A | |
| abs_rat [Rat] |
Returns the absolute value of a rational.
|
| add_poly [Rat_poly_common] |
Adds two polynomials.
|
| add_rat [Rat] |
Adds two rationals.
|
C | |
| cons_list [List_1] |
Constructs from the given pair (x, lx) the list whose first element is x and
whose rest is lx.
|
| cons_open [Rat_inter] |
Constructs the non-empty open interval (a, b) from the given pair (a, b) of rationals.
|
| cons_poly [Rat_poly] |
Constructs the polynomial a + Xp given the pair (a, p) where a is a rational and p is a
polynomial.
|
| cons_rat [Rat] |
Constructs the rational a / b from the given pair of strings whose first and second
elements denote the integers a and b respectively.
|
| cons_single [Rat_inter] |
Constructs the singleton that contains the given rational.
|
| cut_in_half [Rat_inter] |
Returns given the interval (a, b) the pair ((a, c), (c, b)) of intervals where c is the
midpoint of (a, b).
|
D | |
| degree [Rat_poly_common] |
Returns the degree of a non-zero polynomial.
|
| deriv_poly [Rat_poly_common] |
Returns the derivative of a polynomial.
|
| div_rat [Rat] |
Returns the rational a / b given the pair (a, b) of rationals.
|
E | |
| empty_list [List_1] |
Constructs an empty list.
|
| eq_poly [Rat_poly_common] |
Tests two polynomials for equality.
|
| eq_rat [Rat] |
Tests two rationals for equality.
|
| euclid_div_poly [Rat_poly_common] |
Returns the pair (q, r) where q is the quotient and r is the remainder in the euclidean
division of p1 by p2, given the pair (p1, p2) of polynomials where the divisor p2 is
non-zero.
|
| eval_poly [Rat_poly_common] |
Evaluation of a polynomial.
|
F | |
| function_of_poly [Rat_poly_common] |
Returns the polynomial function associated with a polynomial.
|
G | |
| gcd_poly [Rat_poly_common] |
Returns the gcd (greatest common divisor) of a pair of non-zero polynomials.
|
| ge_rat [Rat] |
Tests whether a >= b given the pair (a, b) of rationals.
|
| gt_rat [Rat] |
Tests whether a > b given the pair (a, b) of rationals.
|
H | |
| head [List_1] |
Returns the first element of a list.
|
I | |
| inv_cons_poly [Rat_poly] |
Returns the pair (a, p) given the polynomial a + Xp.
|
| is_empty_list [List_1] |
Tests whether a list is empty.
|
| is_null_poly [Rat_poly] |
Tests whether a polynomial is the polynomial zero.
|
| is_open [Rat_inter] |
Tests whether an interval is open.
|
L | |
| le_rat [Rat] |
Tests whether a <= b given the pair (a, b) of rationals.
|
| leading_coef [Rat_poly_common] |
Returns the leading coefficient of a non-zero polynomial.
|
| list_of_root_bounding_intervals [Rat_poly_sturm] |
Returns a list of intervals where the number of real roots of p in each interval is 1
and where the width of each interval is less than prec, given a pair (p, prec)
where p is a non-zero polynomial and prec is a rational > 0.
|
| list_of_root_bounding_intervals_aux [Rat_poly_sturm] | |
| low_approximation_of_roots [Rat_poly_sturm] |
Returns a list of rationals where each rational r is either a root of p or such that
a root of p is in the interval (r, r + 10 raised to the power -n), given a pair (p, n)
where p is a non-zero polynomial and n is a non-negative integer.
|
| lower_bound [Rat_inter] |
Returns the lower bound of an interval.
|
| lt_rat [Rat] |
Tests whether a < b given the pair (a, b) of rationals.
|
M | |
| midpoint [Rat_inter] |
Returns the midpoint of an interval.
|
| mod_poly [Rat_poly_common] |
Returns the remainder in the euclidean division of p1 by p2, given the pair (p1, p2) of
polynomials where the divisor p2 is non-zero.
|
| monomial [Rat_poly_common] |
Returns the monomial ap where p is X raised to the power d given the pair (a, d) where a
is a non-zero rational and d is a non-negative integer.
|
| mult_poly [Rat_poly_common] |
Multiply two polynomials.
|
| mult_rat [Rat] |
Multiply two rationals.
|
N | |
| nb_chang [Rat_poly_sturm] | |
| nb_roots [Rat_poly_sturm] |
Returns the number of real roots of a non-zero polynomial.
|
| nb_roots_in [Rat_poly_sturm] |
Returns the number of real roots of a polynomial (which is non-zero, whose degree is
non-zero and which is a square-free polynomial) that are in an open non-empty interval
none of the rational bounds of which is a root of the polynomial.
|
| no_root_out [Rat_poly_sturm] | |
| null_poly [Rat_poly] |
Constructs the polynomial zero.
|
O | |
| opp_rat [Rat] |
Returns the opposite (or additive inverse) of a rational.
|
P | |
| p1 [Ex_rat_poly] |
3 X5 + X4 - 6 X2 + 5 X - 1
|
| p10 [Ex_rat_poly] |
5 X2 - 3 X + 8 has no real root
|
| p11 [Ex_rat_poly] |
X + 1
|
| p12 [Ex_rat_poly] |
X7 + 2 X6 + 2 X5 + 2 X4 - 2 X3 - 2 X2 - 2 X - 2
|
| p13 [Ex_rat_poly] |
- 1200/7 X20 + 1040/7 X19 - 456 X18 + 960/7 X17 - 404 X16 + 316 X15 - 2502/7 X14 - 4208/7 X13 + 14400/7 X12 - 22296/7 X11 + 2238 X10 + 2756/7 X9 - 21048/7 X8 + 25936/7 X7 - 15718/7 X6 + 12/7 X5 + 8948/7 X4 - 8492/7 X3 + 4146/7 X2 - 160 X + 128/7
|
| p2 [Ex_rat_poly] |
2 X3 - X + 1
|
| p3 [Ex_rat_poly] |
X2 - 1
|
| p4 [Ex_rat_poly] |
X - 1
|
| p5 [Ex_rat_poly] |
X2 + 2 X + 1 has a double root (it is -1)
|
| p6 [Ex_rat_poly] |
X3 - 2 X2 - X + 2
|
| p7 [Ex_rat_poly] |
X7 - 1
|
| p8 [Ex_rat_poly] |
7 X + 4
|
| p9 [Ex_rat_poly] |
X2 + 1 has no real root
|
| poly_of_coef_list [Rat_poly_common] |
Returns the polynomial given a list of its coefficients ordered according to decreasing
degrees (we write "a" list rather than "the" list because this list may begin with useless
zeros).
|
| print_rat [Rat] |
Prints a string representation of a rational.
|
| print_rat_inter [Rat_inter] |
Prints a string representation of an interval.
|
| print_rat_poly [Rat_poly] |
Prints a string representation of a polynomial.
|
Q | |
| quotient_poly [Rat_poly_common] |
Returns the quotient in the euclidean division of p1 by p2, given the pair (p1, p2) of
polynomials where the divisor p2 is non-zero.
|
R | |
| rat_times_poly [Rat_poly_common] |
Returns the polynomial ap given the pair (a, p) where a is a rational and p is a
polynomial.
|
| reciproc_rat [Rat] |
Returns the reciprocal (or multiplicative inverse) of a rational.
|
| root_bounding_interval [Rat_poly_sturm] | |
S | |
| square_free_with_the_same_roots [Rat_poly_sturm] |
Returns a square-free polynomial whose set of real roots is equal to the one of the given
polynomial (which is non-zero and whose degree is non-zero).
|
| string_approx_rat [Rat] |
Returns a string representation of a decimal approximation of a rational
according to a given precision (the number of digits after the decimal point).
|
| string_of_poly [Rat_poly] |
Returns a string representation of a polynomial.
|
| string_of_rat [Rat] |
Returns a string representation of a rational, using fractional notation.
|
| string_of_rat_inter [Rat_inter] |
Returns a string representation of an interval.
|
| strings_of_images_of_roots [Rat_poly_sturm] |
In order to test strings_of_roots, this function returns
a list of (string representations of) decimals which approximate the images of the
approximations of the roots of p resulting from applying strings_of_roots to (p, n),
given a pair (p, n) where p is a non-zero polynomial and n is a non-negative integer.
|
| strings_of_roots [Rat_poly_sturm] |
Returns a list of (string representations of) decimals where each decimal is
either a root of p
or such that a root of p is in the interval (d, d + 10 raised to the power -n+1),
given a pair (p, n) where p is a non-zero polynomial and n is a non-negative integer.
|
| sturm_chain [Rat_poly_sturm] |
Returns the Sturm chain (or Sturm sequence) from a polynomial which is non-zero, whose
degree is non-zero and which is a square-free polynomial (i.e.
|
| sub_rat [Rat] |
Returns the rational a - b given the pair (a, b) of rationals.
|
| sum_of_coef_abs_val [Rat_poly_sturm] | |
T | |
| tail [List_1] |
Returns the given list without its first element.
|
U | |
| upper_bound [Rat_inter] |
Returns the upper bound of an interval.
|
V | |
| variation [Rat_poly_sturm] | |
W | |
| w [Rat_poly_sturm] | |
| width [Rat_inter] |
Returns the width of an interval.
|
| with_the_same_roots_except [Rat_poly_sturm] | |
X | |
| x_times [Rat_poly_common] |
Multiply a polynomial by the monomial X.
|