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| """ | |
| Wrapper functions to more user-friendly calling of certain math functions | |
| whose output data-type is different than the input data-type in certain | |
| domains of the input. | |
| For example, for functions like `log` with branch cuts, the versions in this | |
| module provide the mathematically valid answers in the complex plane:: | |
| >>> import math | |
| >>> from numpy.lib import scimath | |
| >>> scimath.log(-math.exp(1)) == (1+1j*math.pi) | |
| True | |
| Similarly, `sqrt`, other base logarithms, `power` and trig functions are | |
| correctly handled. See their respective docstrings for specific examples. | |
| Functions | |
| --------- | |
| .. autosummary:: | |
| :toctree: generated/ | |
| sqrt | |
| log | |
| log2 | |
| logn | |
| log10 | |
| power | |
| arccos | |
| arcsin | |
| arctanh | |
| """ | |
| import numpy.core.numeric as nx | |
| import numpy.core.numerictypes as nt | |
| from numpy.core.numeric import asarray, any | |
| from numpy.core.overrides import array_function_dispatch | |
| from numpy.lib.type_check import isreal | |
| __all__ = [ | |
| 'sqrt', 'log', 'log2', 'logn', 'log10', 'power', 'arccos', 'arcsin', | |
| 'arctanh' | |
| ] | |
| _ln2 = nx.log(2.0) | |
| def _tocomplex(arr): | |
| """Convert its input `arr` to a complex array. | |
| The input is returned as a complex array of the smallest type that will fit | |
| the original data: types like single, byte, short, etc. become csingle, | |
| while others become cdouble. | |
| A copy of the input is always made. | |
| Parameters | |
| ---------- | |
| arr : array | |
| Returns | |
| ------- | |
| array | |
| An array with the same input data as the input but in complex form. | |
| Examples | |
| -------- | |
| First, consider an input of type short: | |
| >>> a = np.array([1,2,3],np.short) | |
| >>> ac = np.lib.scimath._tocomplex(a); ac | |
| array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64) | |
| >>> ac.dtype | |
| dtype('complex64') | |
| If the input is of type double, the output is correspondingly of the | |
| complex double type as well: | |
| >>> b = np.array([1,2,3],np.double) | |
| >>> bc = np.lib.scimath._tocomplex(b); bc | |
| array([1.+0.j, 2.+0.j, 3.+0.j]) | |
| >>> bc.dtype | |
| dtype('complex128') | |
| Note that even if the input was complex to begin with, a copy is still | |
| made, since the astype() method always copies: | |
| >>> c = np.array([1,2,3],np.csingle) | |
| >>> cc = np.lib.scimath._tocomplex(c); cc | |
| array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64) | |
| >>> c *= 2; c | |
| array([2.+0.j, 4.+0.j, 6.+0.j], dtype=complex64) | |
| >>> cc | |
| array([1.+0.j, 2.+0.j, 3.+0.j], dtype=complex64) | |
| """ | |
| if issubclass(arr.dtype.type, (nt.single, nt.byte, nt.short, nt.ubyte, | |
| nt.ushort, nt.csingle)): | |
| return arr.astype(nt.csingle) | |
| else: | |
| return arr.astype(nt.cdouble) | |
| def _fix_real_lt_zero(x): | |
| """Convert `x` to complex if it has real, negative components. | |
| Otherwise, output is just the array version of the input (via asarray). | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Returns | |
| ------- | |
| array | |
| Examples | |
| -------- | |
| >>> np.lib.scimath._fix_real_lt_zero([1,2]) | |
| array([1, 2]) | |
| >>> np.lib.scimath._fix_real_lt_zero([-1,2]) | |
| array([-1.+0.j, 2.+0.j]) | |
| """ | |
| x = asarray(x) | |
| if any(isreal(x) & (x < 0)): | |
| x = _tocomplex(x) | |
| return x | |
| def _fix_int_lt_zero(x): | |
| """Convert `x` to double if it has real, negative components. | |
| Otherwise, output is just the array version of the input (via asarray). | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Returns | |
| ------- | |
| array | |
| Examples | |
| -------- | |
| >>> np.lib.scimath._fix_int_lt_zero([1,2]) | |
| array([1, 2]) | |
| >>> np.lib.scimath._fix_int_lt_zero([-1,2]) | |
| array([-1., 2.]) | |
| """ | |
| x = asarray(x) | |
| if any(isreal(x) & (x < 0)): | |
| x = x * 1.0 | |
| return x | |
| def _fix_real_abs_gt_1(x): | |
| """Convert `x` to complex if it has real components x_i with abs(x_i)>1. | |
| Otherwise, output is just the array version of the input (via asarray). | |
| Parameters | |
| ---------- | |
| x : array_like | |
| Returns | |
| ------- | |
| array | |
| Examples | |
| -------- | |
| >>> np.lib.scimath._fix_real_abs_gt_1([0,1]) | |
| array([0, 1]) | |
| >>> np.lib.scimath._fix_real_abs_gt_1([0,2]) | |
| array([0.+0.j, 2.+0.j]) | |
| """ | |
| x = asarray(x) | |
| if any(isreal(x) & (abs(x) > 1)): | |
| x = _tocomplex(x) | |
| return x | |
| def _unary_dispatcher(x): | |
| return (x,) | |
| def sqrt(x): | |
| """ | |
| Compute the square root of x. | |
| For negative input elements, a complex value is returned | |
| (unlike `numpy.sqrt` which returns NaN). | |
| Parameters | |
| ---------- | |
| x : array_like | |
| The input value(s). | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The square root of `x`. If `x` was a scalar, so is `out`, | |
| otherwise an array is returned. | |
| See Also | |
| -------- | |
| numpy.sqrt | |
| Examples | |
| -------- | |
| For real, non-negative inputs this works just like `numpy.sqrt`: | |
| >>> np.lib.scimath.sqrt(1) | |
| 1.0 | |
| >>> np.lib.scimath.sqrt([1, 4]) | |
| array([1., 2.]) | |
| But it automatically handles negative inputs: | |
| >>> np.lib.scimath.sqrt(-1) | |
| 1j | |
| >>> np.lib.scimath.sqrt([-1,4]) | |
| array([0.+1.j, 2.+0.j]) | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| return nx.sqrt(x) | |
| def log(x): | |
| """ | |
| Compute the natural logarithm of `x`. | |
| Return the "principal value" (for a description of this, see `numpy.log`) | |
| of :math:`log_e(x)`. For real `x > 0`, this is a real number (``log(0)`` | |
| returns ``-inf`` and ``log(np.inf)`` returns ``inf``). Otherwise, the | |
| complex principle value is returned. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| The value(s) whose log is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The log of the `x` value(s). If `x` was a scalar, so is `out`, | |
| otherwise an array is returned. | |
| See Also | |
| -------- | |
| numpy.log | |
| Notes | |
| ----- | |
| For a log() that returns ``NAN`` when real `x < 0`, use `numpy.log` | |
| (note, however, that otherwise `numpy.log` and this `log` are identical, | |
| i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`, and, | |
| notably, the complex principle value if ``x.imag != 0``). | |
| Examples | |
| -------- | |
| >>> np.emath.log(np.exp(1)) | |
| 1.0 | |
| Negative arguments are handled "correctly" (recall that | |
| ``exp(log(x)) == x`` does *not* hold for real ``x < 0``): | |
| >>> np.emath.log(-np.exp(1)) == (1 + np.pi * 1j) | |
| True | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| return nx.log(x) | |
| def log10(x): | |
| """ | |
| Compute the logarithm base 10 of `x`. | |
| Return the "principal value" (for a description of this, see | |
| `numpy.log10`) of :math:`log_{10}(x)`. For real `x > 0`, this | |
| is a real number (``log10(0)`` returns ``-inf`` and ``log10(np.inf)`` | |
| returns ``inf``). Otherwise, the complex principle value is returned. | |
| Parameters | |
| ---------- | |
| x : array_like or scalar | |
| The value(s) whose log base 10 is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The log base 10 of the `x` value(s). If `x` was a scalar, so is `out`, | |
| otherwise an array object is returned. | |
| See Also | |
| -------- | |
| numpy.log10 | |
| Notes | |
| ----- | |
| For a log10() that returns ``NAN`` when real `x < 0`, use `numpy.log10` | |
| (note, however, that otherwise `numpy.log10` and this `log10` are | |
| identical, i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`, | |
| and, notably, the complex principle value if ``x.imag != 0``). | |
| Examples | |
| -------- | |
| (We set the printing precision so the example can be auto-tested) | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.emath.log10(10**1) | |
| 1.0 | |
| >>> np.emath.log10([-10**1, -10**2, 10**2]) | |
| array([1.+1.3644j, 2.+1.3644j, 2.+0.j ]) | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| return nx.log10(x) | |
| def _logn_dispatcher(n, x): | |
| return (n, x,) | |
| def logn(n, x): | |
| """ | |
| Take log base n of x. | |
| If `x` contains negative inputs, the answer is computed and returned in the | |
| complex domain. | |
| Parameters | |
| ---------- | |
| n : array_like | |
| The integer base(s) in which the log is taken. | |
| x : array_like | |
| The value(s) whose log base `n` is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The log base `n` of the `x` value(s). If `x` was a scalar, so is | |
| `out`, otherwise an array is returned. | |
| Examples | |
| -------- | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.lib.scimath.logn(2, [4, 8]) | |
| array([2., 3.]) | |
| >>> np.lib.scimath.logn(2, [-4, -8, 8]) | |
| array([2.+4.5324j, 3.+4.5324j, 3.+0.j ]) | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| n = _fix_real_lt_zero(n) | |
| return nx.log(x)/nx.log(n) | |
| def log2(x): | |
| """ | |
| Compute the logarithm base 2 of `x`. | |
| Return the "principal value" (for a description of this, see | |
| `numpy.log2`) of :math:`log_2(x)`. For real `x > 0`, this is | |
| a real number (``log2(0)`` returns ``-inf`` and ``log2(np.inf)`` returns | |
| ``inf``). Otherwise, the complex principle value is returned. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| The value(s) whose log base 2 is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The log base 2 of the `x` value(s). If `x` was a scalar, so is `out`, | |
| otherwise an array is returned. | |
| See Also | |
| -------- | |
| numpy.log2 | |
| Notes | |
| ----- | |
| For a log2() that returns ``NAN`` when real `x < 0`, use `numpy.log2` | |
| (note, however, that otherwise `numpy.log2` and this `log2` are | |
| identical, i.e., both return ``-inf`` for `x = 0`, ``inf`` for `x = inf`, | |
| and, notably, the complex principle value if ``x.imag != 0``). | |
| Examples | |
| -------- | |
| We set the printing precision so the example can be auto-tested: | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.emath.log2(8) | |
| 3.0 | |
| >>> np.emath.log2([-4, -8, 8]) | |
| array([2.+4.5324j, 3.+4.5324j, 3.+0.j ]) | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| return nx.log2(x) | |
| def _power_dispatcher(x, p): | |
| return (x, p) | |
| def power(x, p): | |
| """ | |
| Return x to the power p, (x**p). | |
| If `x` contains negative values, the output is converted to the | |
| complex domain. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| The input value(s). | |
| p : array_like of ints | |
| The power(s) to which `x` is raised. If `x` contains multiple values, | |
| `p` has to either be a scalar, or contain the same number of values | |
| as `x`. In the latter case, the result is | |
| ``x[0]**p[0], x[1]**p[1], ...``. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The result of ``x**p``. If `x` and `p` are scalars, so is `out`, | |
| otherwise an array is returned. | |
| See Also | |
| -------- | |
| numpy.power | |
| Examples | |
| -------- | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.lib.scimath.power([2, 4], 2) | |
| array([ 4, 16]) | |
| >>> np.lib.scimath.power([2, 4], -2) | |
| array([0.25 , 0.0625]) | |
| >>> np.lib.scimath.power([-2, 4], 2) | |
| array([ 4.-0.j, 16.+0.j]) | |
| """ | |
| x = _fix_real_lt_zero(x) | |
| p = _fix_int_lt_zero(p) | |
| return nx.power(x, p) | |
| def arccos(x): | |
| """ | |
| Compute the inverse cosine of x. | |
| Return the "principal value" (for a description of this, see | |
| `numpy.arccos`) of the inverse cosine of `x`. For real `x` such that | |
| `abs(x) <= 1`, this is a real number in the closed interval | |
| :math:`[0, \\pi]`. Otherwise, the complex principle value is returned. | |
| Parameters | |
| ---------- | |
| x : array_like or scalar | |
| The value(s) whose arccos is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The inverse cosine(s) of the `x` value(s). If `x` was a scalar, so | |
| is `out`, otherwise an array object is returned. | |
| See Also | |
| -------- | |
| numpy.arccos | |
| Notes | |
| ----- | |
| For an arccos() that returns ``NAN`` when real `x` is not in the | |
| interval ``[-1,1]``, use `numpy.arccos`. | |
| Examples | |
| -------- | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.emath.arccos(1) # a scalar is returned | |
| 0.0 | |
| >>> np.emath.arccos([1,2]) | |
| array([0.-0.j , 0.-1.317j]) | |
| """ | |
| x = _fix_real_abs_gt_1(x) | |
| return nx.arccos(x) | |
| def arcsin(x): | |
| """ | |
| Compute the inverse sine of x. | |
| Return the "principal value" (for a description of this, see | |
| `numpy.arcsin`) of the inverse sine of `x`. For real `x` such that | |
| `abs(x) <= 1`, this is a real number in the closed interval | |
| :math:`[-\\pi/2, \\pi/2]`. Otherwise, the complex principle value is | |
| returned. | |
| Parameters | |
| ---------- | |
| x : array_like or scalar | |
| The value(s) whose arcsin is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The inverse sine(s) of the `x` value(s). If `x` was a scalar, so | |
| is `out`, otherwise an array object is returned. | |
| See Also | |
| -------- | |
| numpy.arcsin | |
| Notes | |
| ----- | |
| For an arcsin() that returns ``NAN`` when real `x` is not in the | |
| interval ``[-1,1]``, use `numpy.arcsin`. | |
| Examples | |
| -------- | |
| >>> np.set_printoptions(precision=4) | |
| >>> np.emath.arcsin(0) | |
| 0.0 | |
| >>> np.emath.arcsin([0,1]) | |
| array([0. , 1.5708]) | |
| """ | |
| x = _fix_real_abs_gt_1(x) | |
| return nx.arcsin(x) | |
| def arctanh(x): | |
| """ | |
| Compute the inverse hyperbolic tangent of `x`. | |
| Return the "principal value" (for a description of this, see | |
| `numpy.arctanh`) of ``arctanh(x)``. For real `x` such that | |
| ``abs(x) < 1``, this is a real number. If `abs(x) > 1`, or if `x` is | |
| complex, the result is complex. Finally, `x = 1` returns``inf`` and | |
| ``x=-1`` returns ``-inf``. | |
| Parameters | |
| ---------- | |
| x : array_like | |
| The value(s) whose arctanh is (are) required. | |
| Returns | |
| ------- | |
| out : ndarray or scalar | |
| The inverse hyperbolic tangent(s) of the `x` value(s). If `x` was | |
| a scalar so is `out`, otherwise an array is returned. | |
| See Also | |
| -------- | |
| numpy.arctanh | |
| Notes | |
| ----- | |
| For an arctanh() that returns ``NAN`` when real `x` is not in the | |
| interval ``(-1,1)``, use `numpy.arctanh` (this latter, however, does | |
| return +/-inf for ``x = +/-1``). | |
| Examples | |
| -------- | |
| >>> np.set_printoptions(precision=4) | |
| >>> from numpy.testing import suppress_warnings | |
| >>> with suppress_warnings() as sup: | |
| ... sup.filter(RuntimeWarning) | |
| ... np.emath.arctanh(np.eye(2)) | |
| array([[inf, 0.], | |
| [ 0., inf]]) | |
| >>> np.emath.arctanh([1j]) | |
| array([0.+0.7854j]) | |
| """ | |
| x = _fix_real_abs_gt_1(x) | |
| return nx.arctanh(x) | |