Compute the roots of a Legendre series. Return the roots (a.k.a. "zeros") of the polynomial .. math:: p(x) = \\sum_i c[i] * L_i(x). Parameters ---------- c : 1-D array_like 1-D array of coefficients. Returns ------- out : ndarray Array of the
(c)
| 1457 | |
| 1458 | |
| 1459 | def legroots(c): |
| 1460 | """ |
| 1461 | Compute the roots of a Legendre series. |
| 1462 | |
| 1463 | Return the roots (a.k.a. "zeros") of the polynomial |
| 1464 | |
| 1465 | .. math:: p(x) = \\sum_i c[i] * L_i(x). |
| 1466 | |
| 1467 | Parameters |
| 1468 | ---------- |
| 1469 | c : 1-D array_like |
| 1470 | 1-D array of coefficients. |
| 1471 | |
| 1472 | Returns |
| 1473 | ------- |
| 1474 | out : ndarray |
| 1475 | Array of the roots of the series. If all the roots are real, |
| 1476 | then `out` is also real, otherwise it is complex. |
| 1477 | |
| 1478 | See Also |
| 1479 | -------- |
| 1480 | numpy.polynomial.polynomial.polyroots |
| 1481 | numpy.polynomial.chebyshev.chebroots |
| 1482 | numpy.polynomial.laguerre.lagroots |
| 1483 | numpy.polynomial.hermite.hermroots |
| 1484 | numpy.polynomial.hermite_e.hermeroots |
| 1485 | |
| 1486 | Notes |
| 1487 | ----- |
| 1488 | The root estimates are obtained as the eigenvalues of the companion |
| 1489 | matrix, Roots far from the origin of the complex plane may have large |
| 1490 | errors due to the numerical instability of the series for such values. |
| 1491 | Roots with multiplicity greater than 1 will also show larger errors as |
| 1492 | the value of the series near such points is relatively insensitive to |
| 1493 | errors in the roots. Isolated roots near the origin can be improved by |
| 1494 | a few iterations of Newton's method. |
| 1495 | |
| 1496 | The Legendre series basis polynomials aren't powers of ``x`` so the |
| 1497 | results of this function may seem unintuitive. |
| 1498 | |
| 1499 | Examples |
| 1500 | -------- |
| 1501 | >>> import numpy.polynomial.legendre as leg |
| 1502 | >>> leg.legroots((1, 2, 3, 4)) # 4L_3 + 3L_2 + 2L_1 + 1L_0, all real roots |
| 1503 | array([-0.85099543, -0.11407192, 0.51506735]) # may vary |
| 1504 | |
| 1505 | """ |
| 1506 | # c is a trimmed copy |
| 1507 | [c] = pu.as_series([c]) |
| 1508 | if len(c) < 2: |
| 1509 | return np.array([], dtype=c.dtype) |
| 1510 | if len(c) == 2: |
| 1511 | return np.array([-c[0]/c[1]]) |
| 1512 | |
| 1513 | # rotated companion matrix reduces error |
| 1514 | m = legcompanion(c)[::-1,::-1] |
| 1515 | r = la.eigvals(m) |
| 1516 | r.sort() |
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