Multiply a Legendre series by x. Multiply the Legendre series `c` by x, where x is the independent variable. Parameters ---------- c : array_like 1-D array of Legendre series coefficients ordered from low to high. Returns ------- out : ndarray
(c)
| 406 | |
| 407 | |
| 408 | def legmulx(c): |
| 409 | """Multiply a Legendre series by x. |
| 410 | |
| 411 | Multiply the Legendre series `c` by x, where x is the independent |
| 412 | variable. |
| 413 | |
| 414 | |
| 415 | Parameters |
| 416 | ---------- |
| 417 | c : array_like |
| 418 | 1-D array of Legendre series coefficients ordered from low to |
| 419 | high. |
| 420 | |
| 421 | Returns |
| 422 | ------- |
| 423 | out : ndarray |
| 424 | Array representing the result of the multiplication. |
| 425 | |
| 426 | See Also |
| 427 | -------- |
| 428 | legadd, legmul, legdiv, legpow |
| 429 | |
| 430 | Notes |
| 431 | ----- |
| 432 | The multiplication uses the recursion relationship for Legendre |
| 433 | polynomials in the form |
| 434 | |
| 435 | .. math:: |
| 436 | |
| 437 | xP_i(x) = ((i + 1)*P_{i + 1}(x) + i*P_{i - 1}(x))/(2i + 1) |
| 438 | |
| 439 | Examples |
| 440 | -------- |
| 441 | >>> from numpy.polynomial import legendre as L |
| 442 | >>> L.legmulx([1,2,3]) |
| 443 | array([ 0.66666667, 2.2, 1.33333333, 1.8]) # may vary |
| 444 | |
| 445 | """ |
| 446 | # c is a trimmed copy |
| 447 | [c] = pu.as_series([c]) |
| 448 | # The zero series needs special treatment |
| 449 | if len(c) == 1 and c[0] == 0: |
| 450 | return c |
| 451 | |
| 452 | prd = np.empty(len(c) + 1, dtype=c.dtype) |
| 453 | prd[0] = c[0]*0 |
| 454 | prd[1] = c[0] |
| 455 | for i in range(1, len(c)): |
| 456 | j = i + 1 |
| 457 | k = i - 1 |
| 458 | s = i + j |
| 459 | prd[j] = (c[i]*j)/s |
| 460 | prd[k] += (c[i]*i)/s |
| 461 | return prd |
| 462 | |
| 463 | |
| 464 | def legmul(c1, c2): |