diff options
Diffstat (limited to 'numpy')
| -rw-r--r-- | numpy/polynomial/chebyshev.py | 102 | ||||
| -rw-r--r-- | numpy/polynomial/hermite.py | 101 | ||||
| -rw-r--r-- | numpy/polynomial/hermite_e.py | 100 | ||||
| -rw-r--r-- | numpy/polynomial/laguerre.py | 100 | ||||
| -rw-r--r-- | numpy/polynomial/legendre.py | 100 | ||||
| -rw-r--r-- | numpy/polynomial/polynomial.py | 104 |
6 files changed, 589 insertions, 18 deletions
diff --git a/numpy/polynomial/chebyshev.py b/numpy/polynomial/chebyshev.py index 6212f2bc5..d6ccf25ca 100644 --- a/numpy/polynomial/chebyshev.py +++ b/numpy/polynomial/chebyshev.py @@ -40,6 +40,8 @@ Misc Functions - `chebfromroots` -- create a Chebyshev series with specified roots. - `chebroots` -- find the roots of a Chebyshev series. - `chebvander` -- Vandermonde-like matrix for Chebyshev polynomials. +- `chebvander2d` -- Vandermonde-like matrix for 2D power series. +- `chebvander3d` -- Vandermonde-like matrix for 3D power series. - `chebfit` -- least-squares fit returning a Chebyshev series. - `chebpts1` -- Chebyshev points of the first kind. - `chebpts2` -- Chebyshev points of the second kind. @@ -90,10 +92,10 @@ from polytemplate import polytemplate __all__ = ['chebzero', 'chebone', 'chebx', 'chebdomain', 'chebline', 'chebadd', 'chebsub', 'chebmulx', 'chebmul', 'chebdiv', 'chebpow', - 'chebval', 'chebval2d', 'chebval3d', 'chebgrid2d', 'chebgrid3d', - 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', 'chebfromroots', - 'chebvander', 'chebfit', 'chebtrim', 'chebroots', 'chebpts1', - 'chebpts2', 'Chebyshev'] + 'chebval', 'chebder', 'chebint', 'cheb2poly', 'poly2cheb', + 'chebfromroots', 'chebvander', 'chebfit', 'chebtrim', 'chebroots', + 'chebpts1', 'chebpts2', 'Chebyshev', 'chebval2d', 'chebval3d', + 'chebgrid2d', 'chebgrid3d', 'chebvander2d','chebvander3d'] chebtrim = pu.trimcoef @@ -1315,6 +1317,98 @@ def chebvander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def chebvander2d(x, y, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D Chebyshev series in `x` + and `y`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x` or 'y' are + complex. The maximum degrees of the 2D Chebyshev series in each + variable are specified in the list `deg` in the form ``[xdeg, ydeg]``. + The return array has the shape ``x.shape + (order,)`` if `x`, and `y` + are arrays or ``(1, order) if they are scalars. Here order is the + number of elements in a flattened coefficient array of original shape + ``(xdeg + 1, ydeg + 1)``. The flattening is done so that the resulting + pseudo Vandermonde array can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + chebvander, chebvander3d. chebval2d, chebval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = chebvander(x, degx) + vy = chebvander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def chebvander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D Chebyshev series in `x`, + `y`, or `z`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x`, `y`, or 'z' are + complex. The maximum degrees of the 3D Chebeshev series in each + variable are specified in the list `deg` in the form ``[xdeg, ydeg, + zdeg]``. The return array has the shape ``x.shape + (order,)`` if `x`, + `y`, and `z` are arrays or ``(1, order) if they are scalars. Here order + is the number of elements in a flattened coefficient array of original + shape ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so + that the resulting pseudo Vandermonde array can be easily used in least + squares fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + chebvander, chebvander3d. chebval2d, chebval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = chebvander(x, deg_x) + vy = chebvander(y, deg_y) + vz = chebvander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def chebfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Chebyshev series to data. diff --git a/numpy/polynomial/hermite.py b/numpy/polynomial/hermite.py index 2fd28a3ff..801e8f8d9 100644 --- a/numpy/polynomial/hermite.py +++ b/numpy/polynomial/hermite.py @@ -37,6 +37,8 @@ Misc Functions - `hermfromroots` -- create a Hermite series with specified roots. - `hermroots` -- find the roots of a Hermite series. - `hermvander` -- Vandermonde-like matrix for Hermite polynomials. +- `hermvander2d` -- Vandermonde-like matrix for 2D power series. +- `hermvander3d` -- Vandermonde-like matrix for 3D power series. - `hermfit` -- least-squares fit returning a Hermite series. - `hermtrim` -- trim leading coefficients from a Hermite series. - `hermline` -- Hermite series of given straight line. @@ -62,9 +64,10 @@ from polytemplate import polytemplate __all__ = ['hermzero', 'hermone', 'hermx', 'hermdomain', 'hermline', 'hermadd', 'hermsub', 'hermmulx', 'hermmul', 'hermdiv', 'hermpow', - 'hermval', 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', - 'hermder', 'hermint', 'herm2poly', 'poly2herm', 'hermfromroots', - 'hermvander', 'hermfit', 'hermtrim', 'hermroots', 'Hermite'] + 'hermval', 'hermder', 'hermint', 'herm2poly', 'poly2herm', + 'hermfromroots', 'hermvander', 'hermfit', 'hermtrim', 'hermroots', + 'Hermite', 'hermval2d', 'hermval3d', 'hermgrid2d', 'hermgrid3d', + 'hermvander2d', 'hermvander3d'] hermtrim = pu.trimcoef @@ -1102,6 +1105,98 @@ def hermvander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def hermvander2d(x, y, deg) : + """Pseudo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D Hermite series in `x` and + `y`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x` or 'y' are complex. The + maximum degrees of the 2D Hermite series in each variable are specified + in the list `deg` in the form ``[xdeg, ydeg]``. The return array has + the shape ``x.shape + (order,)`` if `x`, and `y` are arrays or + ``(1, order) if they are scalars. Here order is the number of elements + in a flattened coefficient array of original shape ``(xdeg + 1, ydeg + + 1)``. The flattening is done so that the resulting pseudo Vandermonde + array can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + hermvander, hermvander3d. hermval2d, hermval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = hermvander(x, degx) + vy = hermvander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def hermvander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D Hermite series in `x`, + `y`, or `z`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x`, `y`, or 'z' are + complex. The maximum degrees of the 3D Hermite series in each variable + are specified in the list `deg` in the form ``[xdeg, ydeg, zdeg]``. The + return array has the shape ``x.shape + (order,)`` if `x`, `y`, and `z` + are arrays or ``(1, order) if they are scalars. Here order is the + number of elements in a flattened coefficient array of original shape + ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so that the + resulting pseudo Vandermonde array can be easily used in least squares + fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + hermvander, hermvander3d. hermval2d, hermval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = hermvander(x, deg_x) + vy = hermvander(y, deg_y) + vz = hermvander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def hermfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Hermite series to data. diff --git a/numpy/polynomial/hermite_e.py b/numpy/polynomial/hermite_e.py index 17e285e15..8307cf80c 100644 --- a/numpy/polynomial/hermite_e.py +++ b/numpy/polynomial/hermite_e.py @@ -37,6 +37,8 @@ Misc Functions - `hermefromroots` -- create a Hermite_e series with specified roots. - `hermeroots` -- find the roots of a Hermite_e series. - `hermevander` -- Vandermonde-like matrix for Hermite_e polynomials. +- `hermevander2d` -- Vandermonde-like matrix for 2D power series. +- `hermevander3d` -- Vandermonde-like matrix for 3D power series. - `hermefit` -- least-squares fit returning a Hermite_e series. - `hermetrim` -- trim leading coefficients from a Hermite_e series. - `hermeline` -- Hermite_e series of given straight line. @@ -62,9 +64,11 @@ from polytemplate import polytemplate __all__ = ['hermezero', 'hermeone', 'hermex', 'hermedomain', 'hermeline', 'hermeadd', 'hermesub', 'hermemulx', 'hermemul', 'hermediv', 'hermpow', - 'legval', 'legval2d', 'legval3d', 'leggrid2d', 'leggrid3d', + 'hermeval', 'hermeder', 'hermeint', 'herme2poly', 'poly2herme', 'hermefromroots', - 'hermevander', 'hermefit', 'hermetrim', 'hermeroots', 'HermiteE'] + 'hermevander', 'hermefit', 'hermetrim', 'hermeroots', 'HermiteE', + 'hermeval2d', 'hermeval3d', 'hermegrid2d', 'hermegrid3d', 'hermevander2d', + 'hermevander3d'] hermetrim = pu.trimcoef @@ -1097,6 +1101,98 @@ def hermevander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def hermevander2d(x, y, deg) : + """Pseudo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D Hermite_e series in `x` + and `y`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x` or 'y' are + complex. The maximum degrees of the 2D Hermite_e series in each + variable are specified in the list `deg` in the form ``[xdeg, ydeg]``. + The return array has the shape ``x.shape + (order,)`` if `x`, and `y` + are arrays or ``(1, order) if they are scalars. Here order is the + number of elements in a flattened coefficient array of original shape + ``(xdeg + 1, ydeg + 1)``. The flattening is done so that the resulting + pseudo Vandermonde array can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + hermevander, hermevander3d. hermeval2d, hermeval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = hermevander(x, degx) + vy = hermevander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def hermevander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D Hermite_e series in `x`, + `y`, or `z`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x`, `y`, or 'z' are + complex. The maximum degrees of the 3D Hermite_e series in each + variable are specified in the list `deg` in the form ``[xdeg, ydeg, + zdeg]``. The return array has the shape ``x.shape + (order,)`` if `x`, + `y`, and `z` are arrays or ``(1, order) if they are scalars. Here order + is the number of elements in a flattened coefficient array of original + shape ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so + that the resulting pseudo Vandermonde array can be easily used in least + squares fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + hermevander, hermevander3d. hermeval2d, hermeval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = hermevander(x, deg_x) + vy = hermevander(y, deg_y) + vz = hermevander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def hermefit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Hermite series to data. diff --git a/numpy/polynomial/laguerre.py b/numpy/polynomial/laguerre.py index dbbb2c401..39463b0cb 100644 --- a/numpy/polynomial/laguerre.py +++ b/numpy/polynomial/laguerre.py @@ -37,6 +37,8 @@ Misc Functions - `lagfromroots` -- create a Laguerre series with specified roots. - `lagroots` -- find the roots of a Laguerre series. - `lagvander` -- Vandermonde-like matrix for Laguerre polynomials. +- `lagvander2d` -- Vandermonde-like matrix for 2D power series. +- `lagvander3d` -- Vandermonde-like matrix for 3D power series. - `lagfit` -- least-squares fit returning a Laguerre series. - `lagtrim` -- trim leading coefficients from a Laguerre series. - `lagline` -- Laguerre series of given straight line. @@ -62,9 +64,9 @@ from polytemplate import polytemplate __all__ = ['lagzero', 'lagone', 'lagx', 'lagdomain', 'lagline', 'lagadd', 'lagsub', 'lagmulx', 'lagmul', 'lagdiv', 'lagpow', - 'lagval', 'lagval2d', 'lagval3d', 'laggrid2d', 'laggrid3d', - 'lagder', 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', - 'lagvander', 'lagfit', 'lagtrim', 'lagroots', 'Laguerre'] + 'lagval', 'lagder', 'lagint', 'lag2poly', 'poly2lag', 'lagfromroots', + 'lagvander', 'lagfit', 'lagtrim', 'lagroots', 'Laguerre', 'lagval2d', + 'lagval3d', 'laggrid2d', 'laggrid3d', 'lagvander2d', 'lagvander3d'] lagtrim = pu.trimcoef @@ -1100,6 +1102,98 @@ def lagvander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def lagvander2d(x, y, deg) : + """Pseudo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D Laguerre series in `x` and + `y`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x` or 'y' are complex. The + maximum degrees of the 2D Laguerre series in each variable are specified in + the list `deg` in the form ``[xdeg, ydeg]``. The return array has the + shape ``x.shape + (order,)`` if `x`, and `y` are arrays or ``(1, order) + if they are scalars. Here order is the number of elements in a + flattened coefficient array of original shape ``(xdeg + 1, ydeg + 1)``. + The flattening is done so that the resulting pseudo Vandermonde array + can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + lagvander, lagvander3d. lagval2d, lagval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = lagvander(x, degx) + vy = lagvander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def lagvander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D Laguerre series in `x`, + `y`, or `z`. The sample point coordinates must all have the same shape + after conversion to arrays and the dtype will be converted to either + float64 or complex128 depending on whether any of `x`, `y`, or 'z' are + complex. The maximum degrees of the 3D Laguerre series in each + variable are specified in the list `deg` in the form ``[xdeg, ydeg, + zdeg]``. The return array has the shape ``x.shape + (order,)`` if `x`, + `y`, and `z` are arrays or ``(1, order) if they are scalars. Here order + is the number of elements in a flattened coefficient array of original + shape ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so + that the resulting pseudo Vandermonde array can be easily used in least + squares fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + lagvander, lagvander3d. lagval2d, lagval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = lagvander(x, deg_x) + vy = lagvander(y, deg_y) + vz = lagvander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def lagfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Laguerre series to data. diff --git a/numpy/polynomial/legendre.py b/numpy/polynomial/legendre.py index 5a72217d0..00dbacebe 100644 --- a/numpy/polynomial/legendre.py +++ b/numpy/polynomial/legendre.py @@ -38,6 +38,8 @@ Misc Functions - `legfromroots` -- create a Legendre series with specified roots. - `legroots` -- find the roots of a Legendre series. - `legvander` -- Vandermonde-like matrix for Legendre polynomials. +- `legvander2d` -- Vandermonde-like matrix for 2D power series. +- `legvander3d` -- Vandermonde-like matrix for 3D power series. - `legfit` -- least-squares fit returning a Legendre series. - `legtrim` -- trim leading coefficients from a Legendre series. - `legline` -- Legendre series representing given straight line. @@ -62,10 +64,10 @@ import warnings from polytemplate import polytemplate __all__ = ['legzero', 'legone', 'legx', 'legdomain', 'legline', - 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', - 'legval', 'legval2d', 'legval3d', 'leggrid2d', 'leggrid3d', + 'legadd', 'legsub', 'legmulx', 'legmul', 'legdiv', 'legpow', 'legval', 'legder', 'legint', 'leg2poly', 'poly2leg', 'legfromroots', - 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre'] + 'legvander', 'legfit', 'legtrim', 'legroots', 'Legendre','legval2d', + 'legval3d', 'leggrid2d', 'leggrid3d', 'legvander2d', 'legvander3d'] legtrim = pu.trimcoef @@ -1103,6 +1105,98 @@ def legvander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def legvander2d(x, y, deg) : + """Pseudo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D Legendre series in `x` and + `y`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x` or 'y' are complex. The + maximum degrees of the 2D Legendre series in each variable are specified in + the list `deg` in the form ``[xdeg, ydeg]``. The return array has the + shape ``x.shape + (order,)`` if `x`, and `y` are arrays or ``(1, order) + if they are scalars. Here order is the number of elements in a + flattened coefficient array of original shape ``(xdeg + 1, ydeg + 1)``. + The flattening is done so that the resulting pseudo Vandermonde array + can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + legvander, legvander3d. legval2d, legval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = legvander(x, degx) + vy = legvander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def legvander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D Legendre series in `x`, `y`, + or `z`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x`, `y`, or 'z' are complex. + The maximum degrees of the 3D Legendre series in each variable are + specified in the list `deg` in the form ``[xdeg, ydeg, zdeg]``. The + return array has the shape ``x.shape + (order,)`` if `x`, `y`, and `z` + are arrays or ``(1, order) if they are scalars. Here order is the + number of elements in a flattened coefficient array of original shape + ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so that the + resulting pseudo Vandermonde array can be easily used in least squares + fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + legvander, legvander3d. legval2d, legval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = legvander(x, deg_x) + vy = legvander(y, deg_y) + vz = legvander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def legfit(x, y, deg, rcond=None, full=False, w=None): """ Least squares fit of Legendre series to data. diff --git a/numpy/polynomial/polynomial.py b/numpy/polynomial/polynomial.py index a0c9f903a..bf9c1a216 100644 --- a/numpy/polynomial/polynomial.py +++ b/numpy/polynomial/polynomial.py @@ -37,6 +37,8 @@ Misc Functions - `polyfromroots` -- create a polynomial with specified roots. - `polyroots` -- find the roots of a polynomial. - `polyvander` -- Vandermonde-like matrix for powers. +- `polyvander2d` -- Vandermonde-like matrix for 2D power series. +- `polyvander3d` -- Vandermonde-like matrix for 3D power series. - `polyfit` -- least-squares fit returning a polynomial. - `polytrim` -- trim leading coefficients from a polynomial. - `polyline` -- polynomial representing given straight line. @@ -54,9 +56,9 @@ from __future__ import division __all__ = ['polyzero', 'polyone', 'polyx', 'polydomain', 'polyline', 'polyadd', 'polysub', 'polymulx', 'polymul', 'polydiv', 'polypow', - 'polyval', 'polyval2d', 'polyval3d', 'polygrid2d', 'polygrid3d', - 'polyder', 'polyint', 'polyfromroots', 'polyvander', 'polyfit', - 'polytrim', 'polyroots', 'Polynomial'] + 'polyval', 'polyder', 'polyint', 'polyfromroots', 'polyvander', + 'polyfit', 'polytrim', 'polyroots', 'Polynomial','polyval2d', + 'polyval3d', 'polygrid2d', 'polygrid3d', 'polyvander2d','polyvander3d'] import numpy as np import numpy.linalg as la @@ -891,6 +893,10 @@ def polyvander(x, deg) : The shape of the returned matrix is ``x.shape + (deg+1,)``. The last index is the degree. + See Also + -------- + polyvander2d, polyvander3d + """ ideg = int(deg) if ideg != deg: @@ -908,6 +914,98 @@ def polyvander(x, deg) : return np.rollaxis(v, 0, v.ndim) +def polyvander2d(x, y, deg) : + """Pseudo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 2D polynomials in `x` and + `y`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x` or 'y' are complex. The + maximum degrees of the 2D polynomials in each variable are specified in + the list `deg` in the form ``[xdeg, ydeg]``. The return array has the + shape ``x.shape + (order,)`` if `x`, and `y` are arrays or ``(1, order) + if they are scalars. Here order is the number of elements in a + flattened coefficient array of original shape ``(xdeg + 1, ydeg + 1)``. + The flattening is done so that the resulting pseudo Vandermonde array + can be easily used in least squares fits. + + Parameters + ---------- + x,y : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg]. + + Returns + ------- + vander2d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + polyvander, polyvander3d. polyval2d, polyval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy = deg + x, y = np.array((x, y), copy=0) + 0.0 + + vx = polyvander(x, degx) + vy = polyvander(y, degy) + v = np.einsum("...i,...j->...ij", vx, vy) + return v.reshape(v.shape[:-2] + (-1,)) + + +def polyvander3d(x, y, z, deg) : + """Psuedo Vandermonde matrix of given degree. + + Returns the pseudo Vandermonde matrix for 3D polynomials in `x`, `y`, + or `z`. The sample point coordinates must all have the same shape after + conversion to arrays and the dtype will be converted to either float64 + or complex128 depending on whether any of `x`, `y`, or 'z' are complex. + The maximum degrees of the 3D polynomials in each variable are + specified in the list `deg` in the form ``[xdeg, ydeg, zdeg]``. The + return array has the shape ``x.shape + (order,)`` if `x`, `y`, and `z` + are arrays or ``(1, order) if they are scalars. Here order is the + number of elements in a flattened coefficient array of original shape + ``(xdeg + 1, ydeg + 1, zdeg + 1)``. The flattening is done so that the + resulting pseudo Vandermonde array can be easily used in least squares + fits. + + Parameters + ---------- + x,y,z : array_like + Arrays of point coordinates, each of the same shape. + deg : list + List of maximum degrees of the form [x_deg, y_deg, z_deg]. + + Returns + ------- + vander3d : ndarray + The shape of the returned matrix is described above. + + See Also + -------- + polyvander, polyvander3d. polyval2d, polyval3d + + """ + ideg = [int(d) for d in deg] + is_valid = [id == d and id >= 0 for id, d in zip(ideg, deg)] + if is_valid != [1, 1, 1]: + raise ValueError("degrees must be non-negative integers") + degx, degy, degz = deg + x, y, z = np.array((x, y, z), copy=0) + 0.0 + + vx = polyvander(x, deg_x) + vy = polyvander(y, deg_y) + vz = polyvander(z, deg_z) + v = np.einsum("...i,...j,...k->...ijk", vx, vy, vz) + return v.reshape(v.shape[:-3] + (-1,)) + + def polyfit(x, y, deg, rcond=None, full=False, w=None): """ Least-squares fit of a polynomial to data. |