7ba5c347-beeb-42f2-9b7a-fec81290d9c7 6 6 9 decimal

&[DATE] &[TIME] - &[FILENAME]

x xo f x xo - (x 10^x + 1 + (* e x - 1 -^* :

x xo T.m m x xo f * tanh :

xo π :

m 1 :

x xo f 0 ≡ x 05 solve x xo f 0 ≡ x 3 3.5 solve x xo f 0 ≡ x 3.1 3.2 solve π 41 mat 3.14159265174864 3.14159265357743 3.1415926535166 3.14159265358979 41 mat

the root is exactly 'xo' datum. Dichotomy will find at low ε

x xo f xo 0 o 11 red 1 5 mat 2 1 sys x xo T.m x xo T.m ( x diff 2 1 sys

ULP 0.029 - 0 0.029 - 0.125 0.018 0.125 0.018 0 0.029 - 052 mat :

x Φ x 2 π * / 2 π * x / tanh * sqrt :

In the ULP region: error is erroneous "Division by zero"

x Φ ULP 2 1 sys

Canvas plot

x Φ

0.029 - Φ 0.018 Φ 21 mat 0.067937 0.053524 21 mat

Mathcad 11 starts 0.0001

x datum 0.075 :

datum(x):=0.05 "NO real roots"is misleading.
Should read: Φ(x) does not compute [ULP region]. x Φ x datum - 0 ≡ x 1 - 1 solve 0.035343 - 0.035343 21 mat

Numbers on a computer comprise a discrete set. There are gaps between the numbers and when you do arithmetic the result often is not representable. When the result falls between two representable numbers the best that can be done is to use the closest representable number as the result instead of the correct result. The set of numbers that can be represented on a computer in floatingpoint format are commonly referred to as the set of machine numbers. If you want to investigate how arithmetic is done on computers in a general sense you should use the package NumericalMath` ComputerArithmetic`. That package allows you to vary certain fundamental parameters that control the arithmetic. The difference between two consecutive machine numbers is called an ulp (unit in the last place, i.e., one digit in the least significant place). The size of an ulp varies depending on where you are in the set of machine numbers. Between a and a an ulp is equal to $MachineEpsilon. Between a and a it is equal to 2 $MachineEpsilon. Ideally no function should ever return a result with error exceeding half of an ulp since the distance from the true result to the nearest machine number is always less than half of an ulp, the worst case being when it is exactly halfway between two machine numbers. It is relatively easy to achieve this ideal for the four arithmetic operations and for the square root function. For the elementary transcendental functions it becomes more difficult due to the fact that they cannot be evaluated in a single operation and the combined effects of all the rounding errors would have to be made less than half an ulp. Nevertheless, it is quite possible to design algorithms for these functions such that the error is never more than one ulp.

MachineError[Tan[Sqrt[Exp[Sin[Log[x]]]]], x -> N[Pi]] ... -26.8911 Ulps

ε.f 108 -^:

ε.x 108 -^:

x f α β N SI p α β + 2 / : δ β α - 2 N * / : x F x f sign : p p δ α F * p 2 j * N - (δ * + F j 1 N 1 - sum * + eval : k 0 : k 100 < r p : p x u x f z f' z f (z diff : x f' 21 line / : r r u r r u - f r f - 2 r r u - f * r f - / * - eval 21 line : p f abs ε.f < (p r - abs ε.x < (| break k k 1 + : if 31 line while p eval 71 line :

x f x sin :

R x f 0.3 π * 3 π * 1000 SI : 6.2832

returns zeros (1st output) and  maximum/minimum points (2nd output) in a range x f a b N.0 SIG ε 107 -^: h b a - N.0 / eval : x f.ε ε x f (2^* x ε x f * + f x f - ε 3^+ / : x F.ε x f.ε sign : x invf.ε x ε x f * + f x f - ε x f (2^* / : j m n 0 : (: (: p q 01 matrix : (: a f 0 ≡ m m 1 + : p m el a : 21 line 0 if 41 line j N.0 < j j 1 + : α a j 1 - (h * + : β a j h * + : β F.ε α F.ε - 2 ≡ (β f.ε α f.ε - 2 h * < (& m m 1 + : p m el x f.ε α β 10 SI : 21 line 0 if β F.ε α F.ε - abs 2 ≡ (β f.ε α f.ε - abs 2 h * > (& n n 1 + : q n el x invf.ε α β 10 SI : 21 line 0 if 51 line while p q 21 mat 71 line :

R 1 f plotPts m 05 matrix : i 1 R rows range m m R i el R i el f . 14 red 15 mat stack : for m eval 31 line :

x f 26 x π - (* sin * exp x + 1 - :

R x f 1.5 - 241 SIG : 0.9887140323 - 0.5610243645 - 4.19066 1010 -^* 0.6038168459 0.8458544288 51 mat 1.3926546143 - 0.7835185079 - 0.3454571158 - 0.2636789909 0.7017403839 1.3108764898 1.7489378818 71 mat 21 mat

x f R 1 el X f plotPts R 2 el X f plotPts 3 1 sys

x xo f x xo - (x 10^x + 1 + (* e x - 1 -^* :

x xo T.m m x xo f * tanh :

xo π :

m 1 :

x f x xo T.m :

roots x f 0.875 - 141 SIG :

x f roots 2 el X f plotPts 2 1 sys

roots 01 mat 0.732556 - 0.228825 - 0.828374 31 mat 21 mat