Struct TauSampling 

pub struct TauSampling<S>
where
    S: StatisticsType,
{ /* private fields */ }

Sparse sampling in imaginary time

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

Implementations

impl<S> TauSampling<S>

where S: StatisticsType,

pub fn new(basis: &impl Basis<S>) -> Result<TauSampling<S>, Error>

where S: 'static,

Create a new TauSampling with default sampling points

The default sampling points are the roots of the first discarded basis function u_L (the extrema of u_{L-1} when u_L is not available), which gives near-optimal conditioning. SVD is computed lazily on first call to fit or fit_nd.

Arguments

• basis - Any basis implementing the Basis trait

Returns

A new TauSampling object

Errors

The errors of Basis::default_tau_sampling_points (e.g. NotSupported for a DLR, whose IR basis has the default points)

pub fn with_sampling_points( basis: &impl Basis<S>, sampling_points: Vec<f64>, ) -> Result<TauSampling<S>, Error>

where S: 'static,

Create a new TauSampling with custom sampling points

SVD is computed lazily on first call to fit or fit_nd.

Arguments

• basis - Any basis implementing the Basis trait
• sampling_points - Custom sampling points in τ ∈ [-β, β]

Returns

A new TauSampling object

The points are kept in the given order, and duplicates are accepted; they only raise the condition number.

Errors

• Error::EmptyInput if sampling_points is empty
• Error::OutOfDomain if a point is outside [-β, β] or NaN (from Basis::evaluate_tau)

pub fn from_matrix( sampling_points: Vec<f64>, matrix: &TypedTensor<f64, Rank<2>>, ) -> Result<TauSampling<S>, Error>

Create a new TauSampling with custom sampling points and pre-computed matrix

This constructor is useful when the sampling matrix is already computed (e.g., from external sources or for testing).

Arguments

• sampling_points - Imaginary times τ that label the rows of matrix, in any order. There is no β to check them against, so any finite value is accepted and kept as given.
• matrix - Pre-computed sampling matrix (n_points × basis_size); row i belongs to sampling_points[i]

Duplicate points are accepted; they only raise the condition number.

Errors

• Error::EmptyInput if sampling_points is empty, or matrix has no columns
• Error::ShapeMismatch of the input if matrix does not have one row per point
• Error::NonFiniteInput for the first NaN or infinite point, then for the first NaN or infinite entry of matrix

pub fn sampling_points(&self) -> &[f64]

Get the sampling points

pub fn n_sampling_points(&self) -> usize

Get the number of sampling points

pub fn basis_size(&self) -> usize

Get the basis size

pub fn matrix(&self) -> &TypedTensor<f64, Rank<2>>

Get the sampling matrix

pub fn condition_number(&self) -> Result<f64, Error>

Condition number of the sampling matrix, which fitting solves with

Returns σ_max / σ_min, the ratio of the largest to the smallest of the min(n_sampling_points, basis_size) singular values of the real n_sampling_points × basis_size matrix Self::matrix. It bounds how much Self::fit can amplify relative errors in the values.

Returns f64::INFINITY if the smallest singular value is below 1e-15 (numerically singular matrix). The singular value decomposition is the one fitting uses: it is computed by the first call to this method or to a fit, then cached.

Errors

Error::DecompositionFailed if the singular value decomposition fails, which a matrix of finite entries does not cause in practice (the constructors reject non-finite entries)

pub fn evaluate(&self, coeffs: &[f64]) -> Result<Vec<f64>, Error>

Evaluate basis coefficients at sampling points

Computes g(τ_i) = Σ_l a_l * u_l(τ_i) for all sampling points

Arguments

• coeffs - Basis coefficients (length = basis_size)

Returns

Values at sampling points (length = n_sampling_points)

Errors

• Error::ShapeMismatch of the input if coeffs does not have length basis_size

pub fn evaluate_to(&self, coeffs: &[f64], out: &mut [f64]) -> Result<(), Error>

Evaluate basis coefficients at sampling points, writing to output slice

Errors

• Error::ShapeMismatch of the input if coeffs does not have length basis_size
• Error::ShapeMismatch of the output if out does not have length n_sampling_points

Nothing is written to out on an error.

pub fn fit(&self, values: &[f64]) -> Result<Vec<f64>, Error>

Fit values at sampling points to basis coefficients

Errors

• Error::ShapeMismatch of the input if values does not have length n_sampling_points
• Error::DecompositionFailed if the singular value decomposition fails

pub fn fit_to(&self, values: &[f64], out: &mut [f64]) -> Result<(), Error>

Fit values at sampling points to basis coefficients, writing to output slice

Errors

• Error::ShapeMismatch of the input if values does not have length n_sampling_points
• Error::ShapeMismatch of the output if out does not have length basis_size
• Error::DecompositionFailed if the singular value decomposition fails

Nothing is written to out on an error.

pub fn evaluate_zz( &self, coeffs: &[Complex<f64>], ) -> Result<Vec<Complex<f64>>, Error>

Evaluate complex basis coefficients at sampling points

Errors

• Error::ShapeMismatch of the input if coeffs does not have length basis_size

pub fn evaluate_zz_to( &self, coeffs: &[Complex<f64>], out: &mut [Complex<f64>], ) -> Result<(), Error>

Evaluate complex basis coefficients, writing to output slice

Errors

• Error::ShapeMismatch of the input if coeffs does not have length basis_size
• Error::ShapeMismatch of the output if out does not have length n_sampling_points

Nothing is written to out on an error.

pub fn fit_zz( &self, values: &[Complex<f64>], ) -> Result<Vec<Complex<f64>>, Error>

Fit complex values at sampling points to basis coefficients

Errors

• Error::ShapeMismatch of the input if values does not have length n_sampling_points
• Error::DecompositionFailed if the singular value decomposition fails

pub fn fit_zz_to( &self, values: &[Complex<f64>], out: &mut [Complex<f64>], ) -> Result<(), Error>

Fit complex values, writing to output slice

Errors

• Error::ShapeMismatch of the input if values does not have length n_sampling_points
• Error::ShapeMismatch of the output if out does not have length basis_size
• Error::DecompositionFailed if the singular value decomposition fails

Nothing is written to out on an error.

pub fn evaluate_nd( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensor<f64>, dim: usize, ) -> Result<TypedTensor<f64>, Error>

Evaluate N-D real coefficients at sampling points

Arguments

• coeffs - N-dimensional array with coeffs.shape().dim(dim) == basis_size
• dim - Dimension along which to evaluate (0-indexed)

Returns

N-dimensional array with result.shape().dim(dim) == n_sampling_points

Errors

• Error::AxisOutOfRange if dim is not an axis of coeffs
• Error::ShapeMismatch of the input if coeffs does not have basis_size along dim

pub fn evaluate_nd_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Evaluate N-D real coefficients, writing to a mutable view

out must have the shape of coeffs with n_sampling_points along dim.

Errors

• Error::AxisOutOfRange if dim is not an axis of coeffs
• Error::ShapeMismatch of the input if coeffs does not have basis_size along dim, and of the output if out does not have the shape of coeffs with n_sampling_points along dim

Nothing is written to out then.

pub fn fit_nd( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensor<f64>, dim: usize, ) -> Result<TypedTensor<f64>, Error>

Fit N-D real values at sampling points to basis coefficients

Arguments

• values - N-dimensional array with values.shape().dim(dim) == n_sampling_points
• dim - Dimension along which to fit (0-indexed)

Returns

N-dimensional array with result.shape().dim(dim) == basis_size

Errors

• Error::AxisOutOfRange if dim is not an axis of values
• Error::ShapeMismatch of the input if values does not have n_sampling_points along dim
• Error::DecompositionFailed if the singular value decomposition fails

pub fn fit_nd_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Fit N-D real values, writing to a mutable view

out must have the shape of values with basis_size along dim.

Errors

• Error::AxisOutOfRange if dim is not an axis of values
• Error::ShapeMismatch of the input if values does not have n_sampling_points along dim, and of the output if out does not have the shape of values with basis_size along dim
• Error::DecompositionFailed if the singular value decomposition fails

Nothing is written to out then.

pub fn evaluate_nd_zz( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensor<Complex<f64>>, dim: usize, ) -> Result<TypedTensor<Complex<f64>>, Error>

Evaluate N-D complex coefficients at sampling points

Arguments

• coeffs - N-dimensional complex array with coeffs.shape().dim(dim) == basis_size
• dim - Dimension along which to evaluate (0-indexed)

Returns

N-dimensional complex array with result.shape().dim(dim) == n_sampling_points

Errors

• Error::AxisOutOfRange if dim is not an axis of coeffs
• Error::ShapeMismatch of the input if coeffs does not have basis_size along dim

pub fn evaluate_nd_zz_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Evaluate N-D complex coefficients, writing to a mutable view

out must have the shape of coeffs with n_sampling_points along dim.

Errors

• Error::AxisOutOfRange if dim is not an axis of coeffs
• Error::ShapeMismatch of the input if coeffs does not have basis_size along dim, and of the output if out does not have the shape of coeffs with n_sampling_points along dim

Nothing is written to out then.

pub fn fit_nd_zz( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensor<Complex<f64>>, dim: usize, ) -> Result<TypedTensor<Complex<f64>>, Error>

Fit N-D complex values at sampling points to basis coefficients

Arguments

• values - N-dimensional complex array with values.shape().dim(dim) == n_sampling_points
• dim - Dimension along which to fit (0-indexed)

Returns

N-dimensional complex array with result.shape().dim(dim) == basis_size

Errors

• Error::AxisOutOfRange if dim is not an axis of values
• Error::ShapeMismatch of the input if values does not have n_sampling_points along dim
• Error::DecompositionFailed if the singular value decomposition fails

pub fn fit_nd_zz_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Fit N-D complex values, writing to a mutable view

out must have the shape of values with basis_size along dim.

Errors

• Error::AxisOutOfRange if dim is not an axis of values
• Error::ShapeMismatch of the input if values does not have n_sampling_points along dim, and of the output if out does not have the shape of values with basis_size along dim
• Error::DecompositionFailed if the singular value decomposition fails

Nothing is written to out then.

Trait Implementations

impl<S> InplaceFitter for TauSampling<S>

where S: StatisticsType,

InplaceFitter implementation for TauSampling

Delegates to RealMatrixFitter which supports dd and zz operations.

fn n_points(&self) -> usize

Number of sampling points.

fn basis_size(&self) -> usize

Number of basis functions.

fn evaluate_nd_dd_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Evaluate: f64 coefficients to f64 values.

fn evaluate_nd_zz_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Evaluate: Complex<f64> coefficients to Complex<f64> values.

fn fit_nd_dd_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Fit: f64 values to f64 coefficients.

fn fit_nd_zz_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Fit: Complex<f64> values to Complex<f64> coefficients.

fn evaluate_nd_dz_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Evaluate: f64 coefficients to Complex<f64> values.

fn evaluate_nd_zd_to( &self, backend: Option<&GemmBackendHandle>, coeffs: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Evaluate: Complex<f64> coefficients to f64 values.

fn fit_nd_dz_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, f64>, dim: usize, out: &mut TypedTensorViewMut<'_, Complex<f64>>, ) -> Result<(), Error>

Fit: f64 values to Complex<f64> coefficients.

fn fit_nd_zd_to( &self, backend: Option<&GemmBackendHandle>, values: &TypedTensorView<'_, Complex<f64>>, dim: usize, out: &mut TypedTensorViewMut<'_, f64>, ) -> Result<(), Error>

Fit: Complex<f64> values to f64 coefficients.