density!(
     ρ::AbstractVector{T},
     ϵ::AbstractVector{T},
     μ::AbstractVector{T},
     W::T,
     r2rplan = FFTW.plan_r2r!(ρ, FFTW.REDFT01),
-) where {T<:AbstractFloat}
Given the KPM moments $\mu$, calculate the spectral density $\rho(\epsilon)$ and corresponding energies $\epsilon$ where it is evaluated. Here $W$ is the spectral bandwidth.
source
SmoQyKPMCore.kpm_mul — Function
kpm_mul(
+) where {T<:AbstractFloat}
Given the KPM moments $\mu$, calculate the spectral density $\rho(\epsilon)$ and corresponding energies $\epsilon$ where it is evaluated. Here $W$ is the spectral bandwidth.
source
SmoQyKPMCore.kpm_mul — Function
kpm_mul(
     A, coefs::AbstractVector, bounds, v::T, tmp = zeros(eltype(v), size(v)..., 3)
-) where {T<:AbstractVecOrMat}
Evaluate and return the vector $v^\prime = F(A) \cdot v$ where $F(A)$ is represented by the Chebyshev expansion. For more information refer to kpm_mul!.
source
kpm_mul(A, kpm_expansion::KPMExpansion, v::T) where {T<:AbstractVector}
Evaluate and return the vector $v^\prime = F(A) \cdot v$ where $F(A)$ is represented by the Chebyshev expansion. For more information refer to kpm_mul!.
source
SmoQyKPMCore.kpm_mul! — Function
kpm_mul!(
+) where {T<:AbstractVecOrMat}
Evaluate and return the vector $v^\prime = F(A) \cdot v$ where $F(A)$ is represented by the Chebyshev expansion. For more information refer to kpm_mul!.
source
kpm_mul(A, kpm_expansion::KPMExpansion, v::T) where {T<:AbstractVector}
Evaluate and return the vector $v^\prime = F(A) \cdot v$ where $F(A)$ is represented by the Chebyshev expansion. For more information refer to kpm_mul!.
source
SmoQyKPMCore.kpm_mul! — Function
kpm_mul!(
     v′::T, A, coefs::AbstractVector, bounds, v::T,
     tmp = zeros(eltype(v), size(v)..., 3)
-) where {T<:AbstractVecOrMat}
Evaluates $v^\prime = F(A) \cdot v$, writing the result to v′, where $F(A)$ is represented by the Chebyshev expansion. Here A is either a function that can be called as A(u,v) to evaluate $u = A\cdot v$, modifying u in-place, or is a type for which the operation mul!(u, A, v) is defined. The vector coefs contains Chebyshev expansion coefficients to approximate $F(A)$, where the eigenspectrum of $A$ is contained in the interval (bounds[1], bounds[2]) specified by the bounds argument. The vector v is vector getting multiplied by the Chebyshev expansion for $F(A)$. Lastly, tmp is an array used to avoid dynamic memory allocations.
source
kpm_mul!(
+) where {T<:AbstractVecOrMat}
Evaluates $v^\prime = F(A) \cdot v$, writing the result to v′, where $F(A)$ is represented by the Chebyshev expansion. Here A is either a function that can be called as A(u,v) to evaluate $u = A\cdot v$, modifying u in-place, or is a type for which the operation mul!(u, A, v) is defined. The vector coefs contains Chebyshev expansion coefficients to approximate $F(A)$, where the eigenspectrum of $A$ is contained in the interval (bounds[1], bounds[2]) specified by the bounds argument. The vector v is vector getting multiplied by the Chebyshev expansion for $F(A)$. Lastly, tmp is an array used to avoid dynamic memory allocations.
source
kpm_mul!(
     v′::T, A, kpm_expansion::KPMExpansion, v::T,
     tmp = zeros(eltype(v), size(v)..., 3)
-) where {T<:AbstractVecOrMat}
Evaluates $v^\prime = F(A) \cdot v$, writing the result to v′, where $F(A)$ is represented by the Chebyshev expansion. Here A is either a function that can be called as A(u,v) to evaluate $u = A\cdot v$, modifying u in-place, or is a type for which the operation mul!(u, A, v) is defined. Lastly, the array tmp is used to avoid dynamic memory allocations.
source
SmoQyKPMCore.kpm_eval — Function
kpm_eval(x::AbstractFloat, coefs, bounds)
Evaluate $F(x)$ where $x$ is real number in the interval bounds[1] < x < bound[2], and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(A::AbstractMatrix, coefs, bounds)
Evaluate and return the matrix $F(A),$ where $A$ is an operator with strictly real eigenvalues that fall in the interval (bounds[1], bounds[2]) specified by the bounds argument, and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(x::T, kpm_expansion::KPMExpansion{T}) where {T<:AbstractFloat}
Evaluate $F(x)$ where $x$ is real number in the interval bounds[1] < x < bound[2], and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(A::AbstractMatrix{T}, kpm_expansion::KPMExpansion{T}) where {T<:AbstractFloat}
Evaluate and return the matrix $F(A),$ where $A$ is an operator with strictly real eigenvalues and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
SmoQyKPMCore.kpm_eval! — Function
kpm_eval!(
+) where {T<:AbstractVecOrMat}
Evaluates $v^\prime = F(A) \cdot v$, writing the result to v′, where $F(A)$ is represented by the Chebyshev expansion. Here A is either a function that can be called as A(u,v) to evaluate $u = A\cdot v$, modifying u in-place, or is a type for which the operation mul!(u, A, v) is defined. Lastly, the array tmp is used to avoid dynamic memory allocations.
source
SmoQyKPMCore.kpm_eval — Function
kpm_eval(x::AbstractFloat, coefs, bounds)
Evaluate $F(x)$ where $x$ is real number in the interval bounds[1] < x < bound[2], and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(A::AbstractMatrix, coefs, bounds)
Evaluate and return the matrix $F(A),$ where $A$ is an operator with strictly real eigenvalues that fall in the interval (bounds[1], bounds[2]) specified by the bounds argument, and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(x::T, kpm_expansion::KPMExpansion{T}) where {T<:AbstractFloat}
Evaluate $F(x)$ where $x$ is real number in the interval bounds[1] < x < bound[2], and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
kpm_eval(A::AbstractMatrix{T}, kpm_expansion::KPMExpansion{T}) where {T<:AbstractFloat}
Evaluate and return the matrix $F(A),$ where $A$ is an operator with strictly real eigenvalues and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs.
source
SmoQyKPMCore.kpm_eval! — Function
kpm_eval!(
     F::AbstractMatrix, A, coefs::AbstractVector, bounds,
     tmp = zeros(eltype(F), size(F)..., 3)
-)
Evaluate and write the matrix $F(A)$ to F, where $A$ is an operator with strictly real eigenvalues that fall in the interval (bounds[1], bounds[2]) specified by the bounds argument, and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs. Lastly, tmp is used to avoid dynamic memory allocations.
source
kpm_eval!(
+)
Evaluate and write the matrix $F(A)$ to F, where $A$ is an operator with strictly real eigenvalues that fall in the interval (bounds[1], bounds[2]) specified by the bounds argument, and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs. Lastly, tmp is used to avoid dynamic memory allocations.
source
kpm_eval!(
     F::AbstractMatrix, A, kpm_expansion::KPMExpansion,
     tmp = zeros(eltype(F), size(F)..., 3)
-)
Evaluate and write the matrix $F(A)$ to F, where $A$ is an operator with strictly real eigenvalues and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs. Lastly, the array tmp is used to avoid dynamic memory allocations.
source
SmoQyKPMCore.apply_jackson_kernel! — Function
apply_jackson_kernel!(coefs)
Modify the Chebyshev expansion coefficients by applying the Jackson kernel to them.
source
apply_jackson_kernel!(kpm_expansion::KPMExpansion)
Modify the Chebyshev expansion coefficients by applying the Jackson kernel to them.
source
SmoQyKPMCore.lanczos — Function
lanczos(niters, v, A, S = I, rng = Random.default_rng())
Use niters Lanczos iterations to find a truncated tridiagonal representation of $A\cdot S$, up to similarity transformation. Here, $A$ is any Hermitian matrix, while $S$ is both Hermitian and positive definite. Traditional Lanczos uses the identity matrix, $S = I$. The extension to non-identity matrices $S$ is as follows: Each matrix-vector product $A\cdot v$ becomes $(A S) \cdot v$, and each vector inner product $w^\dagger \cdot v$ becomes $w^\dagger \cdot S \cdot v$. The implementation below follows Wikipedia, and is the most stable of the four variants considered by Paige [1]. This implementation introduces additional vector storage so that each Lanczos iteration requires only one matrix-vector multiplication for $A$ and $S$, respectively.
This function returns a SymTridiagonal matrix. Note that the eigmin and eigmax routines have specialized implementations for a SymTridiagonal matrix type.
Similar generalizations of Lanczos have been considered in [2] and [3].
1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),
https://doi.org/10.1093%2Fimamat%2F10.3.373.
2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),
https://doi.org/10.1090/s0025-5718-1982-0669648-0
3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),
https://doi.org/10.1016/j.commatsci.2011.02.021.
source
SmoQyKPMCore.lanczos! — Function
lanczos!(
+)
Evaluate and write the matrix $F(A)$ to F, where $A$ is an operator with strictly real eigenvalues and the function $F(\bullet)$ is represented by a Chebyshev expansion with coefficients given by the vector coefs. Lastly, the array tmp is used to avoid dynamic memory allocations.
source
SmoQyKPMCore.apply_jackson_kernel! — Function
apply_jackson_kernel!(coefs)
Modify the Chebyshev expansion coefficients by applying the Jackson kernel to them.
source
apply_jackson_kernel!(kpm_expansion::KPMExpansion)
Modify the Chebyshev expansion coefficients by applying the Jackson kernel to them.
source
SmoQyKPMCore.lanczos — Function
lanczos(niters, v, A, S = I, rng = Random.default_rng())
Use niters Lanczos iterations to find a truncated tridiagonal representation of $A\cdot S$, up to similarity transformation. Here, $A$ is any Hermitian matrix, while $S$ is both Hermitian and positive definite. Traditional Lanczos uses the identity matrix, $S = I$. The extension to non-identity matrices $S$ is as follows: Each matrix-vector product $A\cdot v$ becomes $(A S) \cdot v$, and each vector inner product $w^\dagger \cdot v$ becomes $w^\dagger \cdot S \cdot v$. The implementation below follows Wikipedia, and is the most stable of the four variants considered by Paige [1]. This implementation introduces additional vector storage so that each Lanczos iteration requires only one matrix-vector multiplication for $A$ and $S$, respectively.
This function returns a SymTridiagonal matrix. Note that the eigmin and eigmax routines have specialized implementations for a SymTridiagonal matrix type.
Similar generalizations of Lanczos have been considered in [2] and [3].
1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),
https://doi.org/10.1093%2Fimamat%2F10.3.373.
2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),
https://doi.org/10.1090/s0025-5718-1982-0669648-0
3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),
https://doi.org/10.1016/j.commatsci.2011.02.021.
source
SmoQyKPMCore.lanczos! — Function
lanczos!(
     αs::AbstractVector, βs::AbstractVector, v::AbstractVector,
     A, S = I,
     tmp::AbstractMatrix = zeros(eltype(v), length(v), 5);
@@ -57,4 +57,4 @@
     A, S = I,
     tmp::AbstractArray = zeros(eltype(v), size(v)..., 5);
     rng = Random.default_rng()
-)
Use Lanczos iterations to find a truncated tridiagonal representation of $A\cdot S$, up to similarity transformation. Here, $A$ is any Hermitian matrix, while $S$ is both Hermitian and positive definite. Traditional Lanczos uses the identity matrix, $S = I$. The extension to non-identity matrices $S$ is as follows: Each matrix-vector product $A\cdot v$ becomes $(A S) \cdot v$, and each vector inner product $w^\dagger \cdot v$ becomes $w^\dagger \cdot S \cdot v$. The implementation below follows Wikipedia, and is the most stable of the four variants considered by Paige [1]. This implementation introduces additional vector storage so that each Lanczos iteration requires only one matrix-vector multiplication for $A$ and $S$, respectively.
The number of Lanczos iterations performed equals niters = length(αs), and niters - 1 == length(βs). This function returns a SymTridiagonal matrix based on the contents of the vectors αs and βs. Note that the eigmin and eigmax routines have specialized implementations for a SymTridiagonal matrix type.
Note that if αs, βs and v are all matrices, then each column of v is treated as a seperate vector, and a vector of SymTridiagonal of length size(v,2) will be returned.
Similar generalizations of Lanczos have been considered in [2] and [3].
1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),
https://doi.org/10.1093%2Fimamat%2F10.3.373.
2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),
https://doi.org/10.1090/s0025-5718-1982-0669648-0
3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),
https://doi.org/10.1016/j.commatsci.2011.02.021.
source

+)

Use Lanczos iterations to find a truncated tridiagonal representation of $A\cdot S$, up to similarity transformation. Here, $A$ is any Hermitian matrix, while $S$ is both Hermitian and positive definite. Traditional Lanczos uses the identity matrix, $S = I$. The extension to non-identity matrices $S$ is as follows: Each matrix-vector product $A\cdot v$ becomes $(A S) \cdot v$, and each vector inner product $w^\dagger \cdot v$ becomes $w^\dagger \cdot S \cdot v$. The implementation below follows Wikipedia, and is the most stable of the four variants considered by Paige [1]. This implementation introduces additional vector storage so that each Lanczos iteration requires only one matrix-vector multiplication for $A$ and $S$, respectively.

The number of Lanczos iterations performed equals niters = length(αs), and niters - 1 == length(βs). This function returns a SymTridiagonal matrix based on the contents of the vectors αs and βs. Note that the eigmin and eigmax routines have specialized implementations for a SymTridiagonal matrix type.

Note that if αs, βs and v are all matrices, then each column of v is treated as a seperate vector, and a vector of SymTridiagonal of length size(v,2) will be returned.

Similar generalizations of Lanczos have been considered in [2] and [3].

1. C. C. Paige, IMA J. Appl. Math., 373-381 (1972),

https://doi.org/10.1093%2Fimamat%2F10.3.373.

2. H. A. van der Vorst, Math. Comp. 39, 559-561 (1982),

https://doi.org/10.1090/s0025-5718-1982-0669648-0

3. M. Grüning, A. Marini, X. Gonze, Comput. Mater. Sci. 50, 2148-2156 (2011),

https://doi.org/10.1016/j.commatsci.2011.02.021.

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diff --git a/dev/index.html b/dev/index.html index 0b652d5..8430f34 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · SmoQyKPMCore.jl
GitHub

SmoQyKPMCore

Documentation for SmoQyKPMCore package. The SmoQyKPMCore package implements and exports an optimized, low-level implementation of the Kernel Polynomial Method (KPM) algorithm for approximating functions of operators with strictly real, bounded eigenvalues via a Chebyshev polynomial expansion.

Funding

The development of this code was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC0022311.

+Home · SmoQyKPMCore.jl
GitHub

SmoQyKPMCore

Documentation for SmoQyKPMCore package. The SmoQyKPMCore package implements and exports an optimized, low-level implementation of the Kernel Polynomial Method (KPM) algorithm for approximating functions of operators with strictly real, bounded eigenvalues via a Chebyshev polynomial expansion.

Funding

The development of this code was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, under Award Number DE-SC0022311.

diff --git a/dev/usage/28f7b5d4.svg b/dev/usage/86e74be1.svg similarity index 53% rename from dev/usage/28f7b5d4.svg rename to dev/usage/86e74be1.svg index b1dbe3f..dd35e62 100644 --- a/dev/usage/28f7b5d4.svg +++ b/dev/usage/86e74be1.svg @@ -2,278 +2,278 @@ - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - - + + - + - - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - + - + - - + + - - + + - + - + - + - + - - + + - + - + - + - - + + - + - + - + - + - - + + - + - - + + - + - + - + - + - + - + - + - + - + - + - + - + - - + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - - + + - + @@ -285,261 +285,261 @@ - - - - + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + - - + + @@ -548,59 +548,59 @@ - - - - + + + + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + - + diff --git a/dev/usage/index.html b/dev/usage/index.html index c461439..6151663 100644 --- a/dev/usage/index.html +++ b/dev/usage/index.html @@ -89,7 +89,7 @@ kpm_mul!(ρv_kpm, H, kpm_expansion, v, vtmp) # Check how good the approximation is. -println("Vector Error = ", norm(ρv_kpm - ρv) / norm(ρv) )
Vector Error = 0.029305358354320143

Let us now provide the KPM approximation with better bounds on the eigenspectrum and also increase the order the of the expansion and see how the result improves. We will use the kpm_update! function to update the KPMExpansion in-place.

# Define eigenspectrum bounds.
+println("Vector Error = ", norm(ρv_kpm - ρv) / norm(ρv) )
Vector Error = 0.015238739082477082

Let us now provide the KPM approximation with better bounds on the eigenspectrum and also increase the order the of the expansion and see how the result improves. We will use the kpm_update! function to update the KPMExpansion in-place.

# Define eigenspectrum bounds.
 bounds = (-2.5t, 2.5t)
 
 # Define order of Chebyshev expansion used in KPM approximation.
@@ -109,7 +109,7 @@
 
 # Check how good the approximation is.
 println("Vector Error = ", norm(ρv_kpm - ρv) / norm(ρv) )
Matrix Error = 1.031960839804449e-14
-Vector Error = 1.2157363237100364e-14

Density of States Approximation

Next let us demonstrate how we may approximate the density of states $\mathcal{N}(\epsilon)$ for a 1D chain tight-binding model. The first step a very larger Hamiltonian matrix $H,$ which we represent as a spare matrix.

# Size of 1D chain considered
+Vector Error = 1.127015333309712e-14

Density of States Approximation

Next let us demonstrate how we may approximate the density of states $\mathcal{N}(\epsilon)$ for a 1D chain tight-binding model. The first step a very larger Hamiltonian matrix $H,$ which we represent as a spare matrix.

# Size of 1D chain considered
 L = 10_000
 
 # Construct sparse Hamiltonian matrix.
@@ -206,4 +206,4 @@
     ax, halign = :center, valign = :top, labelsize = 30
 )
 
-fig
Example block output +figExample block output