The secant method is a
root-finding algorithm that uses a succession of roots of secant lines to
better approximate a root of a function. The secant method can be thought of
as a finite-difference approximation of Newton's method. If the objective
function's value becomes NaN in a position, NaN is returned in that position
as the estimated root.
Args
objective_fn
Python callable for which roots are searched. It must be a
callable of a single variable. objective_fn must return a Tensor of
the same shape and dtype as initial_position.
initial_position
Tensor or Python float representing the starting
position. The function will search for roots in the neighborhood of each
point. The shape of initial_position should match that of the input to
objective_fn.
next_position
Optional Tensor representing the next position in the
search. If specified, this argument must broadcast with the shape of
initial_position and have the same dtype. It will be used to compute the
first step to take when searching for roots. If not specified, a default
value will be used instead.
Default value: initial_position * (1 + 1e-4) + sign(initial_position) *
1e-4.
value_at_position
Optional Tensor or Python float representing the value
of objective_fn at initial_position. If specified, this argument must
have the same shape and dtype as initial_position. If not specified, the
value will be evaluated during the search.
Default value: None.
position_tolerance
Optional Tensor representing the tolerance for the
estimated roots. If specified, this argument must broadcast with the shape
of initial_position and have the same dtype.
Default value: 1e-8.
value_tolerance
Optional Tensor representing the tolerance used to check
for roots. If the absolute value of objective_fn is smaller than
value_tolerance at a given position, then that position is considered a
root for the function. If specified, this argument must broadcast with the
shape of initial_position and have the same dtype.
Default value: 1e-8.
max_iterations
Optional Tensor or Python integer specifying the maximum
number of steps to perform for each initial position. Must broadcast with
the shape of initial_position.
Default value: 50.
stopping_policy_fn
Python callable controlling the algorithm termination.
It must be a callable accepting a Tensor of booleans with the shape of
initial_position (each denoting whether the search is finished for each
starting point), and returning a scalar boolean Tensor (indicating
whether the overall search should stop). Typical values are
tf.reduce_all (which returns only when the search is finished for all
points), and tf.reduce_any (which returns as soon as the search is
finished for any point).
Default value: tf.reduce_all (returns only when the search is finished
for all points).
validate_args
Python bool indicating whether to validate arguments such
as position_tolerance, value_tolerance, and max_iterations.
Default value: False.
name
Python str name prefixed to ops created by this function.
Returns
root_search_results
A Python namedtuple containing the following items:
estimated_root: Tensor containing the last position explored. If the
search was successful within the specified tolerance, this position is
a root of the objective function.
objective_at_estimated_root: Tensor containing the value of the
objective function at position. If the search was successful within
the specified tolerance, then this is close to 0.
num_iterations: The number of iterations performed.
Raises
ValueError
if a non-callable stopping_policy_fn is passed.
Examples
import tensorflow as tf
import tensorflow_probability as tfp
tf.enable_eager_execution()
# Example 1: Roots of a single function from two different starting points.
f = lambda x: (63 * x**5 - 70 * x**3 + 15 * x) / 8.
x = tf.constant([-1, 10], dtype=tf.float64)
tfp.math.secant_root(objective_fn=f, initial_position=x))
# ==> RootSearchResults(
estimated_root=array([-0.90617985, 0.90617985]),
objective_at_estimated_root=array([-4.81727769e-10, 7.44957651e-10]),
num_iterations=array([ 7, 24], dtype=int32))
tfp.math.secant_root(objective_fn=f,
initial_position=x,
stopping_policy_fn=tf.reduce_any)
# ==> RootSearchResults(
estimated_root=array([-0.90617985, 3.27379206]),
objective_at_estimated_root=array([-4.81727769e-10, 2.66058312e+03]),
num_iterations=array([7, 8], dtype=int32))
# Example 2: Roots of a multiplex function from a single starting point.
def f(x):
return tf.constant([0., 63. / 8], dtype=tf.float64) * x**5 \
+ tf.constant([5. / 2, -70. / 8], dtype=tf.float64) * x**3 \
+ tf.constant([-3. / 2, 15. / 8], dtype=tf.float64) * x
x = tf.constant([-1, -1], dtype=tf.float64)
tfp.math.secant_root(objective_fn=f, initial_position=x)
# ==> RootSearchResults(
estimated_root=array([-0.77459667, -0.90617985]),
objective_at_estimated_root=array([-7.81339438e-11, -4.81727769e-10]),
num_iterations=array([7, 7], dtype=int32))
# Example 3: Roots of a multiplex function from two starting points.
def f(x):
return tf.constant([0., 63. / 8], dtype=tf.float64) * x**5 \
+ tf.constant([5. / 2, -70. / 8], dtype=tf.float64) * x**3 \
+ tf.constant([-3. / 2, 15. / 8], dtype=tf.float64) * x
x = tf.constant([[-1, -1], [10, 10]], dtype=tf.float64)
tfp.math.secant_root(objective_fn=f, initial_position=x)
# ==> RootSearchResults(
estimated_root=array([
[-0.77459667, -0.90617985],
[ 0.77459667, 0.90617985]]),
objective_at_estimated_root=array([
[-7.81339438e-11, -4.81727769e-10],
[6.66025013e-11, 7.44957651e-10]]),
num_iterations=array([
[7, 7],
[16, 24]], dtype=int32))
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