grid.rtransform module
Transformation from 1D intervals [a, b] to other 1D intervals [c, d].
- class BaseTransform[source]
Bases:
ABCAbstract class for transformation.
- __init__()
- property codomain
Transformation codomain.
- Type
tuple
- property domain
Transformation domain.
- Type
tuple
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class BeckeRTransform(rmin, R, trim_inf=True)[source]
Bases:
BaseTransformBecke Transformation.
The Becke transformation transforms from \([-1, 1]\) to \([r_{min}, \infty)\) according to
\[r(x) = R \frac{1 + x}{1 - x} + r_{min}.\]The inverse transformation is given by
\[x(r) = \frac{r - r_{min} - R} {r - r_{min} + R}.\]References
- 1
Becke, Axel D. “A multicenter numerical integration scheme for polyatomic molecules.” The Journal of chemical physics 88.4 (1988): 2547-2553.
- property R
the scale factor for the transformed array.
- Type
float
- __init__(rmin, R, trim_inf=True)[source]
Construct Becke transform, \([-1, 1]\) to :math`[r_{min}, infty)`.
- Parameters
rmin (float) – The minimum coordinate \(r_{min}\) in the transformed interval \([r_{min}, \infty)\).
R (float) – The scale factor used in the transformation.
trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of Becke transformation.
\[\frac{dr_i}{dx_i} = 2R \frac{1}{(1-x)^2}\]- Parameters
x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).
- Returns
First derivative of Becke transformation at each point.
- Return type
np.ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of Becke transformation.
\[\frac{d^2r}{dx^2} = 4R \frac{1}{1-x^3}\]- Parameters
x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Second derivative of Becke transformation at each point.
- Return type
np.ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of Becke transformation.
\[\frac{d^3r}{dx^3} = 12R \frac{1}{1 - x^4}\]- Parameters
x (np.array(N,)) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Third derivative of Becke transformation at each point.
- Return type
np.ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- static find_parameter(array, rmin, radius)[source]
Compute R such that half of the points in \([r_{min}, \infty)\) are within radius.
- Parameters
array (np.ndarray(N,)) – One-dimensional array in the domain \([-1, 1]\).
rmin (float) – Minimum value for transformed array.
radius (float) – Atomic radius of interest.
- Returns
The optimal value of scale factor R.
- Return type
float
- inverse(r)[source]
Transform \([r_{mi}n, \infty)\) back to original \([-1, 1]\).
\[x_i = \frac{r_i - r_{min} - R} {r_i - r_{min} + R}\]- Parameters
r (np.ndarray(N,)) – One-dimensional array in the codomain \([r_{min}, \infty)\).
- Returns
One dimensional array in \([-1, 1]\).
- Return type
np.ndarray(N,)
- property rmin
the minimum value for the transformed array.
- Type
float
- transform(x)[source]
Transform from \([-1, 1]\) to \([r_{min}, \infty)\).
\[r_i = R \frac{1 + x_i}{1 - x_i} + r_{min}\]- Parameters
x (np.ndarray(N,)) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Transformed array located between \([r_min, \infty)\).
- Return type
np.ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class ExpRTransform(rmin, rmax, b=None)[source]
Bases:
BaseTransformExponential transform from \([0, \infty)\) to \([r_{min}, r_{max}]\).
This transformation is given by
\[r(x) = r_{min} e^{x \log\bigg(\frac{r_{max}}{r_{min} / b} \bigg)},\]- where \(b\) maps to rmax. If None, then the \(b\) is taken to be the maximum
from the first grid that is being transformed. This transformation always maps zero to \(r_{min}\).
The inverse transformation is given by
\[x(r) = \frac{\log\big(\frac{r}{r_{min}} \big) b}{\log(\frac{r_{max}}{r_{min}})}\]- __init__(rmin, rmax, b=None)[source]
Initialize exp transform instance.
- Parameters
rmin (float) – Minimum value for transformed points.
rmax (float) – Maximum value for transformed points.
b (float) – Maximum \(b\) of a prespecified radial grid \([0, b]\) such that \(b\) maps to rmax. If None, then the maximum is taken and stored from the grid that is transformed initially.
- property b
Parameter \(b\) that maps/transforms to \(r_{max}\).
- Type
float
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of exponential transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation.
- Returns
First derivative of transformation at x.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of exponential transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation.
- Returns
Second derivative of transformation at x.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of exponential transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation.
- Returns
Third derivative of transformation at x.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of exponential transform.
\[x(r_i) = \frac{\log\big(\frac{r}{r_{min}} \big) b}{\log(\frac{r_{max}}{r_{min}})},\]where \(b\) is a prespecified parameter
- Parameters
r (ndarray(N,)) – One-dimensional array in the domain of the transformation.
- Returns
Inverse transformation at r.
- Return type
ndarray(N,)
- property rmax
the value of rmax.
- Type
float
- property rmin
the value of rmin.
- Type
float
- transform(x)[source]
Perform exponential transform.
\[r = r_{min} e^{x \log\bigg(\frac{r_{max}}{r_{min} / b} \bigg)},\]where \(b\) is a prespecified parameter that maps to \(r_{max}\).
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation.
- Returns
The transformation of x in the co-domain of the transformation.
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class HandyModRTransform(rmin, rmax, m, trim_inf=True)[source]
Bases:
BaseTransformModified Handy Transformation class from \([-1, 1]\) to \([r_{min}, r_{max}]\).
This transformation is given by
\[r(x) = \frac{(1+x)^m (r_{max} - r_{min})} { 2^m (1 - 2^m + r_{max} - r_{min}) - (1 + x)^m (r_{max} - r_{min} - 2^m )} + r_{min},\]where \(m > 0\).
The inverse transformation is given by
\[x(r) = 2 \sqrt[m]{ \frac{(r - r_{min})(r_{max} - r_{min} - 2^m + 1)} {(r - r_{min})(r_{max} - r_{min} - 2^m) + r_{max} - r_{min}} } - 1.\]- __init__(rmin, rmax, m, trim_inf=True)[source]
Construct a modified Handy transform from \([-1, 1]\) to \([r_{min}, r_{max}]\).
- Parameters
rmin (float) – The minimum coordinate for transformed radial array.
rmax (float) – The maximum coordinate for transformed radial array.
m (integer m > 0) – Free parameter, m must be > 0.
trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of modified Handy transformation.
\[\frac{dr}{dx} = -\frac{ 2^m m (r_{max}-r_{min})(2^m-r_{max}+r_{min}-1)(1+x)^{m-1}} {\left( 2^m (2^m-1-r_{max}+r_{min})-(2^m-r_{max} + r_{min})(1 + x)^m\right)^2}\]- Parameters
x (ndarrray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The first derivative of Handy transformation at each point.
- Return type
ndarrray(N,)
- deriv2(x)[source]
Compute the second derivative of modified Handy transformation.
- Parameters
x (ndarrray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The second derivative of Handy transformation at each point.
- Return type
ndarrray(N,)
- deriv3(x)[source]
Compute the third derivative of modified Handy transformation.
- Parameters
x (ndarrray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The third derivative of Handy transformation at each point.
- Return type
ndarrray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Inverse transform from \([r_{min},r_{max}]\) to \([-1,1]\).
\[x_i = 2 \sqrt[m]{ \frac{(r_i - r_{min})(r_{max} - r_{min} - 2^m + 1)} {(r_i - r_{min})(r_{max} - r_{min} - 2^m) + r_{max} - r_{min}} } - 1\]- Parameters
r (ndarrray(N,)) – One dimensional array in \([r_{min},\infty)\).
- Returns
The original one dimensional array in \([-1,1]\).
- Return type
ndarrray(N,)
- property m
Free and extra parameter, m must be > 0.
- Type
integer
- property rmax
The maximum value for the transformed radial array.
- Type
float
- property rmin
The minimum value for the transformed radial array.
- Type
float
- transform(x)[source]
Transform given array \([-1,1]\) to radial array \([r_{min},r_{max}]\).
\[r_i = \frac{(1+x_i)^m (r_{max} - r_{min})} { 2^m (1 - 2^m + r_{max} - r_{min}) - (1 + x_i)^m (r_{max} - r_{min} - 2^m )} + r_{min}\]- Parameters
x (ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
One dimensional array in \([r_{min},r_{max}]\).
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class HandyRTransform(rmin, R, m, trim_inf=True)[source]
Bases:
BaseTransformHandy Transformation class from \([-1, 1]\) to \([r_{min}, \infty)\).
This transformation is given by
\[r(x) = R \left( \frac{1+x}{1-x} \right)^m + r_{min}.\]The inverse transformations is given by
\[x(r) = \frac{\sqrt[m]{r-r_{min}} - \sqrt[m]{R}} {\sqrt[m]{r-r_{min}} + \sqrt[m]{R}}.\]- property R
The scale factor for the transformed radial array.
- Type
float
- __init__(rmin, R, m, trim_inf=True)[source]
Construct Handy transformation.
- Parameters
rmin (float) – The minimum coordinate for transformed radial array.
R (float) – The scale factor for transformed radial array.
m (integer m > 0) – Free parameter, m must be > 0.
trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of Handy transformation.
\[\frac{dr}{dx} = 2mR \frac{(1+x)^{m-1}}{(1-x)^{m+1}}\]- Parameters
x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).
- Returns
The first derivative of Handy transformation at each point.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of Handy transformation.
\[\frac{d^2r}{dx^2} = 4mR (m + x) \frac{(1+x)^{m-2}}{(1-x)^{m+2}}\]- Parameters
x (ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The second derivative of Handy transformation at each point.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of Handy transformation.
\[\frac{d^3r}{dx^3} = 4mR ( 1 + 6 m x + 2 m^2 + 3 x^2) \frac{(1+x)^{m-3}}{(1-x)^{m+3}}\]- Parameters
array (np.ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The third derivative of Handy transformation at each point.
- Return type
np.ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Inverse transform from \([r_{min},\infty)\) to \([-1,1]\).
\[x_i = \frac{\sqrt[m]{r_i-r_{min}} - \sqrt[m]{R}} {\sqrt[m]{r_i-r_{min}} + \sqrt[m]{R}}\]- Parameters
r (np.ndarray(N,)) – One dimensional array in \([r_{min},\infty)\).
- Returns
One-dimensional array in \([-1,1]\).
- Return type
np.ndarray(N,)
- property m
Free and extra parameter, m must be > 0.
- Type
integer
- property rmin
The minimum value for the transformed radial array.
- Type
float
- transform(x)[source]
Transform from \([-1,1]\) to \([r_{min},\infty)\).
\[r_i = R \left( \frac{1+x_i}{1-x_i} \right)^m + r_{min}\]- Parameters
x (np.ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
One dimensional array in \([r_{min},\infty)\).
- Return type
np.ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class HyperbolicRTransform(a, b)[source]
Bases:
BaseTransformHyperbolic transform from :math`[0, infty)` to \([0, \infty)\).
The transformation is given by
\[r(x) = \frac{a x}{(1 - bx)},\]where \(b ( N - 1) \geq 1\), and \(N\) is the number of points in x.
The inverse transformation is given by
\[x(r) = \frac{r}{a + br}\]- __init__(a, b)[source]
Hyperbolic transform class.
- Parameters
a (float) – parameter a to determine hyperbolic function
b (float) – parameter b to determine hyperbolic function
- property a
value of parameter a.
- Type
float
- property b
value of parameter b.
- Type
float
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of hyperbolic transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
First derivative of transformation at x.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of hyperbolic transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
Second derivative of transformation at x.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of hyperbolic transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
Third derivative of transformation at x.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of hyperbolic transformation.
\[x(r) = \frac{r}{a + br}\]- Parameters
r (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
Inverse transformation at r.
- Return type
ndarray(N,)
- transform(x)[source]
Perform hyperbolic transformation.
\[r_i = \frac{a x_i}{(1 - bx_i)},\]where \(b ( N - 1) \geq 1\), and \(N\) is the number of points in x.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
The transformation of x to the co-domain of the transformation.
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class IdentityRTransform[source]
Bases:
BaseTransformIdentity Transform class.
The identity transform class trivially transforms from \([0, \infty)\) to \([0, \infty)\) given by
\[r(x) = x.\]The inverse transformation is given by
\[x(r) = r.\]- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of identity transform.
- Parameters
x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).
- Returns
First derivative of identity transformation at x.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of identity transform.
- Parameters
x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).
- Returns
Second derivative of identity transformation at x.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of identity transform.
- Parameters
x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).
- Returns
Third derivative of identity transformation at x.
- Return type
np.ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of identity transform.
- Parameters
r (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).
- Returns
Inverse transformation of the identity transformation at x.
- Return type
np.ndarray(N,)
- transform(x)[source]
Perform given array into itself.
\[r_i = x_i\]- Parameters
x (ndarray(N,)) – One dimension numpy array located in \([0, \infty)\).
- Returns
Identity transformation at each point x.
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class InverseRTransform(transform)[source]
Bases:
BaseTransformInverse transformation class for any general transformation.
- __init__(transform)[source]
Construct InverseRTransform instance.
- Parameters
transform (BaseTransform) – One-dimension transformation instance.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(r)[source]
Compute the first derivative of inverse transformation.
\[\frac{dx}{dr} = (\frac{dr}{dx})^{-1}\]- Parameters
r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.
- Returns
First derivative of the inverse transformation at each point r.
- Return type
np.ndarray(N,)
- deriv2(r)[source]
Compute the second derivative of inverse transformation.
\[\frac{d^2 x}{dr^2} = - \frac{d^2 r}{dx^2} (\frac{dx}{dr})^3\]- Parameters
r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.
- Returns
Second derivative of the inverse transformation at each point r.
- Return type
np.ndarray(N,)
- deriv3(r)[source]
Compute the third derivative of inverse transformation.
\[\frac{d^3 x}{dr^3} = -\frac{d^3 r}{dx^3} (\frac{dx}{dr})^4 + 3 (\frac{d^2 r}{dx^2})^2 (\frac{dx}{dr})^5\]- Parameters
r (np.array(N,)) – One-dimensional array in the co-domain r of the original transformation.
- Returns
Third derivative of inverse transformation at each point r.
- Return type
np.ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(x)[source]
Transform array to the co-domain of the provided transformation.
This transformation is equivalent to the transformation function of the original transformation (i.e. OriginTF.transform).
- Parameters
x (np.ndarray) – One-dimension array in the domain x of the original transformation.
- Returns
One-dimensional array in the co-domain r of the original transformation.
- Return type
np.ndarray
- transform(r)[source]
Transform array back to the original, domain of the provided transformation.
This transformation is equivalent to the inverse function of the original transformation (i.e. OriginTF.inverse).
- Parameters
r (np.ndarray(N,)) – One-dimensional array in the co-domain r of the original transformation.
- Returns
Original one-dimensional array in the domain x of the original transformation.
- Return type
np.ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class KnowlesRTransform(rmin, R, k, trim_inf=True)[source]
Bases:
BaseTransformKnowles Transformation from \([-1, 1]\) to \([r_{min}, \infty)\).
The transformation is given by
\[r(x) = r_{min} - R \log \left( 1 - 2^{-k} (x + 1)^k \right),\]where \(k > 0\) and \(R\) is the scaling parameter.
The inverse transformation is given by
\[x(r) = 2 \sqrt[k]{1-\exp \left( -\frac{r-r_{min}}{R}\right)}-1\]- property R
The scale factor for the transformed radial array.
- Type
float
- __init__(rmin, R, k, trim_inf=True)[source]
Construct Knowles transformation class.
- Parameters
rmin (float) – The minimum coordinate for transformed radial array.
R (float) – The scale factor for transformed radial array.
k (integer k > 0) – Free parameter, k must be > 0.
trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of Knowles transformation.
\[\frac{dr}{dx} = kR \frac{(1+x_i)^{k-1}}{2^k-(1+x_i)^k}\]- Parameters
x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).
- Returns
The first derivative of Knowles transformation at each point.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of Knowles transformation.
\[\frac{d^2r}{dx^2} = kR \frac{(1+x_i)^{k-2} \left(2^k(k-1) + (1+x_i)^k \right)}{\left( 2^k-(1+x_i)^k\right)^2}\]- Parameters
x (ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The second derivative of Knowles transformation at each point.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of Knowles transformation.
\[\frac{d^3r}{dx^3} = kR \frac{(1+x_i)^{k-3} \left( 4^k (k-1) (k-2) + 2^k (k-1)(k+4)(1+x_i)^k +2(1+x_i)^k \right)}{\left( 2^k - (1+x_i)^k \right)^3}\]- Parameters
x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).
- Returns
The third derivative of Knowles transformation at each point.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Inverse of transformation from \([r_{min},\infty)\) to \([-1,1]\).
\[x_i = 2 \sqrt[k]{1-\exp \left( -\frac{r_i-r_{min}}{R}\right)}-1\]- Parameters
r (ndarray(N,)) – One-dimensional array in \([r_{min},\infty)\).
- Returns
The inverse transformation in \([-1,1]\).
- Return type
ndarray(N,)
- property k
Free and extra parameter, k must be > 0.
- Type
float
- property rmin
The minimum value for the transformed radial array.
- Type
float
- transform(x)[source]
Transform from \([-1,1]\) to \([r_{min},\infty)\).
\[r_i = r_{min} - R \log \left( 1 - 2^{-k} (x_i + 1)^k \right)\]- Parameters
x (np.ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
One dimensional array in \([r_{min},\infty)\).
- Return type
np.ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class LinearFiniteRTransform(rmin, rmax)[source]
Bases:
BaseTransformLinear finite transformation from \([-1, 1]\) to \([r_{min}, r_{max}]\).
The Linear transformation from finite interval \([-1, 1]\) to finite interval \([r_{min}, r_{max}]\) is given by
\[r(x) = \frac{r_{max} - r_{min}}{2} (1 + x) + r_{min}.\]The inverse transformation is given by
\[x(r) = \frac{2 r - (r_{max} + r_{min})}{r_{max} - r_{min}}\]- __init__(rmin, rmax)[source]
Construct linear transformation instance.
- Parameters
rmin (float) – Minimum value for transformed interval.
rmax (float) – Maximum value for transformed interval.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the 1st order derivative.
\[\frac{dr}{dx} = \frac{r_{max} - r_{min}}{2}\]- Parameters
x (ndarray) – One-dimensional array in the domain \([-1, 1]\).
- Returns
First order derivative at given points.
- Return type
float or np.ndarray
- deriv2(x)[source]
Compute the second order derivative.
\[\frac{d^2 r}{dx^2} = 0\]- Parameters
x (ndarray) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Second order derivative at given points.
- Return type
float or np.ndarray
- deriv3(x)[source]
Compute the third order derivative.
\[\frac{d^2 r}{dx^2} = 0\]- Parameters
x (ndarray) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Third order derivative at given points.
- Return type
ndarray
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of the transformation.
\[x_i = \frac{2 r_i - (r_{max} + r_{min})}{r_{max} - r_{min}}\]- Parameters
r (ndarray) – One-dimensional array in the co-domain \([r_{min}, r_{max}]\).
- Returns
One-dimensional array in the domain \([-1, 1]\).
- Return type
ndarray
- transform(x)[source]
Transform from interval \([-1, 1]\) to \([r_{min}, r_{max}]\).
\[r_i = \frac{r_{max} - r_{min}}{2} (1 + x_i) + r_{min}.\]This transformation maps \(r_i(-1) = r_{min}\) and \(r_i(1) = r_{max}\).
- Parameters
x (ndarray) – One-dimensional array in the domain \([-1, 1]\).
- Returns
Transformed points between \([r_{min}, r_{max}]\)
- Return type
float or np.ndarray
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class LinearInfiniteRTransform(rmin, rmax, b=None)[source]
Bases:
BaseTransformLinear transform from interval \([0, \infty)\) to \([r_{min}, r_{max})\).
This transformation linearly maps the infinite interval \([0, \infty)\) to a finite interval \([r_{min}, r_{max}]\) given by
\[r(x) = \frac{(r_{max} - r_{min})}{b} x + r_{min},\]where \(r(b) = r_{max}\). If None, then the \(b\) is taken to be the maximum from the first grid that is being transformed. This transformation always maps zero to \(r_{min}\).
The original goal is to transform the UniformGrid, equally-spaced integers from 0 to N-1, to \([r_{min}, r_{max}]\).
The inverse is given by
\[x(r) = (r - r_{min}) \frac{\max_i (r_i)}{r_{max} - r_{min}}\]- __init__(rmin, rmax, b=None)[source]
Initialize linear transform class.
- Parameters
rmin (float) – Define the lower end of the linear transform
rmax (float) – Define the upper end of the linear transform
b (float) – Maximum \(b\) of a prespecified radial grid \([0, b]\) such that \(b\) maps to rmax. If None, then the maximum is taken and stored from the grid that is transformed initially.
- property b
Parameter such that \(r(b) = r_{max}\).
- Type
float
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of linear transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).
- Returns
First derivative of transformation at x.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of linear transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).
- Returns
Second derivative of transformation at x.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of linear transformation.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).
- Returns
Third derivative of transformation at x.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of linear transformation.
\[x_i = (r - r_{min}) / frac{b}{r_{max} - r_{min}}\]- Parameters
r (ndarray(N,)) – One-dimensional array in the domain \([r_{min}, r_{max}]\).
- Returns
Inverse of transformation from coordinate \(r\) to \(x\).
- Return type
ndarray(N,)
- property rmax
rmax value of the tf.
- Type
float
- property rmin
rmin value of the tf.
- Type
float
- set_maximum_parameter_b(x)[source]
Sets up the parameter b from taken the maximum over some grid x.
- transform(x)[source]
Transform from interval \([0, \infty)\) to \([r_{min}, r_{max}]\).
\[r_i = \frac{(r_{max} - r_{min})}{\max_i x_i} x_i + r_{min},\]where \(N\) is the number of points. The goal is to transform equally-spaced integers from 0 to N-1, to \([r_{min}, r_{max}]\).
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain \([0, \infty)\).
- Returns
Transformed points between \([r_{min}, r_{max}]\).
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class MultiExpRTransform(rmin, R, trim_inf=True)[source]
Bases:
BaseTransformMultiExp Transformation class from \([-1,1]\) to \([r_{min}, \infty)\).
The transformation is given by
\[r(x) = -R \log \left( \frac{x + 1}{2} \right) + r_{min}\]The inverse of this transformation is given by
\[x(r) = 2 \exp \left( \frac{-(r - r_{min})}{R} \right) - 1\]References
- 2
Gill, Peter MW, and Siu-Hung Chien. “Radial quadrature for multiexponential integrands.” Journal of computational chemistry 24.6 (2003): 732-740.
- property R
The scale factor for the transformed radial array.
- Type
float
- __init__(rmin, R, trim_inf=True)[source]
Construct MultiExp transform from \([-1,1]\) to \([r_{min}, \infty)\).
- Parameters
rmin (float) – The minimum coordinate for transformed radial array.
R (float) – The scale factor for transformed radial array.
trim_inf (bool, optional) – Flag to trim infinite value in transformed array. If True, it will replace np.inf with 1e16. This may cause unexpected errors in the following operations.
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute the first derivative of MultiExp transformation.
\[\frac{dr}{dx} = -\frac{R}{1+x}\]- Parameters
x (ndarray(N,)) – One dimensional in \([-1, 1]\).
- Returns
The first derivative of MultiExp transformation at each point.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute the second derivative of MultiExp transformation.
\[\frac{d^2r}{dx^2} = \frac{R}{(1 + x)^2}\]- Parameters
x (ndarray(N,)) – One dimensional in \([-1,1]\).
- Returns
The second derivative of MultiExp transformation at each point.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute the third derivative of MultiExp transformation.
\[\frac{d^3r}{dx^3} = -\frac{2R}{(1 + x)^3}\]- Parameters
x (ndarray(N,)) – One dimensional array in \([-1,1]\).
- Returns
The third derivative of MultiExp transformation at each point.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Inverse of transform from \([r_{min},\infty)\) to \([-1,1]\).
\[x_i = 2 \exp \left( \frac{-(r_i - r_{min})}{R} \right) - 1\]- Parameters
r (np.ndarray(N,)) – One-dimensional array in \([r_{min}, \infty)\).
- Returns
The inverse of transformation in \([-1, 1]\).
- Return type
np.ndarray(N,)
- property rmin
The minimum value for the transformed radial array.
- Type
float
- transform(x)[source]
Transform from [-1,1] to \([r_{min},\infty)\).
\[r_i = -R \log \left( \frac{x_i + 1}{2} \right) + r_{min}\]- Parameters
x (ndarray(N,)) – One dimensional array with values between \([-1,1]\).
- Returns
Transformed array located between \([r_{min},\infty)\).
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]
- class PowerRTransform(rmin, rmax, b=None)[source]
Bases:
BaseTransformPower transform class from \([0, \infty)\) to \([r_{min}, r_{max}]\).
This transformations is given by
\[r(x) = r_{min} (x + 1)^{\frac{\log(r_{max} - \log(r_{min}}{\log(b + 1)}},\]such that \(r(b) = r_{max}\).
The inverse of the transformation is given by
\[x(r) = \frac{r}{r_{min}}^{\frac{\log(N)}{\log(r_{max}) - \log(r_{min})}} - 1.\]- __init__(rmin, rmax, b=None)[source]
Initialize power transform instance.
- Parameters
rmin (float) – Minimum value for transformed points
rmax (float) – Maximum value for transformed points
b (float) – The parameter b that maps to \(r_{max}\).
- property b
Parameter \(b\) that maps/transforms to \(r_{max}\).
- Type
float
- property codomain
Transformation codomain.
- Type
tuple
- deriv(x)[source]
Compute first derivative of power transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
First derivative of transformation at x.
- Return type
ndarray(N,)
- deriv2(x)[source]
Compute second derivative of power transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
Second derivative of transformation at x.
- Return type
ndarray(N,)
- deriv3(x)[source]
Compute third derivative of power transform.
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
Third derivative of transformation at x.
- Return type
ndarray(N,)
- property domain
Transformation domain.
- Type
tuple
- inverse(r)[source]
Compute the inverse of power transform.
\[x(r) = \frac{r}{r_{min}}^{\frac{\log(b + 1)}{\log(r_{max}) - \log(r_{min})}} - 1\]such that \(r(b) = r_{max}\).
- Parameters
r (ndarray(N,)) – One-dimensional array in the co-domain of the transformation.
- Returns
Inverse of transformation at r.
- Return type
ndarray(N,)
- property rmax
the value of rmax.
- Type
float
- property rmin
the value of rmin.
- Type
float
- transform(x)[source]
Perform power transform.
\[r = r_{min} (x + 1)^{\frac{\log(r_{max} - \log(r_{min}}{b + 1}},\]such that \(r(b) = r_{max}\).
- Parameters
x (ndarray(N,)) – One-dimensional array in the domain of the transformation \([0,\infty)\).
- Returns
The transformation of x to the co-domain of the transformation.
- Return type
ndarray(N,)
- transform_1d_grid(oned_grid)[source]
Generate a new integral grid by transforming the provided grid.
\[\begin{split}\int^{\inf}_0 g(r) d r &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &= \int^{r(\infty)}_{r(0)} g(r(x)) \frac{dr}{dx} dx \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w_i \\ &\approx \sum_{i=1}^N g(r(x_i)) \frac{dr}{dx}(x_i) w^r_n \\ w^r_n &= w^x_n \cdot \frac{dr}{dx}\end{split}\]