Minmaxline

The Min-Max Line for the HP-41

Overview:

1°) Program#1
2°) Program#2

Latest Update: paragraph 2°)

1°) Program#1

-Let be a set of n points ( x₁ ; y₁ ) ..... ( x_n ; y_n )
The min-max line: y = m.x + p minimizes the maximum "error" h = sup_i | y_i - m x_i - p | over the entire set of points.

-This program uses the "exchange method":

1°) The Chebyshev line is found for the first 3 points.
2°) The greatest error is computed at all data points for this Chebyshev line.
3°) One of the first 3 points is exchanged with a data point at which the greatest error occurs ( so that errors are of alternating sign for this new triple )

-The process is repeated until the greatest error occurs at one of the first 3 points: this is always achieved and only a few exchanges are usually needed.

Data Registers: Registers R00 thru R2n are used and all of them are to be initialized before executing MML

R00 = n = the number of point

R01 = x₁ ; R03 = x₂ ; .......... ; R2n-1 = x_n ( it's necessary to have x₁ < x₂ < x₃ )
R02 = y₁ ; R04 = y₂ ; .......... ; R2n = y_n

Flags and subroutines: none.

-Synthetic registers M N O P can be replaced by any unused standard registers:
for instance R41 R42 R43 R44 if n < 21.

-Lines 52-106-120 are three-byte GTOs

01 LBL "MML"
02 LBL 01
03 RCL 02
04 RCL 04
05 +
06 RCL 06
07 RCL 02
08 -
09 RCL 05
10 RCL 01
11 -
12 /
13 STO M
14 RCL 01
15 RCL 03
16 +
17 *
18 -
19 2
20 /
21 STO N
22 RCL 00
23 ST+ X
24 0
25 LBL 02

26 DSE Y
27 RCL IND Y
28 RCL M
29 *
30 RCL N
31 +
32 ISG T
33 CLX
34 RCL IND T
35 -
36 ABS
37 X<Y?
38 GTO 03
39 R^
40 STO O
41 X<>Y
42 LAST X
43 STO P
44 LBL 03
45 RDN
46 DSE Y
47 GTO 02
48 RCL O
49 PI
50 ST+ X

51 X>Y?
52 GTO 10
53 RCL IND O
54 ISG O
55 CLX
56 RCL P
57 LAST X
58 *
59 X<0?
60 GTO 06
61 CLX
62 RCL 03
63 X>Y?
64 GTO 05
65 RCL 06
66 X<> IND O
67 STO 06
68 DSE O
69 RCL 05
70 LBL 04
71 X<> IND O
72 STO 05
73 GTO 01
74 LBL 05
75 RCL 02

76 X<> IND O
77 STO 02
78 DSE O
79 RCL 01
80 GTO 09
81 LBL 06
82 CLX
83 RCL 05
84 X>Y?
85 GTO 07
86 RCL 06
87 X<> 04
88 X<> IND O
89 STO 06
90 DSE O
91 RCL 05
92 X<> 03
93 GTO 04
94 LBL 07
95 CLX
96 RCL 01
97 X>Y?
98 GTO 08
99 RCL 04
100 X<> IND O

101 STO 04
102 DSE O
103 RCL 03
104 X<> IND O
105 STO 03
106 GTO 01
107 LBL 08
108 RCL 02
109 X<> 04
110 X<> 06
111 X<> IND O
112 STO 02
113 DSE O
114 RCL 01
115 X<> 03
116 X<> 05
117 LBL 09
118 X<> IND O
119 STO 01
120 GTO 01
121 LBL 10
122 RCL N
123 RCL M
124 CLA
125 END

( 186 bytes / SIZE 2n+1 )

STACK	INPUTS	OUTPUTS
T	/	1 or 3 or 5
Z	/	2*pi
Y	/	p
X	/	m
L	/	+h or -h

Example: Find the min-max line for the following data:

x_i	1	3	4	7	10	12	16	19	24	37	41	49	51	61
y_i	1	2	3	6	8	9	12	14	20	31	38	41	48	57

1- SIZE 029 ( or greater )

2- 14 STO 00 ( there are 14 points )

3-   1 STO 01       STO 02
       3 STO 03   2 STO 04
       ..............................
     61 STO 27   57 STO 28

4- XEQ "MML" >>>> m = 0.94 ---Execution time = 36s---
RDN p = -2.5

-The min-max line is y = 0.94 x - 2.5

and the maximum error is h = 2.56
( L-register contains -2.56 )

-Note that the least squares line is 0.9275 x - 1.4785 in this example.

2°) Program#2

-The following program doesn't use synthetic registers and the registers Rbb to Ree are not modified.

Data Registers: R00 = bbb.eee R01 to R09: temp bb > 09 ( Registers Rbb thru Ree are to be initialized before executing "MML" )

• Rbb = x₁ • Rb+2 = x₂ ....... • Re-1 = x_n ( it's necessary to have x₁ < x₂ < x₃ )
• Rb+1 = y₁ • Rb+3 = y₂ ....... • Reee = y_n
Flags: /
Subroutines: /

-Lines 107-113-117 are three-byte GTOs

01 LBL "MML"
02 STO 00
03 STO 01
04 2
05 +
06 STO 02
07 LASTX
08 +
09 STO 03
10 STO 09
11 LBL 01
12 RCL 01
13 RCL 02
14 RCL 03
15 1
16 ST+ T
17 ST+ Z
18 +
19 RCL IND Y
20 RCL IND T
21 +
22 RCL IND Y
23 LASTX
24 -

25 RCL IND 03
26 RCL IND 01
27 -
28 /
29 STO 04
30 RCL IND 01
31 RCL IND 02
32 +
33 *
34 -
35 2
36 /
37 STO 05
38 RCL 01
39 1
40 +
41 RCL 00
42 RCL IND 01
43 RCL 04
44 *
45 RCL 05
46 +
47 RCL IND Z
48 -

49 STO 08
50 ABS
51 LBL 02
52 RCL IND Y
53 RCL 04
54 *
55 RCL 05
56 +
57 ISG Z
58 RCL IND Z
59 -
60 ABS
61 X<=Y?
62 GTO 02
63 R^
64 STO 06
65 X<>Y
66 LASTX
67 STO 07
68 DSE 06
69 LBL 02
70 RDN
71 ISG Y
72 GTO 02

73 RCL 09
74 RCL 06
75 STO 09
76 X=Y?
77 GTO 05
78 RCL 08
79 RCL 07
80 STO 08
81 *
82 X<0?
83 GTO 04
84 RCL IND 06
85 RCL IND 02
86 X>Y?
87 GTO 03
88 RCL 06
89 STO 03
90 GTO 01
91 LBL 03
92 RCL 06
93 STO 01
94 GTO 01
95 LBL 04
96 RCL IND 01
97 RCL IND 06

98 X<Y?
99 GTO 03
100 RCL IND 03
101 X>Y?
102 GTO 04
103 RCL 06
104 X<> 03
105 X<> 02
106 STO 01
107 GTO 01
108 LBL 03
109 RCL 06
110 X<> 01
111 X<> 02
112 STO 03
113 GTO 01
114 LBL 04
115 RCL 06
116 STO 02
117 GTO 01
118 LBL 05
119 RCL 08
120 RCL 05
121 RCL 04
122 END

( 170 bytes / SIZE eee+1 )

STACK	INPUTS	OUTPUTS
Z	/	+h or -h
Y	/	p
X	bbb.eee	m

Example: Find the min-max line for the following data:

x_i	1	3	4	7	10	12	16	19	24	37	41	49	51	61
y_i	1	2	3	6	8	9	12	14	20	31	38	41	48	57

-If we choose to store the data in R11 , R12 , .....

   1 STO 11       STO 12
   3 STO 13   2 STO 14
       ..............................

61 STO 37 57 STO 38

11.038 XEQ "MML" >>>> m = 0.94 = R04              ---Execution time = 36s---
                             RDN   p = -2.5 = R05
   RDN -h = -2.56 = R08

-The min-max line is y = 0.94 x - 2.5 and the maximum error is h = 2.56

Reference:

[1] "Numerical Analysis" - Francis Scheid - McGraw-Hill ISBN 07-055197-9

hp41programs

The Min-Max Line for the HP-41