chebyshev module¶
Chebyshev interpolation and integration in 1, 2, and 4 dimensions
Note
if the math looks strange in the documentation, just reload the page.
Interval
,Rectangle
: basic classes to define the integration domainmove_from1m1, move_to1m1
: rescale to and from the \([-1,1]\) intervalcheb_get_nodes_1d
: get Chebyshev nodes and weights on an intervalcheb_eval_fun_at_nodes_1d
: evaluates a function at is nodes on an intervalcheb_get_coefficients_1d
: get the Chebyshev coefficients for a functioncheb_interp_1d
: interpolate a function on an interval given its definition or its coefficientscheb_interp_1d_from_nodes
: interpolate a function on an interval given its values at the nodescheb_find_root
: finds the roots of a function in an intervalcheb_integrate_from_coeffs_1d
: integrates a function given its coefficientscheb_integrate_from_nodes_1d
: integrates a function given its values at the nodes (less precise)cheb_get_nodes_2d
: get Chebyshev nodes and weights on a rectanglecheb_eval_fun_at_nodes_2d
: evaluates a function at is nodes on a rectanglecheb_get_coefficients_2d
: get the Chebyshev coefficients for a function of 2 argumentscheb_interp_2d
: interpolate a function on a rectangle given its definition or its coefficientscheb_interp_2d_from_nodes
: interpolate a function on a rectangle given its values at the nodescheb_integrate_from_nodes_4d
: integrate over a product of rectangles given values at the tensor products of the 2d nodes.
Interval
dataclass
¶
a real interval \([x_0,x_1]\)
Source code in bs_python_utils/chebyshev.py
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|
Rectangle
dataclass
¶
a product interval \([x_0,x_1] imes [y_0, y_1]\)
Source code in bs_python_utils/chebyshev.py
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cheb_eval_fun_at_nodes_1d(fun, nodes=None, interval=None, degree=None)
¶
evaluate a function at the Chebyshev nodes on an interval
Parameters:
Name | Type | Description | Default |
---|---|---|---|
fun
|
ArrayFunctionOfArray
|
the function to evaluate on an interval |
required |
nodes
|
ndarray | None
|
the Chebyshev nodes on that interval, if precomputed |
None
|
interval
|
Interval | None
|
the Interval |
None
|
degree
|
int | None
|
number of Chebyshev nodes used for evaluation |
None
|
Notes
interval
, degree
are required if nodes
is not provided
Returns:
Type | Description |
---|---|
ndarray
|
the values of the function at the Chebyshev nodes |
Source code in bs_python_utils/chebyshev.py
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cheb_eval_fun_at_nodes_2d(fun, nodes=None, rectangle=None, degree=None)
¶
evaluate a function at the Chebyshev nodes on a rectangle $
Parameters:
Name | Type | Description | Default |
---|---|---|---|
fun
|
ArrayFunctionOfArray
|
the function to evaluate on the rectangle |
required |
nodes
|
ndarray | None
|
the Chebyshev nodes on that rectangle, if precomputed |
None
|
rectangle
|
Rectangle | None
|
the Rectangle |
None
|
degree
|
int | None
|
number of Chebyshev nodes in each dimension |
None
|
Notes
rectangle
and degree
are required if nodes
is not provided
Returns:
Type | Description |
---|---|
ndarray
|
the values of the function at the Chebyshev nodes |
Source code in bs_python_utils/chebyshev.py
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cheb_find_root(f, degree, interval=None)
¶
find the roots of \(f(x)=0\) in \([0,1]\); also return the one(s) within the interval, if given
Parameters:
Name | Type | Description | Default |
---|---|---|---|
f
|
ArrayFunctionOfArray
|
the function |
required |
degree
|
int
|
the degree of the Chebyshev expansion |
required |
interval
|
Interval | None
|
the interval where we want the root |
None
|
Returns:
Type | Description |
---|---|
ndarray | tuple[ndarray, ndarray | float | None]
|
the roots in \([0,1]\); and the one(s) in |
Source code in bs_python_utils/chebyshev.py
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|
cheb_get_coefficients_1d(fun, interval, degree)
¶
get the Chebyshev coefficients for fun
on an interval
Parameters:
Name | Type | Description | Default |
---|---|---|---|
fun
|
ArrayFunctionOfArray
|
the function |
required |
interval
|
Interval
|
the Interval |
required |
degree
|
int
|
the degree of the Chebyshev expansion |
required |
Returns:
Type | Description |
---|---|
ndarray
|
a |
Source code in bs_python_utils/chebyshev.py
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|
cheb_get_coefficients_2d(rectangle, degree, vals_at_nodes=None, fun=None)
¶
get the Chebyshev coefficients for fun
on a rectangle,
using an OLS fit on the values on the grid of nodes
Parameters:
Name | Type | Description | Default |
---|---|---|---|
rectangle
|
Rectangle
|
the Rectangle |
required |
degree
|
int
|
the number of Chebyshev nodes per dimension |
required |
vals_at_nodes
|
ndarray | None
|
the values on the grid, if precomputed (either length |
None
|
fun
|
ArrayFunctionOfArray | None
|
the function |
None
|
Notes
if vals_at_nodes
is not provided then fun
must be.
Returns:
Type | Description |
---|---|
ndarray
|
the Chebyshev coefficients of the OLS Chebyshev fit, an |
ndarray
|
the approximation is \(f(x_1,x_2) = \sum_{k,l} c_{kl} T_k(x_1)T_l(x_2)\) |
Source code in bs_python_utils/chebyshev.py
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|
cheb_get_nodes_1d(interval, n_nodes)
¶
get the Chebyshev nodes and weights on the interval \([x0, x1]\)
Parameters:
Name | Type | Description | Default |
---|---|---|---|
interval
|
Interval
|
the Interval \([x_0, x_1]\) |
required |
n_nodes
|
int
|
number of Chebyshev nodes used for quadrature |
required |
Returns:
Type | Description |
---|---|
TwoArrays
|
two |
TwoArrays
|
so that |
Source code in bs_python_utils/chebyshev.py
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cheb_get_nodes_2d(rectangle, n_nodes)
¶
get the Chebyshev nodes and weights on a rectangle
Parameters:
Name | Type | Description | Default |
---|---|---|---|
rectangle
|
Rectangle
|
the Rectangle |
required |
n_nodes
|
int
|
number of Chebyshev nodes per dimension |
required |
Returns:
Type | Description |
---|---|
TwoArrays
|
two \(( ext{n_nodes}^2, 2)\)-matrices of Chebyshev nodes and weights |
TwoArrays
|
on the rectangle \([x0, x1] imes [y0, y1]\) |
Source code in bs_python_utils/chebyshev.py
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cheb_integrate_from_coeffs_1d(c, interval)
¶
integrate a function on an interval using the coefficients of its Chebyshev expansion
Parameters:
Name | Type | Description | Default |
---|---|---|---|
c
|
ndarray
|
the Chebyshev coefficients for |
required |
interval
|
Interval
|
the Interval |
required |
Returns:
Type | Description |
---|---|
float
|
the value of the integral over the interval |
Source code in bs_python_utils/chebyshev.py
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|
cheb_integrate_from_coeffs_2d(c, rectangle)
¶
integrate a function on an interval using the coefficients of its Chebyshev expansion
Parameters:
Name | Type | Description | Default |
---|---|---|---|
c
|
ndarray
|
the Chebyshev coefficients for |
required |
rectangle
|
Rectangle
|
the Rectangle |
required |
Returns:
Type | Description |
---|---|
float
|
the value of the integral over the interval |
Source code in bs_python_utils/chebyshev.py
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cheb_integrate_from_nodes_1d(vals_at_nodes, weights)
¶
integrate a function given its values at the Chebyshev nodes
Parameters:
Name | Type | Description | Default |
---|---|---|---|
vals_at_nodes
|
ndarray
|
the values of the function at these nodes |
required |
weights
|
ndarray
|
the Chebyshev nodes |
required |
Returns:
Type | Description |
---|---|
float
|
the value of the integral |
Notes
this is much less precise than cheb_integrate_from_coeffs_1d
Source code in bs_python_utils/chebyshev.py
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|
cheb_integrate_from_nodes_2d(vals_at_nodes, weights)
¶
integrate a function given its values at the Chebyshev nodes
Parameters:
Name | Type | Description | Default |
---|---|---|---|
vals_at_nodes
|
ndarray
|
the values of the function on the grid of 2d nodes |
required |
weights
|
ndarray
|
the Chebyshev weights in 2d |
required |
Returns:
Type | Description |
---|---|
float
|
the value of the integral |
Warning
this is much less precise than cheb_integrate_from_coeffs_2d
Source code in bs_python_utils/chebyshev.py
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cheb_integrate_from_nodes_4d(vals_at_nodes4d, weights2d)
¶
integrate a function on the square of a rectangle given its values at the 4d Chebyshev nodes
Parameters:
Name | Type | Description | Default |
---|---|---|---|
vals_at_nodes4d
|
ndarray
|
the values of the function on the square of the grid of 2d nodes, an \((M^2, M^2)\) matrix |
required |
weights2d
|
ndarray
|
the Chebyshev weights on the rectangular grid, an \(M^2\)-vector |
required |
Returns:
Type | Description |
---|---|
float
|
the value of the integral |
Warning
it would be better to have a cheb_integrate_from_coeffs_4d
Source code in bs_python_utils/chebyshev.py
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cheb_interp_1d(x_vals, interval, c=None, fun=None, degree=None)
¶
interpolate a function on on interval using Chebyshev polynomials
Parameters:
Name | Type | Description | Default |
---|---|---|---|
x_vals
|
ndarray
|
the values at which to interpolate |
required |
interval
|
Interval
|
the Interval |
required |
c
|
ndarray | None
|
the Chebyshev coefficients for |
None
|
fun
|
ArrayFunctionOfArray | None
|
the function to interpolate |
None
|
degree
|
int | None
|
number of Chebyshev nodes per dimension (required if |
None
|
Notes
fun
and degree
are required if c
is not provided
Returns:
Type | Description |
---|---|
TwoArrays
|
the values of the interpolation at |
Source code in bs_python_utils/chebyshev.py
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cheb_interp_1d_from_nodes(f_vals_at_nodes, x, interval=None)
¶
interpolate \(f(x)\) given the values \(f(x_m)\) for \(m=1,\ldots,M^2\) at the Chebyshev nodes on an intervak
Parameters:
Name | Type | Description | Default |
---|---|---|---|
f_vals_at_nodes
|
ndarray
|
an \(M^2\) vector of values \(f(x_m)\) |
required |
x
|
ndarray
|
a scalar where we want \(f(x)\) |
required |
interval
|
Interval | None
|
the interval on which the function acts; by default, \([0,1]\) |
None
|
Returns:
Type | Description |
---|---|
float
|
the interpolated value of \(f(x)\). |
Source code in bs_python_utils/chebyshev.py
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cheb_interp_2d(xy_vals, rectangle, c=None, fun=None, degree=None, vals_at_nodes=None)
¶
Interpolate a function on a rectangle using Chebyshev polynomials.
Parameters:
Name | Type | Description | Default |
---|---|---|---|
xy_vals
|
ndarray
|
Evaluation points, either an |
required |
rectangle
|
Rectangle
|
Domain rectangle. |
required |
c
|
ndarray | None
|
Chebyshev coefficient matrix for the function, if already available. |
None
|
fun
|
ArrayFunctionOfArray | None
|
Callable used to compute coefficients when |
None
|
degree
|
int | None
|
Number of Chebyshev nodes per dimension (required if |
None
|
vals_at_nodes
|
ndarray | None
|
Optional function values on the tensor grid when |
None
|
Notes
degree
is required if c
is not supplied, alongside either fun
or
vals_at_nodes
.
Returns:
Type | Description |
---|---|
TwoArrays | tuple[float, ndarray]
|
A pair |
TwoArrays | tuple[float, ndarray]
|
scalar is returned when |
TwoArrays | tuple[float, ndarray]
|
Chebyshev coefficient matrix used for the evaluation. |
Source code in bs_python_utils/chebyshev.py
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cheb_interp_2d_from_nodes(f_vals_at_nodes, x, rectangle=None)
¶
interpolate \(f(x)\) given the values \(f(x_m)\) for \(m=1,\ldots,M^2\) at the Chebyshev nodes on a rectangle
Parameters:
Name | Type | Description | Default |
---|---|---|---|
f_vals_at_nodes
|
ndarray
|
an \(M^2\) vector of values \(f(x_m)\) |
required |
x
|
ndarray
|
a 2-vector where we want \(f(x)\) |
required |
rectangle
|
Rectangle | None
|
the rectangle on which the function acts; by default, \([0,1]^2\) |
None
|
Returns:
Type | Description |
---|---|
float
|
the interpolated value of \(f(x)\). |
Source code in bs_python_utils/chebyshev.py
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move_from1m1(t, interval)
¶
get the position of t
in \([-1,1]\) and move it in the same position in interval
Parameters:
Name | Type | Description | Default |
---|---|---|---|
t
|
FloatOrArray
|
position(s) within `\([-1,1]\) |
required |
interval
|
Interval
|
the Interval |
required |
Returns:
Type | Description |
---|---|
FloatOrArray
|
|
Source code in bs_python_utils/chebyshev.py
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|
move_to1m1(x, interval)
¶
get the position of x
in interval
and move it to the same position in \([-1,1]\)
Parameters:
Name | Type | Description | Default |
---|---|---|---|
x
|
FloatOrArray
|
position(s) within |
required |
interval
|
Interval
|
the Interval |
required |
Returns:
Type | Description |
---|---|
FloatOrArray
|
|
Source code in bs_python_utils/chebyshev.py
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