FixedPointList

FixedPointList(f, expr)

starting with expr, iteratively applies f until the result no longer changes, and returns a list of all intermediate results.

FixedPointList(f, expr, n)

performs at most n iterations.

See:

• Wikipedia - Fixed-point iteration
• Wikipedia - Newton’s method

Examples

>> FixedPointList(Cos, 1.0, 4)
{1.0,0.5403023058681398,0.8575532158463934,0.6542897904977791,0.7934803587425656}

Observe the convergence of Newton’s method for approximating square roots:

>> newton(n_) := FixedPointList(.5(# + n/#) &, 1.);
>> newton(9)
{1.0,5.0,3.4,3.023529411764706,3.00009155413138,3.000000001396984,3.0,3.0}

Get the “hailstone” sequence of a number:

>> collatz(1) := 1;
>> collatz(x_ ? EvenQ) := x / 2;
>> collatz(x_) := 3*x + 1;
>> FixedPointList(collatz, 14)
{14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1,1}

>> FixedPointList(f, x, 0)
{x}

Non-negative integer expected.

>> FixedPointList(f, x, -1)
FixedPointList(f,x,-1)

>> Last(FixedPointList(Cos, 1.0, Infinity))
0.7390851332151607

Related terms

FixedPoint, Nest, NestWhile

Implementation status

• ✅ - full supported

Github

• Implementation of FixedPointList