diff --git a/dev/.documenter-siteinfo.json b/dev/.documenter-siteinfo.json index 2cfad4a..c991276 100644 --- a/dev/.documenter-siteinfo.json +++ b/dev/.documenter-siteinfo.json @@ -1 +1 @@ -{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-06T17:06:34","documenter_version":"1.8.0"}} \ No newline at end of file +{"documenter":{"julia_version":"1.11.2","generation_timestamp":"2024-12-12T07:57:51","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/dev/guide/index.html b/dev/guide/index.html index 3f23eb0..04ccfa3 100644 --- a/dev/guide/index.html +++ b/dev/guide/index.html @@ -207,4 +207,4 @@ fit!(Gl, siω, Giω) end return basis, Σl -end +end diff --git a/dev/index.html b/dev/index.html index 1d3024e..1ff7b63 100644 --- a/dev/index.html +++ b/dev/index.html @@ -1,2 +1,2 @@ -Home · SparseIR.jl
GitHub
+Home · SparseIR.jl
GitHub
diff --git a/dev/private/index.html b/dev/private/index.html index b3421ea..b30cbb3 100644 --- a/dev/private/index.html +++ b/dev/private/index.html @@ -1,23 +1,23 @@ -Private · SparseIR.jl
GitHub

Private names index

These are not considered API and therefore not covered by any semver promises.

Core.Int — Method

Get prefactor n for the Matsubara frequency ω = n*π/β

source
Core.Union — Method
(polyFT::PiecewiseLegendreFT)(ω)

Obtain Fourier transform of polynomial for given MatsubaraFreq ω.

source
SparseIR.AbstractAugmentation — Type
AbstractAugmentation

Scalar function in imaginary time/frequency.

This represents a single function in imaginary time and frequency, together with some auxiliary methods that make it suitable for augmenting a basis.

See also: AugmentedBasis

source
SparseIR.AbstractBasis — Type
AbstractBasis

Abstract base class for bases on the imaginary-time axis.

Let basis be an abstract basis. Then we can expand a two-point propagator G(τ), where τ is imaginary time, into a set of basis functions:

G(τ) == sum(basis.u[l](τ) * g[l] for l in 1:length(basis)) + ϵ(τ),

where basis.u[l] is the l-th basis function, g[l] is the associated expansion coefficient and ϵ(τ) is an error term. Similarly, the Fourier transform Ĝ(n), where n is now a Matsubara frequency, can be expanded as follows:

Ĝ(n) == sum(basis.uhat[l](n) * g[l] for l in 1:length(basis)) + ϵ(n),

where basis.uhat[l] is now the Fourier transform of the basis function.

source
SparseIR.AbstractKernel — Type
(kernel::AbstractKernel)(x, y[, x₊, x₋])

Evaluate kernel at point (x, y).

The parameters x₊ and x₋, if given, shall contain the values of x - xₘᵢₙ and xₘₐₓ - x, respectively. This is useful if either difference is to be formed and cancellation expected.

source
SparseIR.AbstractKernel — Type
AbstractKernel

Integral kernel K(x, y).

Abstract base type for an integral kernel, i.e. a AbstractFloat binary function $K(x, y)$ used in a Fredhold integral equation of the first kind:

\[ u(x) = ∫ K(x, y) v(y) dy\]

where $x ∈ [x_\mathrm{min}, x_\mathrm{max}]$ and $y ∈ [y_\mathrm{min}, y_\mathrm{max}]$. For its SVE to exist, the kernel must be square-integrable, for its singular values to decay exponentially, it must be smooth.

In general, the kernel is applied to a scaled spectral function $ρ'(y)$ as:

\[ ∫ K(x, y) ρ'(y) dy,\]

where $ρ'(y) = w(y) ρ(y)$.

source
SparseIR.AbstractSampling — Type
AbstractSampling

Abstract type for sparse sampling.

Encodes the "basis transformation" of a propagator from the truncated IR basis coefficients G_ir[l] to time/frequency sampled on sparse points G(x[i]) together with its inverse, a least squares fit:

     ________________                   ___________________
+Private · SparseIR.jl
GitHub
Private names index
These are not considered API and therefore not covered by any semver promises.
Core.Int — Method
Get prefactor n for the Matsubara frequency ω = n*π/β
source
Core.Union — Method
(polyFT::PiecewiseLegendreFT)(ω)
Obtain Fourier transform of polynomial for given MatsubaraFreq ω.
source
SparseIR.AbstractAugmentation — Type
AbstractAugmentation
Scalar function in imaginary time/frequency.
This represents a single function in imaginary time and frequency, together with some auxiliary methods that make it suitable for augmenting a basis.
See also: AugmentedBasis
source
SparseIR.AbstractBasis — Type
AbstractBasis
Abstract base class for bases on the imaginary-time axis.
Let basis be an abstract basis. Then we can expand a two-point propagator  G(τ), where τ is imaginary time, into a set of basis functions:
G(τ) == sum(basis.u[l](τ) * g[l] for l in 1:length(basis)) + ϵ(τ),
where basis.u[l] is the l-th basis function, g[l] is the associated expansion coefficient and ϵ(τ) is an error term. Similarly, the Fourier transform Ĝ(n), where n is now a Matsubara frequency, can be expanded as follows:
Ĝ(n) == sum(basis.uhat[l](n) * g[l] for l in 1:length(basis)) + ϵ(n),
where basis.uhat[l] is now the Fourier transform of the basis function.
source
SparseIR.AbstractKernel — Type
(kernel::AbstractKernel)(x, y[, x₊, x₋])
Evaluate kernel at point (x, y).
The parameters x₊ and x₋, if given, shall contain the values of x - xₘᵢₙ and xₘₐₓ - x, respectively. This is useful if either difference is to be formed and cancellation expected.
source
SparseIR.AbstractKernel — Type
AbstractKernel
Integral kernel K(x, y).
Abstract base type for an integral kernel, i.e. a AbstractFloat binary function $K(x, y)$ used in a Fredhold integral equation of the first kind:
\[    u(x) = ∫ K(x, y) v(y) dy\]
where $x ∈ [x_\mathrm{min}, x_\mathrm{max}]$ and $y ∈ [y_\mathrm{min}, y_\mathrm{max}]$. For its SVE to exist, the kernel must be square-integrable, for its singular values to decay exponentially, it must be smooth.
In general, the kernel is applied to a scaled spectral function $ρ'(y)$ as:
\[    ∫ K(x, y) ρ'(y) dy,\]
where $ρ'(y) = w(y) ρ(y)$.
source
SparseIR.AbstractSampling — Type
AbstractSampling
Abstract type for sparse sampling.
Encodes the "basis transformation" of a propagator from the truncated IR basis coefficients G_ir[l] to time/frequency sampled on sparse points G(x[i]) together with its inverse, a least squares fit:
     ________________                   ___________________
     |                |    evaluate     |                   |
     |     Basis      |---------------->|     Value on      |
     |  coefficients  |<----------------|  sampling points  |
-    |________________|      fit        |___________________|
source
SparseIR.CentrosymmSVE — Type
CentrosymmSVE <: AbstractSVE
SVE of centrosymmetric kernel in block-diagonal (even/odd) basis.
For a centrosymmetric kernel K, i.e., a kernel satisfying: K(x, y) == K(-x, -y), one can make the following ansatz for the singular functions:
u[l](x) = ured[l](x) + sign[l] * ured[l](-x)
+    |________________|      fit        |___________________|
source
SparseIR.CentrosymmSVE — Type
CentrosymmSVE <: AbstractSVE
SVE of centrosymmetric kernel in block-diagonal (even/odd) basis.
For a centrosymmetric kernel K, i.e., a kernel satisfying: K(x, y) == K(-x, -y), one can make the following ansatz for the singular functions:
u[l](x) = ured[l](x) + sign[l] * ured[l](-x)
 v[l](y) = vred[l](y) + sign[l] * ured[l](-y)
where sign[l] is either +1 or -1. This means that the singular value expansion can be block-diagonalized into an even and an odd part by (anti-)symmetrizing the kernel:
K_even = K(x, y) + K(x, -y)
-K_odd  = K(x, y) - K(x, -y)
The lth basis function, restricted to the positive interval, is then the singular function of one of these kernels. If the kernel generates a Chebyshev system [1], then even and odd basis functions alternate.
[1]: A. Karlin, Total Positivity (1968).
source
SparseIR.LogisticKernelOdd — Type
LogisticKernelOdd <: AbstractReducedKernel
Fermionic analytical continuation kernel, odd.
In dimensionless variables $x = 2τ/β - 1$, $y = βω/Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = -\frac{\sinh(Λ x y / 2)}{\cosh(Λ y / 2)}\]
source
SparseIR.PiecewiseLegendreFT — Type
PiecewiseLegendreFT <: Function
Fourier transform of a piecewise Legendre polynomial.
For a given frequency index n, the Fourier transform of the Legendre function is defined as:
    p̂(n) == ∫ dx exp(im * π * n * x / (xmax - xmin)) p(x)
The polynomial is continued either periodically (freq=:even), in which case n must be even, or antiperiodically (freq=:odd), in which case n must be odd.
source
SparseIR.PiecewiseLegendrePoly — Type
PiecewiseLegendrePoly <: Function
Piecewise Legendre polynomial.
Models a function on the interval $[xmin, xmax]$ as a set of segments on the intervals $S[i] = [a[i], a[i+1]]$, where on each interval the function is expanded in scaled Legendre polynomials.
source
SparseIR.PowerModel — Type
PowerModel
Model from a high-frequency series expansion::
A(iω) == sum(A[n] / (iω)^(n+1) for n in 1:N)
where $iω == i * π/2 * wn$ is a reduced imaginary frequency, i.e., $wn$ is an odd/even number for fermionic/bosonic frequencies.
source
SparseIR.ReducedKernel — Type
ReducedKernel
Restriction of centrosymmetric kernel to positive interval.
For a kernel $K$ on $[-1, 1] × [-1, 1]$ that is centrosymmetric, i.e. $K(x, y) = K(-x, -y)$, it is straight-forward to show that the left/right singular vectors can be chosen as either odd or even functions.
Consequentially, they are singular functions of a reduced kernel $K_\mathrm{red}$ on $[0, 1] × [0, 1]$ that is given as either:
\[    K_\mathrm{red}(x, y) = K(x, y) \pm K(x, -y)\]
This kernel is what this type represents. The full singular functions can be reconstructed by (anti-)symmetrically continuing them to the negative axis.
source
SparseIR.RegularizedBoseKernelOdd — Type
RegularizedBoseKernelOdd <: AbstractReducedKernel
Bosonic analytical continuation kernel, odd.
In dimensionless variables $x = 2 τ / β - 1$, $y = β ω / Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = -y \frac{\sinh(Λ x y / 2)}{\sinh(Λ y / 2)}\]
source
SparseIR.Rule — Type
Rule{T<:AbstractFloat}
Quadrature rule.
Approximation of an integral over [a, b] by a sum over discrete points x with weights w:
\[    ∫ f(x) ω(x) dx ≈ ∑_i f(x_i) w_i\]
where we generally have superexponential convergence for smooth $f(x)$ in  the number of quadrature points.
source
SparseIR.SVEResult — Method
SVEResult(kernel::AbstractKernel;
+K_odd  = K(x, y) - K(x, -y)
The lth basis function, restricted to the positive interval, is then the singular function of one of these kernels. If the kernel generates a Chebyshev system [1], then even and odd basis functions alternate.
[1]: A. Karlin, Total Positivity (1968).
source
SparseIR.LogisticKernelOdd — Type
LogisticKernelOdd <: AbstractReducedKernel
Fermionic analytical continuation kernel, odd.
In dimensionless variables $x = 2τ/β - 1$, $y = βω/Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = -\frac{\sinh(Λ x y / 2)}{\cosh(Λ y / 2)}\]
source
SparseIR.PiecewiseLegendreFT — Type
PiecewiseLegendreFT <: Function
Fourier transform of a piecewise Legendre polynomial.
For a given frequency index n, the Fourier transform of the Legendre function is defined as:
    p̂(n) == ∫ dx exp(im * π * n * x / (xmax - xmin)) p(x)
The polynomial is continued either periodically (freq=:even), in which case n must be even, or antiperiodically (freq=:odd), in which case n must be odd.
source
SparseIR.PiecewiseLegendrePoly — Type
PiecewiseLegendrePoly <: Function
Piecewise Legendre polynomial.
Models a function on the interval $[xmin, xmax]$ as a set of segments on the intervals $S[i] = [a[i], a[i+1]]$, where on each interval the function is expanded in scaled Legendre polynomials.
source
SparseIR.PowerModel — Type
PowerModel
Model from a high-frequency series expansion::
A(iω) == sum(A[n] / (iω)^(n+1) for n in 1:N)
where $iω == i * π/2 * wn$ is a reduced imaginary frequency, i.e., $wn$ is an odd/even number for fermionic/bosonic frequencies.
source
SparseIR.ReducedKernel — Type
ReducedKernel
Restriction of centrosymmetric kernel to positive interval.
For a kernel $K$ on $[-1, 1] × [-1, 1]$ that is centrosymmetric, i.e. $K(x, y) = K(-x, -y)$, it is straight-forward to show that the left/right singular vectors can be chosen as either odd or even functions.
Consequentially, they are singular functions of a reduced kernel $K_\mathrm{red}$ on $[0, 1] × [0, 1]$ that is given as either:
\[    K_\mathrm{red}(x, y) = K(x, y) \pm K(x, -y)\]
This kernel is what this type represents. The full singular functions can be reconstructed by (anti-)symmetrically continuing them to the negative axis.
source
SparseIR.RegularizedBoseKernelOdd — Type
RegularizedBoseKernelOdd <: AbstractReducedKernel
Bosonic analytical continuation kernel, odd.
In dimensionless variables $x = 2 τ / β - 1$, $y = β ω / Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = -y \frac{\sinh(Λ x y / 2)}{\sinh(Λ y / 2)}\]
source
SparseIR.Rule — Type
Rule{T<:AbstractFloat}
Quadrature rule.
Approximation of an integral over [a, b] by a sum over discrete points x with weights w:
\[    ∫ f(x) ω(x) dx ≈ ∑_i f(x_i) w_i\]
where we generally have superexponential convergence for smooth $f(x)$ in  the number of quadrature points.
source
SparseIR.SVEResult — Method
SVEResult(kernel::AbstractKernel;
     Twork=nothing, ε=nothing, lmax=typemax(Int),
     n_gauss=nothing, svd_strat=:auto,
     sve_strat=iscentrosymmetric(kernel) ? CentrosymmSVE : SamplingSVE
-)
Perform truncated singular value expansion of a kernel.
Perform a truncated singular value expansion (SVE) of an integral kernel kernel : [xmin, xmax] x [ymin, ymax] -> ℝ:
kernel(x, y) == sum(s[l] * u[l](x) * v[l](y) for l in (1, 2, 3, ...)),
where s[l] are the singular values, which are ordered in non-increasing fashion, u[l](x) are the left singular functions, which form an orthonormal system on [xmin, xmax], and v[l](y) are the right singular functions, which form an orthonormal system on [ymin, ymax].
The SVE is mapped onto the singular value decomposition (SVD) of a matrix by expanding the kernel in piecewise Legendre polynomials (by default by using a collocation).
Arguments
  • K::AbstractKernel: Integral kernel to take SVE from.
  • ε::Real: Accuracy target for the basis: attempt to have singular values down to a relative magnitude of ε, and have each singular value and singular vector be accurate to ε. A Twork with a machine epsilon of ε^2 or lower is required to satisfy this. Defaults to 2.2e-16 if xprec is available, and 1.5e-8 otherwise.
  • cutoff::Real: Relative cutoff for the singular values. A Twork with machine epsilon of cutoff is required to satisfy this. Defaults to a small multiple of the machine epsilon.
    Note that cutoff and ε serve distinct purposes. cutoff reprsents the accuracy to which the kernel is reproduced, whereas ε is the accuracy to which the singular values and vectors are guaranteed.
  • lmax::Integer: Maximum basis size. If given, only at most the lmax most significant singular values and associated singular functions are returned.
  • `n_gauss (int): Order of Legendre polynomials. Defaults to kernel hinted value.
  • Twork: Working data type. Defaults to a data type with machine epsilon of at  mostε^2and at mostcutoff`, or otherwise most accurate data type available.
  • sve_strat::AbstractSVE: SVE to SVD translation strategy. Defaults to SamplingSVE, optionally wrapped inside of a CentrosymmSVE if the kernel is centrosymmetric.
  • svd_strat ('fast' or 'default' or 'accurate'): SVD solver. Defaults to fast (ID/RRQR) based solution when accuracy goals are moderate, and more accurate Jacobi-based algorithm otherwise.
Returns: An SVEResult containing the truncated singular value expansion.
source
SparseIR.SamplingSVE — Type
SamplingSVE <: AbstractSVE
SVE to SVD translation by sampling technique [1].
Maps the singular value expansion (SVE) of a kernel kernel onto the singular value decomposition of a matrix A. This is achieved by choosing two sets of Gauss quadrature rules: (x, wx) and (y, wy) and approximating the integrals in the SVE equations by finite sums. This implies that the singular values of the SVE are well-approximated by the singular values of the following matrix:
A[i, j] = √(wx[i]) * K(x[i], y[j]) * √(wy[j])
and the values of the singular functions at the Gauss sampling points can be reconstructed from the singular vectors u and v as follows:
u[l,i] ≈ √(wx[i]) u[l](x[i])
-v[l,j] ≈ √(wy[j]) u[l](y[j])
[1] P. Hansen, Discrete Inverse Problems, Ch. 3.1
source
SparseIR.accuracy — Function
accuracy(basis::AbstractBasis)
Accuracy of the basis.
Upper bound to the relative error of reprensenting a propagator with the given number of basis functions (number between 0 and 1).
source
SparseIR.basis — Method
basis(sampling::AbstractSampling)
Return the IR basis associated with sampling.
source
SparseIR.canonicalize! — Method
canonicalize!(u, v)
Canonicalize basis.
Each SVD (u[l], v[l]) pair is unique only up to a global phase, which may differ from implementation to implementation and also platform. We fix that gauge by demanding u[l](1) > 0. This ensures a diffeomorphic connection to the Legendre polynomials as Λ → 0.
source
SparseIR.conv_radius — Function
conv_radius(kernel)
Convergence radius of the Matsubara basis asymptotic model.
For improved relative numerical accuracy, the IR basis functions on the Matsubara axis uhat(basis, n) can be evaluated from an asymptotic expression for abs(n) > conv_radius. If isinf(conv_radius), then the asymptotics are unused (the default).
source
SparseIR.default_matsubara_sampling_points — Function
default_matsubara_sampling_points(basis::AbstractBasis; positive_only=false)
Default sampling points on the imaginary frequency axis.
Arguments
  • positive_only::Bool: Only return non-negative frequencies. This is useful if the object to be fitted is symmetric in Matsubura frequency, ĝ(ω) == conj(ĝ(-ω)), or, equivalently, real in imaginary time.
source
SparseIR.eval_matrix — Function
eval_matrix(T, basis, x)
Return evaluation matrix from coefficients to sampling points. T <: AbstractSampling.
source
SparseIR.find_extrema — Method
find_extrema(polyFT::PiecewiseLegendreFT; part=nothing, grid=DEFAULT_GRID)
Obtain extrema of Fourier-transformed polynomial.
source
SparseIR.finite_temp_bases — Function
finite_temp_bases(β::Real, ωmax::Real, ε=nothing;
-                  kernel=LogisticKernel(β * ωmax), sve_result=SVEResult(kernel; ε))
Construct FiniteTempBasis objects for fermion and bosons using the same LogisticKernel instance.
source
SparseIR.from_IR — Function
from_IR(dlr::DiscreteLehmannRepresentation, gl::AbstractArray, dims=1)
From IR to DLR. gl`: Expansion coefficients in IR.
source
SparseIR.get_symmetrized — Method
get_symmetrized(kernel, sign)
Construct a symmetrized version of kernel, i.e. kernel(x, y) + sign * kernel(x, -y).
Beware!
By default, this returns a simple wrapper over the current instance which naively performs the sum. You may want to override this to avoid cancellation.
source
SparseIR.get_tnl — Method
get_tnl(l, w)
Fourier integral of the l-th Legendre polynomial::
Tₗ(ω) == ∫ dx exp(iωx) Pₗ(x)
source
SparseIR.giw — Method
giw(polyFT, wn)
Return model Green's function for reduced frequencies
source
SparseIR.iscentrosymmetric — Function
iscentrosymmetric(kernel)
Return true if kernel(x, y) == kernel(-x, -y) for all values of x and y in range. This allows the kernel to be block-diagonalized, speeding up the singular value expansion by a factor of 4. Defaults to false.
source
SparseIR.matop! — Method
matop!(buffer, mat, arr::AbstractArray, op, dim)
Apply the operator op to the matrix mat and to the array arr along the first dimension (dim=1) or the last dimension (dim=N).
source
SparseIR.matop_along_dim! — Method
matop_along_dim!(buffer, mat, arr::AbstractArray, dim::Integer, op)
Apply the operator op to the matrix mat and to the array arr along the dimension dim, writing the result to buffer.
source
SparseIR.movedim — Method
movedim(arr::AbstractArray, src => dst)
Move arr's dimension at src to dst while keeping the order of the remaining dimensions unchanged.
source
SparseIR.nsvals — Method
nsvals(hints)
Upper bound for number of singular values.
Upper bound on the number of singular values above the given threshold, i.e. where s[l] ≥ ε * first(s).
source
SparseIR.phase_stable — Method
phase_stable(poly, wn)
Phase factor for the piecewise Legendre to Matsubara transform.
Compute the following phase factor in a stable way:
exp.(iπ/2 * wn * cumsum(Δx(poly)))
source
SparseIR.piecewise — Method
piecewise(rule, edges)
Piecewise quadrature with the same quadrature rule, but scaled.
source
SparseIR.rescale — Method
rescale(basis::FiniteTempBasis, new_β)
Return a basis for different temperature.
Uses the same kernel with the same $ε$, but a different temperature. Note that this implies a different UV cutoff $ωmax$, since $Λ == β * ωmax$ stays constant.
source
SparseIR.segments_x — Method
segments_x(sve_hints::AbstractSVEHints[, T])
Segments for piecewise polynomials on the $x$ axis.
List of segments on the $x$ axis for the associated piecewise polynomial. Should reflect the approximate position of roots of a high-order singular function in $x$.
source
SparseIR.segments_y — Method
segments_y(sve_hints::AbstractSVEHints[, T])
Segments for piecewise polynomials on the $y$ axis.
List of segments on the $y$ axis for the associated piecewise polynomial. Should reflect the approximate position of roots of a high-order singular function in $y$.
source
SparseIR.shift_xmid — Method
shift_xmid(knots, Δx)
Return midpoint relative to the nearest integer plus a shift.
Return the midpoints xmid of the segments, as pair (diff, shift), where shift is in (0, 1, -1) and diff is a float such that xmid == shift + diff to floating point accuracy.
source
SparseIR.significance — Function
significance(basis::AbstractBasis)
Return vector σ, where 0 ≤ σ[i] ≤ 1 is the significance level of the i-th basis function. If ϵ is the desired accuracy to which to represent a propagator, then any basis function where σ[i] < ϵ can be neglected.
For the IR basis, we simply have that σ[i] = s[i] / first(s).
source
SparseIR.split — Method
split(poly, x)
Split segment.
Find segment of poly's domain that covers x.
source
SparseIR.statistics — Method
statistics(basis::AbstractBasis)
Quantum statistic (Statistics instance, Fermionic() or Bosonic()).
source
SparseIR.sve_hints — Function
sve_hints(kernel, ε)
Provide discretisation hints for the SVE routines.
Advises the SVE routines of discretisation parameters suitable in tranforming the (infinite) SVE into an (finite) SVD problem.
See also AbstractSVEHints.
source
SparseIR.to_IR — Function
to_IR(dlr::DiscreteLehmannRepresentation, g_dlr::AbstractArray, dims=1)
From DLR to IR. g_dlr`: Expansion coefficients in DLR.
source
SparseIR.truncate — Method
truncate(u, s, v; rtol=0.0, lmax=typemax(Int))
Truncate singular value expansion.
Arguments
- `u`, `s`, `v`: Thin singular value expansion
+)
Perform truncated singular value expansion of a kernel.
Perform a truncated singular value expansion (SVE) of an integral kernel kernel : [xmin, xmax] x [ymin, ymax] -> ℝ:
kernel(x, y) == sum(s[l] * u[l](x) * v[l](y) for l in (1, 2, 3, ...)),
where s[l] are the singular values, which are ordered in non-increasing fashion, u[l](x) are the left singular functions, which form an orthonormal system on [xmin, xmax], and v[l](y) are the right singular functions, which form an orthonormal system on [ymin, ymax].
The SVE is mapped onto the singular value decomposition (SVD) of a matrix by expanding the kernel in piecewise Legendre polynomials (by default by using a collocation).
Arguments
  • K::AbstractKernel: Integral kernel to take SVE from.
  • ε::Real: Accuracy target for the basis: attempt to have singular values down to a relative magnitude of ε, and have each singular value and singular vector be accurate to ε. A Twork with a machine epsilon of ε^2 or lower is required to satisfy this. Defaults to 2.2e-16 if xprec is available, and 1.5e-8 otherwise.
  • cutoff::Real: Relative cutoff for the singular values. A Twork with machine epsilon of cutoff is required to satisfy this. Defaults to a small multiple of the machine epsilon.
    Note that cutoff and ε serve distinct purposes. cutoff reprsents the accuracy to which the kernel is reproduced, whereas ε is the accuracy to which the singular values and vectors are guaranteed.
  • lmax::Integer: Maximum basis size. If given, only at most the lmax most significant singular values and associated singular functions are returned.
  • `n_gauss (int): Order of Legendre polynomials. Defaults to kernel hinted value.
  • Twork: Working data type. Defaults to a data type with machine epsilon of at  mostε^2and at mostcutoff`, or otherwise most accurate data type available.
  • sve_strat::AbstractSVE: SVE to SVD translation strategy. Defaults to SamplingSVE, optionally wrapped inside of a CentrosymmSVE if the kernel is centrosymmetric.
  • svd_strat ('fast' or 'default' or 'accurate'): SVD solver. Defaults to fast (ID/RRQR) based solution when accuracy goals are moderate, and more accurate Jacobi-based algorithm otherwise.
Returns: An SVEResult containing the truncated singular value expansion.
source
SparseIR.SamplingSVE — Type
SamplingSVE <: AbstractSVE
SVE to SVD translation by sampling technique [1].
Maps the singular value expansion (SVE) of a kernel kernel onto the singular value decomposition of a matrix A. This is achieved by choosing two sets of Gauss quadrature rules: (x, wx) and (y, wy) and approximating the integrals in the SVE equations by finite sums. This implies that the singular values of the SVE are well-approximated by the singular values of the following matrix:
A[i, j] = √(wx[i]) * K(x[i], y[j]) * √(wy[j])
and the values of the singular functions at the Gauss sampling points can be reconstructed from the singular vectors u and v as follows:
u[l,i] ≈ √(wx[i]) u[l](x[i])
+v[l,j] ≈ √(wy[j]) u[l](y[j])
[1] P. Hansen, Discrete Inverse Problems, Ch. 3.1
source
SparseIR.accuracy — Function
accuracy(basis::AbstractBasis)
Accuracy of the basis.
Upper bound to the relative error of reprensenting a propagator with the given number of basis functions (number between 0 and 1).
source
SparseIR.basis — Method
basis(sampling::AbstractSampling)
Return the IR basis associated with sampling.
source
SparseIR.canonicalize! — Method
canonicalize!(u, v)
Canonicalize basis.
Each SVD (u[l], v[l]) pair is unique only up to a global phase, which may differ from implementation to implementation and also platform. We fix that gauge by demanding u[l](1) > 0. This ensures a diffeomorphic connection to the Legendre polynomials as Λ → 0.
source
SparseIR.conv_radius — Function
conv_radius(kernel)
Convergence radius of the Matsubara basis asymptotic model.
For improved relative numerical accuracy, the IR basis functions on the Matsubara axis uhat(basis, n) can be evaluated from an asymptotic expression for abs(n) > conv_radius. If isinf(conv_radius), then the asymptotics are unused (the default).
source
SparseIR.default_matsubara_sampling_points — Function
default_matsubara_sampling_points(basis::AbstractBasis; positive_only=false)
Default sampling points on the imaginary frequency axis.
Arguments
  • positive_only::Bool: Only return non-negative frequencies. This is useful if the object to be fitted is symmetric in Matsubura frequency, ĝ(ω) == conj(ĝ(-ω)), or, equivalently, real in imaginary time.
source
SparseIR.eval_matrix — Function
eval_matrix(T, basis, x)
Return evaluation matrix from coefficients to sampling points. T <: AbstractSampling.
source
SparseIR.find_extrema — Method
find_extrema(polyFT::PiecewiseLegendreFT; part=nothing, grid=DEFAULT_GRID)
Obtain extrema of Fourier-transformed polynomial.
source
SparseIR.finite_temp_bases — Function
finite_temp_bases(β::Real, ωmax::Real, ε=nothing;
+                  kernel=LogisticKernel(β * ωmax), sve_result=SVEResult(kernel; ε))
Construct FiniteTempBasis objects for fermion and bosons using the same LogisticKernel instance.
source
SparseIR.from_IR — Function
from_IR(dlr::DiscreteLehmannRepresentation, gl::AbstractArray, dims=1)
From IR to DLR. gl`: Expansion coefficients in IR.
source
SparseIR.get_symmetrized — Method
get_symmetrized(kernel, sign)
Construct a symmetrized version of kernel, i.e. kernel(x, y) + sign * kernel(x, -y).
Beware!
By default, this returns a simple wrapper over the current instance which naively performs the sum. You may want to override this to avoid cancellation.
source
SparseIR.get_tnl — Method
get_tnl(l, w)
Fourier integral of the l-th Legendre polynomial::
Tₗ(ω) == ∫ dx exp(iωx) Pₗ(x)
source
SparseIR.giw — Method
giw(polyFT, wn)
Return model Green's function for reduced frequencies
source
SparseIR.iscentrosymmetric — Function
iscentrosymmetric(kernel)
Return true if kernel(x, y) == kernel(-x, -y) for all values of x and y in range. This allows the kernel to be block-diagonalized, speeding up the singular value expansion by a factor of 4. Defaults to false.
source
SparseIR.matop! — Method
matop!(buffer, mat, arr::AbstractArray, op, dim)
Apply the operator op to the matrix mat and to the array arr along the first dimension (dim=1) or the last dimension (dim=N).
source
SparseIR.matop_along_dim! — Method
matop_along_dim!(buffer, mat, arr::AbstractArray, dim::Integer, op)
Apply the operator op to the matrix mat and to the array arr along the dimension dim, writing the result to buffer.
source
SparseIR.movedim — Method
movedim(arr::AbstractArray, src => dst)
Move arr's dimension at src to dst while keeping the order of the remaining dimensions unchanged.
source
SparseIR.nsvals — Method
nsvals(hints)
Upper bound for number of singular values.
Upper bound on the number of singular values above the given threshold, i.e. where s[l] ≥ ε * first(s).
source
SparseIR.phase_stable — Method
phase_stable(poly, wn)
Phase factor for the piecewise Legendre to Matsubara transform.
Compute the following phase factor in a stable way:
exp.(iπ/2 * wn * cumsum(Δx(poly)))
source
SparseIR.piecewise — Method
piecewise(rule, edges)
Piecewise quadrature with the same quadrature rule, but scaled.
source
SparseIR.rescale — Method
rescale(basis::FiniteTempBasis, new_β)
Return a basis for different temperature.
Uses the same kernel with the same $ε$, but a different temperature. Note that this implies a different UV cutoff $ωmax$, since $Λ == β * ωmax$ stays constant.
source
SparseIR.segments_x — Method
segments_x(sve_hints::AbstractSVEHints[, T])
Segments for piecewise polynomials on the $x$ axis.
List of segments on the $x$ axis for the associated piecewise polynomial. Should reflect the approximate position of roots of a high-order singular function in $x$.
source
SparseIR.segments_y — Method
segments_y(sve_hints::AbstractSVEHints[, T])
Segments for piecewise polynomials on the $y$ axis.
List of segments on the $y$ axis for the associated piecewise polynomial. Should reflect the approximate position of roots of a high-order singular function in $y$.
source
SparseIR.shift_xmid — Method
shift_xmid(knots, Δx)
Return midpoint relative to the nearest integer plus a shift.
Return the midpoints xmid of the segments, as pair (diff, shift), where shift is in (0, 1, -1) and diff is a float such that xmid == shift + diff to floating point accuracy.
source
SparseIR.significance — Function
significance(basis::AbstractBasis)
Return vector σ, where 0 ≤ σ[i] ≤ 1 is the significance level of the i-th basis function. If ϵ is the desired accuracy to which to represent a propagator, then any basis function where σ[i] < ϵ can be neglected.
For the IR basis, we simply have that σ[i] = s[i] / first(s).
source
SparseIR.split — Method
split(poly, x)
Split segment.
Find segment of poly's domain that covers x.
source
SparseIR.statistics — Method
statistics(basis::AbstractBasis)
Quantum statistic (Statistics instance, Fermionic() or Bosonic()).
source
SparseIR.sve_hints — Function
sve_hints(kernel, ε)
Provide discretisation hints for the SVE routines.
Advises the SVE routines of discretisation parameters suitable in tranforming the (infinite) SVE into an (finite) SVD problem.
See also AbstractSVEHints.
source
SparseIR.to_IR — Function
to_IR(dlr::DiscreteLehmannRepresentation, g_dlr::AbstractArray, dims=1)
From DLR to IR. g_dlr`: Expansion coefficients in DLR.
source
SparseIR.truncate — Method
truncate(u, s, v; rtol=0.0, lmax=typemax(Int))
Truncate singular value expansion.
Arguments
- `u`, `s`, `v`: Thin singular value expansion
 - `rtol`: Only singular values satisfying `s[l]/s[1] > rtol` are retained.
-- `lmax`: At most the `lmax` most significant singular values are retained.
source
SparseIR.weight_func — Function
weight_func(kernel, statistics::Statistics)
Return the weight function for the given statistics.
  • Fermion: w(x) == 1
  • Boson: w(y) == 1/tanh(Λ*y/2)
source
SparseIR.xrange — Function
xrange(kernel)
Return a tuple $(x_\mathrm{min}, x_\mathrm{max})$ delimiting the range of allowed x values.
source
SparseIR.yrange — Function
yrange(kernel)
Return a tuple $(y_\mathrm{min}, y_\mathrm{max})$ delimiting the range of allowed y values.
source
SparseIR.Λ — Function
Λ(basis::AbstractBasis)
-lambda(basis::AbstractBasis)
Basis cutoff parameter, Λ = β * ωmax, or None if not present
source
SparseIR.β — Method
β(basis::AbstractBasis)
-beta(basis::AbstractBasis)
Inverse temperature or nothing if unscaled basis.
source
SparseIR.ωmax — Function
ωmax(basis::AbstractBasis)
-wmax(basis::AbstractBasis)
Real frequency cutoff or nothing if unscaled basis.
source
SparseIR._LinAlg.rrqr! — Method
Truncated rank-revealing QR decomposition with full column pivoting.
Decomposes a (m, n) matrix A into the product:
A[:,piv] == Q * R
where Q is an (m, k) isometric matrix, R is a (k, n) upper triangular matrix, piv is a permutation vector, and k is chosen such that the relative tolerance tol is met in the equality above.
source
SparseIR._LinAlg.svd2x2 — Method
Perform the SVD of an arbitrary two-by-two matrix:
  [ a11  a12 ]  =  [  cu  -su ] [ smax     0 ] [  cv   sv ]
-  [ a21  a22 ]     [  su   cu ] [    0  smin ] [ -sv   cv ]
Note that smax and smin can be negative.
source
SparseIR._LinAlg.svd2x2 — Method
Perform the SVD of upper triangular two-by-two matrix:
  [ f    g   ]  =  [  cu  -su ] [ smax     0 ] [  cv   sv ]
-  [ 0    h   ]     [  su   cu ] [    0  smin ] [ -sv   cv ]
Note that smax and smin can be negative.
source
SparseIR._LinAlg.tsvd! — Method
Truncated singular value decomposition.
Decomposes an (m, n) matrix A into the product:
A == U * (s .* VT)
where U is a (m, k) matrix with orthogonal columns, VT is a (k, n) matrix with orthogonal rows and s are the singular values, a set of k nonnegative numbers in non-ascending order. The SVD is truncated in the sense that singular values below tol are discarded.
source

+- `lmax`: At most the `lmax` most significant singular values are retained.
source
SparseIR.weight_func — Function
weight_func(kernel, statistics::Statistics)

Return the weight function for the given statistics.

  • Fermion: w(x) == 1
  • Boson: w(y) == 1/tanh(Λ*y/2)
source
SparseIR.xrange — Function
xrange(kernel)

Return a tuple $(x_\mathrm{min}, x_\mathrm{max})$ delimiting the range of allowed x values.

source
SparseIR.yrange — Function
yrange(kernel)

Return a tuple $(y_\mathrm{min}, y_\mathrm{max})$ delimiting the range of allowed y values.

source
SparseIR.Λ — Function
Λ(basis::AbstractBasis)
+lambda(basis::AbstractBasis)

Basis cutoff parameter, Λ = β * ωmax, or None if not present

source
SparseIR.β — Method
β(basis::AbstractBasis)
+beta(basis::AbstractBasis)

Inverse temperature or nothing if unscaled basis.

source
SparseIR.ωmax — Function
ωmax(basis::AbstractBasis)
+wmax(basis::AbstractBasis)

Real frequency cutoff or nothing if unscaled basis.

source
SparseIR._LinAlg.rrqr! — Method

Truncated rank-revealing QR decomposition with full column pivoting.

Decomposes a (m, n) matrix A into the product:

A[:,piv] == Q * R

where Q is an (m, k) isometric matrix, R is a (k, n) upper triangular matrix, piv is a permutation vector, and k is chosen such that the relative tolerance tol is met in the equality above.

source
SparseIR._LinAlg.svd2x2 — Method

Perform the SVD of an arbitrary two-by-two matrix:

  [ a11  a12 ]  =  [  cu  -su ] [ smax     0 ] [  cv   sv ]
+  [ a21  a22 ]     [  su   cu ] [    0  smin ] [ -sv   cv ]

Note that smax and smin can be negative.

source
SparseIR._LinAlg.svd2x2 — Method

Perform the SVD of upper triangular two-by-two matrix:

  [ f    g   ]  =  [  cu  -su ] [ smax     0 ] [  cv   sv ]
+  [ 0    h   ]     [  su   cu ] [    0  smin ] [ -sv   cv ]

Note that smax and smin can be negative.

source
SparseIR._LinAlg.tsvd! — Method

Truncated singular value decomposition.

Decomposes an (m, n) matrix A into the product:

A == U * (s .* VT)

where U is a (m, k) matrix with orthogonal columns, VT is a (k, n) matrix with orthogonal rows and s are the singular values, a set of k nonnegative numbers in non-ascending order. The SVD is truncated in the sense that singular values below tol are discarded.

source
diff --git a/dev/public/index.html b/dev/public/index.html index f3309ed..62651a8 100644 --- a/dev/public/index.html +++ b/dev/public/index.html @@ -1,8 +1,8 @@ -Public · SparseIR.jl
GitHub

Public names index

SparseIR.AugmentedBasis — Type
AugmentedBasis <: AbstractBasis

Augmented basis on the imaginary-time/frequency axis.

Groups a set of additional functions, augmentations, with a given basis. The augmented functions then form the first basis functions, while the rest is provided by the regular basis, i.e.:

u[l](x) == l < naug ? augmentations[l](x) : basis.u[l-naug](x),

where naug = length(augmentations) is the number of added basis functions through augmentation. Similar expressions hold for Matsubara frequencies.

Augmentation is useful in constructing bases for vertex-like quantities such as self-energies [wallerberger2021] and when constructing a two-point kernel that serves as a base for multi-point functions [shinaoka2018].

Warning

Bases augmented with TauConst and TauLinear tend to be poorly conditioned. Care must be taken while fitting and compactness should be enforced if possible to regularize the problem.

While vertex bases, i.e. bases augmented with MatsubaraConst, stay reasonably well-conditioned, it is still good practice to treat the Hartree–Fock term separately rather than including it in the basis, if possible.

See also: MatsubaraConst for vertex basis [wallerberger2021], TauConst, TauLinear for multi-point [shinaoka2018]

source
SparseIR.DiscreteLehmannRepresentation — Type
DiscreteLehmannRepresentation <: AbstractBasis

Discrete Lehmann representation (DLR) with poles selected according to extrema of IR.

This class implements a variant of the discrete Lehmann representation (DLR) 1. Instead of a truncated singular value expansion of the analytic continuation kernel $K$ like the IR, the discrete Lehmann representation is based on a "sketching" of $K$. The resulting basis is a linear combination of discrete set of poles on the real-frequency axis, continued to the imaginary-frequency axis:

 G(iv) == sum(a[i] / (iv - w[i]) for i in range(L))

Warning The poles on the real-frequency axis selected for the DLR are based on a rank-revealing decomposition, which offers accuracy guarantees. Here, we instead select the pole locations based on the zeros of the IR basis functions on the real axis, which is a heuristic. We do not expect that difference to matter, but please don't blame the DLR authors if we were wrong :-)

source
SparseIR.FiniteTempBasis — Type
FiniteTempBasis <: AbstractBasis

Intermediate representation (IR) basis for given temperature.

For a continuation kernel K from real frequencies, ω ∈ [-ωmax, ωmax], to imaginary time, τ ∈ [0, β], this type stores the truncated singular value expansion or IR basis:

K(τ, ω) ≈ sum(u[l](τ) * s[l] * v[l](ω) for l in 1:L)

This basis is inferred from a reduced form by appropriate scaling of the variables.

Fields

  • u::PiecewiseLegendrePolyVector: Set of IR basis functions on the imaginary time (tau) axis. These functions are stored as piecewise Legendre polynomials.

    To obtain the value of all basis functions at a point or a array of points x, you can call the function u(x). To obtain a single basis function, a slice or a subset l, you can use u[l].

  • uhat::PiecewiseLegendreFT: Set of IR basis functions on the Matsubara frequency (wn) axis. These objects are stored as a set of Bessel functions.

    To obtain the value of all basis functions at a Matsubara frequency or a array of points wn, you can call the function uhat(wn). Note that we expect reduced frequencies, which are simply even/odd numbers for bosonic/fermionic objects. To obtain a single basis function, a slice or a subset l, you can use uhat[l].

  • s: Vector of singular values of the continuation kernel

  • v::PiecewiseLegendrePoly: Set of IR basis functions on the real frequency (w) axis. These functions are stored as piecewise Legendre polynomials.

    To obtain the value of all basis functions at a point or a array of points w, you can call the function v(w). To obtain a single basis function, a slice or a subset l, you can use v[l].

source
SparseIR.FiniteTempBasis — Method
FiniteTempBasis{S}(β, ωmax, ε=nothing; max_size=nothing, args...)

Construct a finite temperature basis suitable for the given S (Fermionic or Bosonic) and cutoffs β and ωmax.

source
SparseIR.FiniteTempBasisSet — Type
FiniteTempBasisSet

Type for holding IR bases and sparse-sampling objects.

An object of this type holds IR bases for fermions and bosons and associated sparse-sampling objects.

Fields

  • basis_f::FiniteTempBasis: Fermion basis
  • basis_b::FiniteTempBasis: Boson basis
  • tau::Vector{Float64}: Sampling points in the imaginary-time domain
  • wn_f::Vector{Int}: Sampling fermionic frequencies
  • wn_b::Vector{Int}: Sampling bosonic frequencies
  • smpltauf::TauSampling: Sparse sampling for tau & fermion
  • smpltaub::TauSampling: Sparse sampling for tau & boson
  • smplwnf::MatsubaraSampling: Sparse sampling for Matsubara frequency & fermion
  • smplwnb::MatsubaraSampling: Sparse sampling for Matsubara frequency & boson
  • sve_result::Tuple{PiecewiseLegendrePoly,Vector{Float64},PiecewiseLegendrePoly}: Results of SVE

Getters

  • beta::Float64: Inverse temperature
  • ωmax::Float64: Cut-off frequency
source
SparseIR.LogisticKernel — Type
LogisticKernel <: AbstractKernel

Fermionic/bosonic analytical continuation kernel.

In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the integral kernel is a function on $[-1, 1] × [-1, 1]$:

\[ K(x, y) = \frac{e^{-Λ y (x + 1) / 2}}{1 + e^{-Λ y}}\]

LogisticKernel is a fermionic analytic continuation kernel. Nevertheless, one can model the $τ$ dependence of a bosonic correlation function as follows:

\[ ∫ \frac{e^{-Λ y (x + 1) / 2}}{1 - e^{-Λ y}} ρ(y) dy = ∫ K(x, y) ρ'(y) dy,\]

with

\[ ρ'(y) = w(y) ρ(y),\]

where the weight function is given by

\[ w(y) = \frac{1}{\tanh(Λ y/2)}.\]

source
SparseIR.MatsubaraFreq — Type
MatsubaraFreq(n)

Prefactor n of the Matsubara frequency ω = n*π/β

Struct representing the Matsubara frequency ω entering the Fourier transform of a propagator G(τ) on imaginary time τ to its Matsubara equivalent Ĝ(iω) on the imaginary-frequency axis:

        β
+Public · SparseIR.jl
GitHub
Public names index
SparseIR.AugmentedBasis — Type
AugmentedBasis <: AbstractBasis
Augmented basis on the imaginary-time/frequency axis.
Groups a set of additional functions, augmentations, with a given basis. The augmented functions then form the first basis functions, while the rest is provided by the regular basis, i.e.:
u[l](x) == l < naug ? augmentations[l](x) : basis.u[l-naug](x),
where naug = length(augmentations) is the number of added basis functions through augmentation. Similar expressions hold for Matsubara frequencies.
Augmentation is useful in constructing bases for vertex-like quantities such as self-energies [wallerberger2021] and when constructing a two-point kernel that serves as a base for multi-point functions [shinaoka2018].
Warning
Bases augmented with TauConst and TauLinear tend to be poorly conditioned. Care must be taken while fitting and compactness should be enforced if possible to regularize the problem.
While vertex bases, i.e. bases augmented with MatsubaraConst, stay reasonably well-conditioned, it is still good practice to treat the Hartree–Fock term separately rather than including it in the basis, if possible.
See also: MatsubaraConst for vertex basis [wallerberger2021], TauConst, TauLinear for multi-point [shinaoka2018]
source
SparseIR.DiscreteLehmannRepresentation — Type
DiscreteLehmannRepresentation <: AbstractBasis
Discrete Lehmann representation (DLR) with poles selected according to extrema of IR.
This class implements a variant of the discrete Lehmann representation (DLR) 1. Instead of a truncated singular value expansion of the analytic continuation kernel $K$ like the IR, the discrete Lehmann representation is based on a "sketching" of $K$. The resulting basis is a linear combination of discrete set of poles on the real-frequency axis, continued to the imaginary-frequency axis:
 G(iv) == sum(a[i] / (iv - w[i]) for i in range(L))
Warning The poles on the real-frequency axis selected for the DLR are based on a rank-revealing decomposition, which offers accuracy guarantees. Here, we instead select the pole locations based on the zeros of the IR basis functions on the real axis, which is a heuristic. We do not expect that difference to matter, but please don't blame the DLR authors if we were wrong :-)
source
SparseIR.FiniteTempBasis — Type
FiniteTempBasis <: AbstractBasis
Intermediate representation (IR) basis for given temperature.
For a continuation kernel K from real frequencies, ω ∈ [-ωmax, ωmax], to imaginary time, τ ∈ [0, β], this type stores the truncated singular value expansion or IR basis:
K(τ, ω) ≈ sum(u[l](τ) * s[l] * v[l](ω) for l in 1:L)
This basis is inferred from a reduced form by appropriate scaling of the variables.
Fields
  • u::PiecewiseLegendrePolyVector: Set of IR basis functions on the imaginary time (tau) axis. These functions are stored as piecewise Legendre polynomials.
    To obtain the value of all basis functions at a point or a array of points x, you can call the function u(x). To obtain a single basis function, a slice or a subset l, you can use u[l].
  • uhat::PiecewiseLegendreFT: Set of IR basis functions on the Matsubara frequency (wn) axis. These objects are stored as a set of Bessel functions.
    To obtain the value of all basis functions at a Matsubara frequency or a array of points wn, you can call the function uhat(wn). Note that we expect reduced frequencies, which are simply even/odd numbers for bosonic/fermionic objects. To obtain a single basis function, a slice or a subset l, you can use uhat[l].
  • s: Vector of singular values of the continuation kernel
  • v::PiecewiseLegendrePoly: Set of IR basis functions on the real frequency (w) axis. These functions are stored as piecewise Legendre polynomials.
    To obtain the value of all basis functions at a point or a array of points w, you can call the function v(w). To obtain a single basis function, a slice or a subset l, you can use v[l].
source
SparseIR.FiniteTempBasis — Method
FiniteTempBasis{S}(β, ωmax, ε=nothing; max_size=nothing, args...)
Construct a finite temperature basis suitable for the given S (Fermionic or Bosonic) and cutoffs β and ωmax.
source
SparseIR.FiniteTempBasisSet — Type
FiniteTempBasisSet
Type for holding IR bases and sparse-sampling objects.
An object of this type holds IR bases for fermions and bosons and associated sparse-sampling objects.
Fields
  • basis_f::FiniteTempBasis: Fermion basis
  • basis_b::FiniteTempBasis: Boson basis
  • tau::Vector{Float64}: Sampling points in the imaginary-time domain
  • wn_f::Vector{Int}: Sampling fermionic frequencies
  • wn_b::Vector{Int}: Sampling bosonic frequencies
  • smpltauf::TauSampling: Sparse sampling for tau & fermion
  • smpltaub::TauSampling: Sparse sampling for tau & boson
  • smplwnf::MatsubaraSampling: Sparse sampling for Matsubara frequency & fermion
  • smplwnb::MatsubaraSampling: Sparse sampling for Matsubara frequency & boson
  • sve_result::Tuple{PiecewiseLegendrePoly,Vector{Float64},PiecewiseLegendrePoly}: Results of SVE
Getters
  • beta::Float64: Inverse temperature
  • ωmax::Float64: Cut-off frequency
source
SparseIR.LogisticKernel — Type
LogisticKernel <: AbstractKernel
Fermionic/bosonic analytical continuation kernel.
In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = \frac{e^{-Λ y (x + 1) / 2}}{1 + e^{-Λ y}}\]
LogisticKernel is a fermionic analytic continuation kernel. Nevertheless, one can model the $τ$ dependence of a bosonic correlation function as follows:
\[    ∫ \frac{e^{-Λ y (x + 1) / 2}}{1 - e^{-Λ y}} ρ(y) dy = ∫ K(x, y) ρ'(y) dy,\]
with
\[    ρ'(y) = w(y) ρ(y),\]
where the weight function is given by
\[    w(y) = \frac{1}{\tanh(Λ y/2)}.\]
source
SparseIR.MatsubaraFreq — Type
MatsubaraFreq(n)
Prefactor n of the Matsubara frequency ω = n*π/β
Struct representing the Matsubara frequency ω entering the Fourier transform of a propagator G(τ) on imaginary time τ to its Matsubara equivalent Ĝ(iω) on the imaginary-frequency axis:
        β
 Ĝ(iω) = ∫  dτ exp(iωτ) G(τ)      with    ω = n π/β,
-        0
where β is inverse temperature and by convention we include the imaginary unit in the frequency argument, i.e, Ĝ(iω). The frequencies depend on the statistics of the propagator, i.e., we have that:
G(τ - β) = ± G(τ)
where + is for bosons and - is for fermions. The frequencies are restricted accordingly.
  • Bosonic frequency (S == Fermionic): n even (periodic in β)
  • Fermionic frequency (S == Bosonic): n odd (anti-periodic in β)
source
SparseIR.MatsubaraSampling — Type
MatsubaraSampling <: AbstractSampling
Sparse sampling in Matsubara frequencies.
Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary frequencies.
source
SparseIR.MatsubaraSampling — Method
MatsubaraSampling(basis; positive_only=false,
+        0
where β is inverse temperature and by convention we include the imaginary unit in the frequency argument, i.e, Ĝ(iω). The frequencies depend on the statistics of the propagator, i.e., we have that:
G(τ - β) = ± G(τ)
where + is for bosons and - is for fermions. The frequencies are restricted accordingly.
  • Bosonic frequency (S == Fermionic): n even (periodic in β)
  • Fermionic frequency (S == Bosonic): n odd (anti-periodic in β)
source
SparseIR.MatsubaraSampling — Type
MatsubaraSampling <: AbstractSampling
Sparse sampling in Matsubara frequencies.
Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary frequencies.
source
SparseIR.MatsubaraSampling — Method
MatsubaraSampling(basis; positive_only=false,
                   sampling_points=default_matsubara_sampling_points(basis; positive_only),
-                  factorize=true)
Construct a MatsubaraSampling object. If not given, the sampling_points are chosen as the (discrete) extrema of the highest-order basis function in Matsubara. This turns out to be close to optimal with respect to conditioning for this size (within a few percent).
By setting positive_only=true, one assumes that functions to be fitted are symmetric in Matsubara frequency, i.e.:
\[    Ĝ(iν) = conj(Ĝ(-iν))\]
or equivalently, that they are purely real in imaginary time. In this case, sparse sampling is performed over non-negative frequencies only, cutting away half of the necessary sampling space. factorize controls whether the SVD decomposition is computed.
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SparseIR.RegularizedBoseKernel — Type
RegularizedBoseKernel <: AbstractKernel
Regularized bosonic analytical continuation kernel.
In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:
\[    K(x, y) = y \frac{e^{-Λ y (x + 1) / 2}}{e^{-Λ y} - 1}\]
Care has to be taken in evaluating this expression around $y = 0$.
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SparseIR.TauConst — Type
TauConst <: AbstractAugmentation
Constant in imaginary time/discrete delta in frequency.
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SparseIR.TauLinear — Type
TauLinear <: AbstractAugmentation
Linear function in imaginary time, antisymmetric around β/2.
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SparseIR.TauSampling — Type
TauSampling <: AbstractSampling
Sparse sampling in imaginary time.
Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary time.
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SparseIR.TauSampling — Method
TauSampling(basis; sampling_points=default_tau_sampling_points(basis), factorize=true)
Construct a TauSampling object. If not given, the sampling_points are chosen as the extrema of the highest-order basis function in imaginary time. This turns out to be close to optimal with respect to conditioning for this size (within a few percent). factorize controls whether the SVD decomposition is computed.
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SparseIR.evaluate! — Method
evaluate!(buffer::AbstractArray{T,N}, sampling, al; dim=1) where {T,N}
Like evaluate, but write the result to buffer. Please use dim = 1 or N to avoid allocating large temporary arrays internally.
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SparseIR.evaluate — Method
evaluate(sampling, al; dim=1)
Evaluate the basis coefficients al at the sparse sampling points.
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SparseIR.fit! — Method
fit!(buffer::Array{S,N}, smpl::AbstractSampling, al::Array{T,N}; 
-    dim=1, workarr::Vector{S}) where {S,T,N}
Like fit, but write the result to buffer. Use dim = 1 or dim = N to avoid allocating large temporary arrays internally. The length of workarr cannot be smaller than SparseIR.workarrlength(smpl, al).
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SparseIR.fit — Method
fit(sampling, al::AbstractArray{T,N}; dim=1)
Fit basis coefficients from the sparse sampling points Please use dim = 1 or N to avoid allocating large temporary arrays internally.
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SparseIR.overlap — Method
overlap(poly::PiecewiseLegendrePoly, f; 
-    rtol=eps(T), return_error=false, maxevals=10^4, points=T[])
Evaluate overlap integral of poly with arbitrary function f.
Given the function f, evaluate the integral
∫ dx f(x) poly(x)
using adaptive Gauss-Legendre quadrature.
points is a sequence of break points in the integration interval where local difficulties of the integrand may occur (e.g. singularities, discontinuities).
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+                  factorize=true)

Construct a MatsubaraSampling object. If not given, the sampling_points are chosen as the (discrete) extrema of the highest-order basis function in Matsubara. This turns out to be close to optimal with respect to conditioning for this size (within a few percent).

By setting positive_only=true, one assumes that functions to be fitted are symmetric in Matsubara frequency, i.e.:

\[ Ĝ(iν) = conj(Ĝ(-iν))\]

or equivalently, that they are purely real in imaginary time. In this case, sparse sampling is performed over non-negative frequencies only, cutting away half of the necessary sampling space. factorize controls whether the SVD decomposition is computed.

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SparseIR.RegularizedBoseKernel — Type
RegularizedBoseKernel <: AbstractKernel

Regularized bosonic analytical continuation kernel.

In dimensionless variables $x = 2 τ/β - 1$, $y = β ω/Λ$, the fermionic integral kernel is a function on $[-1, 1] × [-1, 1]$:

\[ K(x, y) = y \frac{e^{-Λ y (x + 1) / 2}}{e^{-Λ y} - 1}\]

Care has to be taken in evaluating this expression around $y = 0$.

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SparseIR.TauConst — Type
TauConst <: AbstractAugmentation

Constant in imaginary time/discrete delta in frequency.

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SparseIR.TauLinear — Type
TauLinear <: AbstractAugmentation

Linear function in imaginary time, antisymmetric around β/2.

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SparseIR.TauSampling — Type
TauSampling <: AbstractSampling

Sparse sampling in imaginary time.

Allows the transformation between the IR basis and a set of sampling points in (scaled/unscaled) imaginary time.

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SparseIR.TauSampling — Method
TauSampling(basis; sampling_points=default_tau_sampling_points(basis), factorize=true)

Construct a TauSampling object. If not given, the sampling_points are chosen as the extrema of the highest-order basis function in imaginary time. This turns out to be close to optimal with respect to conditioning for this size (within a few percent). factorize controls whether the SVD decomposition is computed.

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SparseIR.evaluate! — Method
evaluate!(buffer::AbstractArray{T,N}, sampling, al; dim=1) where {T,N}

Like evaluate, but write the result to buffer. Please use dim = 1 or N to avoid allocating large temporary arrays internally.

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SparseIR.evaluate — Method
evaluate(sampling, al; dim=1)

Evaluate the basis coefficients al at the sparse sampling points.

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SparseIR.fit! — Method
fit!(buffer::Array{S,N}, smpl::AbstractSampling, al::Array{T,N}; 
+    dim=1, workarr::Vector{S}) where {S,T,N}

Like fit, but write the result to buffer. Use dim = 1 or dim = N to avoid allocating large temporary arrays internally. The length of workarr cannot be smaller than SparseIR.workarrlength(smpl, al).

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SparseIR.fit — Method
fit(sampling, al::AbstractArray{T,N}; dim=1)

Fit basis coefficients from the sparse sampling points Please use dim = 1 or N to avoid allocating large temporary arrays internally.

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SparseIR.overlap — Method
overlap(poly::PiecewiseLegendrePoly, f; 
+    rtol=eps(T), return_error=false, maxevals=10^4, points=T[])

Evaluate overlap integral of poly with arbitrary function f.

Given the function f, evaluate the integral

∫ dx f(x) poly(x)

using adaptive Gauss-Legendre quadrature.

points is a sequence of break points in the integration interval where local difficulties of the integrand may occur (e.g. singularities, discontinuities).

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