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Example 1. Compute f(x) = X**2 - X + 3 using different formulas.
X is the interval [ -0.5 , 1.0 ]
X**2 gives [ 0.000 , 1.000 ]
X*X gives [ -0.500 , 1.000 ]
X**2 - X + 3 gives [ 2.000 , 4.500 ]. Magnification = 2.500
X*X - X + 3 gives [ 1.500 , 4.500 ]. Magnification = 3.000
X*(X - 1) + 3 gives [ 1.500 , 3.750 ]. Magnification = 2.250
(X - 0.5)**2 + 2.75 gives [ 2.750 , 3.750 ]. Magnification = 1.000
X is the interval [ 0.1 , 1.0 ]
X**2 - X + 3 gives [ 2.010 , 3.900 ]. Magnification = 7.560
X*X - X + 3 gives [ 2.010 , 3.900 ]. Magnification = 7.560
X*(X - 1) + 3 gives [ 2.100 , 3.000 ]. Magnification = 3.600
(X - 0.5)**2 + 2.75 gives [ 2.750 , 3.000 ]. Magnification = 1.000
X is the interval [ 0.9 , 1.0 ]
X**2 - X + 3 gives [ 2.810 , 3.100 ]. Magnification = 3.222
X*X - X + 3 gives [ 2.810 , 3.100 ]. Magnification = 3.222
X*(X - 1) + 3 gives [ 2.900 , 3.000 ]. Magnification = 1.111
(X - 0.5)**2 + 2.75 gives [ 2.910 , 3.000 ]. Magnification = 1.000
X is the interval [ 0.99 , 1.0 ]
X**2 - X + 3 gives [ 2.980 , 3.010 ]. Magnification = 3.020
X*X - X + 3 gives [ 2.980 , 3.010 ]. Magnification = 3.020
X*(X - 1) + 3 gives [ 2.990 , 3.000 ]. Magnification = 1.010
(X - 0.5)**2 + 2.75 gives [ 2.990 , 3.000 ]. Magnification = 1.000
Example 2. Sum 1 / n^7 from 1 to 100,000 =
1.0083492773819228268397975498496300979331385605652M+0
1.0083492773819228268397975498496300979331385605653M+0
The two endpoints agree to about 51 decimal digits.
Example 3. Integrate sin(t) / t from t = 0 to t = 20:
1.5482417010434398401636433421295136922615733621093M+0
1.5482417010434398401636433421295136922615733621094M+0
The two endpoints agree to about 54 decimal digits.
Example 4. Step around the unit circle multiple times using a recurrence
1 trips. The two endpoints for x agree to about 54 decimal digits.
2 trips. The two endpoints for x agree to about 52 decimal digits.
3 trips. The two endpoints for x agree to about 49 decimal digits.
4 trips. The two endpoints for x agree to about 47 decimal digits.
5 trips. The two endpoints for x agree to about 44 decimal digits.
6 trips. The two endpoints for x agree to about 42 decimal digits.
7 trips. The two endpoints for x agree to about 39 decimal digits.
8 trips. The two endpoints for x agree to about 36 decimal digits.
9 trips. The two endpoints for x agree to about 34 decimal digits.
10 trips. The two endpoints for x agree to about 31 decimal digits.
11 trips. The two endpoints for x agree to about 29 decimal digits.
12 trips. The two endpoints for x agree to about 26 decimal digits.
13 trips. The two endpoints for x agree to about 24 decimal digits.
14 trips. The two endpoints for x agree to about 21 decimal digits.
15 trips. The two endpoints for x agree to about 18 decimal digits.
16 trips. The two endpoints for x agree to about 16 decimal digits.
17 trips. The two endpoints for x agree to about 13 decimal digits.
18 trips. The two endpoints for x agree to about 11 decimal digits.
19 trips. The two endpoints for x agree to about 8 decimal digits.
20 trips. The two endpoints for x agree to about 6 decimal digits.
21 trips. The two endpoints for x agree to about 3 decimal digits.
22 trips. The two endpoints for x agree to about 1 decimal digits.
23 trips. The two endpoints for x agree to about 0 decimal digits.
X =
-5.8564010117471012542822698126300121646708464338966M+1
6.0564010117471012542822698126300121646708464338966M+1
Y =
-5.9564010117471012542822698126300121646708464338966M+1
5.9564010117471012542822698126300121646708464338966M+1
The two endpoints for x agree to about 0 decimal digits.
Compare the same calculation in ordinary FM arithmetic.
1 trips. x agrees with 1.0 to about 62 decimal digits.
2 trips. x agrees with 1.0 to about 61 decimal digits.
3 trips. x agrees with 1.0 to about 61 decimal digits.
4 trips. x agrees with 1.0 to about 61 decimal digits.
5 trips. x agrees with 1.0 to about 61 decimal digits.
6 trips. x agrees with 1.0 to about 61 decimal digits.
7 trips. x agrees with 1.0 to about 61 decimal digits.
8 trips. x agrees with 1.0 to about 61 decimal digits.
9 trips. x agrees with 1.0 to about 61 decimal digits.
10 trips. x agrees with 1.0 to about 61 decimal digits.
11 trips. x agrees with 1.0 to about 61 decimal digits.
12 trips. x agrees with 1.0 to about 60 decimal digits.
13 trips. x agrees with 1.0 to about 60 decimal digits.
14 trips. x agrees with 1.0 to about 60 decimal digits.
15 trips. x agrees with 1.0 to about 60 decimal digits.
16 trips. x agrees with 1.0 to about 60 decimal digits.
17 trips. x agrees with 1.0 to about 60 decimal digits.
18 trips. x agrees with 1.0 to about 60 decimal digits.
19 trips. x agrees with 1.0 to about 60 decimal digits.
20 trips. x agrees with 1.0 to about 60 decimal digits.
21 trips. x agrees with 1.0 to about 60 decimal digits.
22 trips. x agrees with 1.0 to about 60 decimal digits.
23 trips. x agrees with 1.0 to about 60 decimal digits.
X =
1.0000000000000000000000000000000000000000000000000M+0
Y =
-2.2745102100000000000000000000000000000000000000000M-55
Example 4b. Step back and forth around the unit circle using a recurrence
Starting point for X =
8.7630668004386358730811590392206258339906423848214M-1
8.7630668004386358730811590392206258339906423848215M-1
Starting point for Y =
4.8175367410171527498719150287212965352854201015207M-1
4.8175367410171527498719150287212965352854201015208M-1
5 trips. The two endpoints for x agree to about 59 decimal digits.
10 trips. The two endpoints for x agree to about 56 decimal digits.
15 trips. The two endpoints for x agree to about 54 decimal digits.
20 trips. The two endpoints for x agree to about 51 decimal digits.
25 trips. The two endpoints for x agree to about 48 decimal digits.
30 trips. The two endpoints for x agree to about 46 decimal digits.
35 trips. The two endpoints for x agree to about 43 decimal digits.
40 trips. The two endpoints for x agree to about 41 decimal digits.
45 trips. The two endpoints for x agree to about 38 decimal digits.
50 trips. The two endpoints for x agree to about 36 decimal digits.
55 trips. The two endpoints for x agree to about 33 decimal digits.
60 trips. The two endpoints for x agree to about 30 decimal digits.
65 trips. The two endpoints for x agree to about 28 decimal digits.
70 trips. The two endpoints for x agree to about 25 decimal digits.
75 trips. The two endpoints for x agree to about 23 decimal digits.
80 trips. The two endpoints for x agree to about 20 decimal digits.
85 trips. The two endpoints for x agree to about 18 decimal digits.
90 trips. The two endpoints for x agree to about 15 decimal digits.
95 trips. The two endpoints for x agree to about 12 decimal digits.
100 trips. The two endpoints for x agree to about 10 decimal digits.
105 trips. The two endpoints for x agree to about 7 decimal digits.
110 trips. The two endpoints for x agree to about 5 decimal digits.
115 trips. The two endpoints for x agree to about 2 decimal digits.
X =
8.7369369964583434983069574632632545487869869160581M-1
8.7891966044189282478553606151779971191942978535848M-1
Y =
4.7914069370368603750977134527639252500817646327574M-1
4.8436665449974451246461166046786678204890755702842M-1
The two endpoints for x agree to about 2 decimal digits.
Example 5. 2 * Product 4*n^2 / (4*n^2-1) from 1 to 10,000. =
3.1415141186819220469785580507138775513425133394700M+0
3.1415141186819220469785580507138775513425133394701M+0
The two endpoints of y(30) agree to about 52 decimal digits.
Example 6. Differential equation. y'' = -y' / 10 - 2 * y / (x+2)
X = 3.0000000000 V = 0.831136520960373 -0.452290751829937 Endpoints of V(1) agree to 53 digits.
X = 6.0000000000 V = -0.765883193156779 -0.352158220002235 Endpoints of V(1) agree to 52 digits.
X = 9.0000000000 V = -0.821640364088937 0.257782946044846 Endpoints of V(1) agree to 52 digits.
X = 12.0000000000 V = 0.193374374107742 0.325345242516926 Endpoints of V(1) agree to 51 digits.
X = 15.0000000000 V = 0.786966466722528 0.052221392312944 Endpoints of V(1) agree to 51 digits.
X = 18.0000000000 V = 0.572088550349089 -0.166039347343631 Endpoints of V(1) agree to 50 digits.
X = 21.0000000000 V = -0.008218454426127 -0.190332029943315 Endpoints of V(1) agree to 48 digits.
X = 24.0000000000 V = -0.440713401930602 -0.086656234642814 Endpoints of V(1) agree to 49 digits.
X = 27.0000000000 V = -0.515393780043246 0.031589999652254 Endpoints of V(1) agree to 48 digits.
X = 30.0000000000 V = -0.307766881676032 0.095343042626880 Endpoints of V(1) agree to 48 digits.
y(30) =
-3.0776688167603169203382987651096461581717811055464M-1
-3.0776688167603169203382987651096461581717811055418M-1
y'(30) =
9.5343042626880286284843246841489833596602442626286M-2
9.5343042626880286284843246841489833596602442626426M-2
The two endpoints of y(30) agree to about 48 decimal digits.
Example 7. Differential equation. y'' = -y' / 10 - 200 * y / (x+2)
X = 3.0000000000 V = -0.102742064000739 -0.215544901796916 Endpoints of V(1) agree to 47 digits.
X = 6.0000000000 V = 0.078485157239314 -0.348572728869748 Endpoints of V(1) agree to 39 digits.
X = 9.0000000000 V = -0.037481204238851 -0.383488535010064 Endpoints of V(1) agree to 33 digits.
X = 12.0000000000 V = 0.013531679745067 -0.333961311141985 Endpoints of V(1) agree to 27 digits.
X = 15.0000000000 V = 0.075496648457680 0.094822963645320 Endpoints of V(1) agree to 23 digits.
X = 18.0000000000 V = -0.072047742909790 0.022294258042798 Endpoints of V(1) agree to 19 digits.
X = 21.0000000000 V = 0.063355820821447 0.032461966986868 Endpoints of V(1) agree to 14 digits.
X = 24.0000000000 V = -0.029168578777506 -0.135277237857925 Endpoints of V(1) agree to 10 digits.
X = 27.0000000000 V = -0.036220481301623 0.094265735252438 Endpoints of V(1) agree to 7 digits.
X = 30.0000000000 V = 0.025275350395373 0.090890872993315 Endpoints of V(1) agree to 3 digits.
y(30) =
2.5268274663776318683290452013354524008325842304305M-2
2.5282426126969610817979865312140233217038151226782M-2
y'(30) =
9.0872770822409973135783151703562888452619638318962M-2
9.0908975164219866493786176735651583524503965546846M-2
The two endpoints of y(30) agree to about 3 decimal digits.
Example 8. Solve a "random" NxN linear system.
For N = 10 the solution elements agree to between 51 and 53 significant digits.
For N = 20 the solution elements agree to between 47 and 50 significant digits.
For N = 30 the solution elements agree to between 44 and 48 significant digits.
For N = 40 the solution elements agree to between 36 and 46 significant digits.
For N = 50 the solution elements agree to between 35 and 43 significant digits.
For N = 60 the solution elements agree to between 28 and 40 significant digits.
For N = 70 the solution elements agree to between 27 and 37 significant digits.
For N = 80 the solution elements agree to between 23 and 34 significant digits.
For N = 90 the solution elements agree to between 16 and 33 significant digits.
For N = 100 the solution elements agree to between 16 and 30 significant digits.
Example 9. Newton's method starting with a single point.
Iteration 1. X =
8.6787944117144232159552377016146086744581113103176M-1
8.6787944117144232159552377016146086744581113103177M-1
Iteration 2. X =
8.5278337341640992137798507086554877127484032504254M-1
8.5278337341640992137798507086554877127484032504255M-1
Iteration 3. X =
8.5260552636892205445692227465314123881418241936054M-1
8.5260552636892205445692227465314123881418241936055M-1
Iteration 4. X =
8.5260550201372594802652409548836209726836629454221M-1
8.5260550201372594802652409548836209726836629454222M-1
Iteration 5. X =
8.5260550201372549134647241469547803262670602201255M-1
8.5260550201372549134647241469547803262670602201256M-1
Iteration 6. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 7. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 8. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 9. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 10. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Example 10. Newton's method starting with an interval containing a root
Iteration 1. X =
8.2582914751966549600358492004459062470952152212196M-1
9.1669483793964048943502242929035570875441708134219M-1
Iteration 2. X =
8.5153653288747945014206990942716504355878954736541M-1
8.5410378857010420828800408165620227158607956580201M-1
Iteration 3. X =
8.5260511282637639422279677005694780814425748090707M-1
8.5260596120050999165916620170701727472347232648131M-1
Iteration 4. X =
8.5260550201370357420197023626581603375273929231293M-1
8.5260550201374929467463703975692934764436548299921M-1
Iteration 5. X =
8.5260550201372549134647241466280563503694077596793M-1
8.5260550201372549134647241472919881356741988716405M-1
Iteration 6. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 7. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 8. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
Iteration 9. X =
8.5260550201372549134647241469531746689845330015140M-1
8.5260550201372549134647241469531746689845330015141M-1
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