Polynomials2

Polynomials(III) for the HP-41

Overview

1°) Cubic Equations
2°) Quartic Equations
3°) Palindromic Polynomials

a) Degree = 4
b) Degree = 5
c) Degree = 6
d) Degree = 7
e) Degree = 8
f) Degree = 9

4°) Integer Roots

a) Real Roots
b) Complex Roots

5°) A few Quintic Equations

a) Program#1
b) Program#2
c) Program#3

6°) A few Sextic Equations
7°) A few Septic Equations

a) Program#1
b) Program#2

8°) A few Octic Equations

-There are already 2 programs listed in "Polynomials for the HP41" to solve cubic & quartic equations
-The following ones are sometimes more accurate... and sometimes less accurate !

-If the polynomial is palindromic, we can find its roots easily, provided its degree is < 10.

-In paragraph 4°), the routines employ the brute force to find the roots if all the roots are integers ( Gaussian integrers with complex polynomials )

-It's impossible to solve all polynomial equations of degree > 4 with similar formulae
-But there are special cases where it's still possible.

1°) Cubic Equations

-"P3" solves a.x³+b.x²+c.x+d = 0

Data Registers: R00 .... R05: temp
Flag: F01 ( indicates complex roots )
Subroutines: /

01 LBL "P3"
02 CF 01
03 R^
04 ST/ T
05 ST/ Z
06 /
07 27
08 *
09 X<>Y
10 STO 02
11 9
12 *
13 R^
14 STO 01
15 X^2
16 ST+ X
17 -
18 RCL 01
19 *
20 -
21 STO 00
22 X^2

23 RCL 01
24 X^2
25 RCL 02
26 3
27 ST/ 01
28 *
29 -
30 STO 04
31 3
32 Y^X
33 4
34 *
35 X<Y?
36 GTO 00
37 DEG
38 X#0?
39 /
40 SQRT
41 RCL 00
42 SIGN
43 CHS
44 *

45 ACOS
46 3
47 /
48 STO 05
49 120
50 STO 03
51 +
52 COS
53 RCL 05
54 RCL 03
55 -
56 COS
57 RCL 04
58 SQRT
59 1.5
60 /
61 RCL 05
62 COS
63 X<>Y
64 ST* T
65 ST* Z
66 *

67 RCL 01
68 ST- T
69 GTO 01
70 LBL 00
71 -
72 SQRT
73 RCL 00
74 ABS
75 +
76 2
77 /
78 3
79 1/X
80 Y^X
81 RCL 00
82 SIGN
83 *
84 CHS
85 STO 05
86 RCL 04
87 RCL 05
88 /

89 STO 03
90 -
91 12
92 SQRT
93 /
94 RCL 03
95 RCL 05
96 +
97 3
98 /
99 STO 03
100 2
101 /
102 CHS
103 RCL 03
104 RCL 01
105 SF 01
106 LBL 01
107 ST- Z
108 -
109 END

( 134 bytes / SIZE 006 )


STACK	INPUTS	OUTPUTS
T	a # 0	/
Z	b	z
Y	c	y
X	d	x

-If CF 01 the 3 solutions are x ; y ; z
-If SF 01 the 3 solutions are x ; y+i.z ; y-i.z

Example: Solve 2.x³-5.x²-7.x+1 = 0 and 2.x³-5.x²+7.x+1 = 0

        2   ENTER^
       -5 ENTER^
       -7 ENTER^
        1   XEQ "P3" >>>>   3.467727065   RDN   0.131206041   RDN   -1.098933110 which are the 3 real solutions since flag F01 is clear.

        2 ENTER^
       -5 ENTER^
        7   ENTER^
        1     R/S         >>>>   -0.130131633 RDN   1.315065816   RDN   -1.453569820

-Flag F01 is set , therefore the 3 solutions are -0.130131633 ; 1.315065816 - 1.453569820.i ; 1.315065816 + 1.453569820.i

Note:

-It's better to replace lines 78-79-80 by the M-Code routine CBRT ( cf "A few M-Code routines for the HP41" )

2°) Quartic Equations

-"P4" solves the equation x⁴+a.x³+b.x²+c.x+d = 0

Data Registers: R00 thru R05: temp
Flags: F01 F02
Subroutines: /

01 LBL "P4"
02 CF 01
03 STO 04
04 X<>Y
05 STO 03
06 R^
07 STO 01
08 *
09 X<>Y
10 4
11 *
12 -
13 STO 00
14 X<>Y
15 STO 02
16 4
17 *
18 RCL 01
19 X^2
20 -
21 RCL 04
22 *
23 RCL 03
24 X^2
25 -
26 27

27 *
28 RCL 02
29 X^2
30 ST+ X
31 RCL 00
32 9
33 *
34 -
35 RCL 02
36 *
37 -
38 STO 05
39 X^2
40 RCL 02
41 X^2
42 RCL 00
43 3
44 *
45 -
46 STO 00
47 3
48 Y^X
49 4
50 *
51 X<Y?
52 GTO 00

53 DEG
54 X#0?
55 /
56 SQRT
57 RCL 05
58 SIGN
59 CHS
60 *
61 ACOS
62 3
63 /
64 COS
65 RCL 00
66 SQRT
67 ST+ X
68 *
69 GTO 01
70 LBL 03
71 STO 02
72 LASTX
73 X=0?
74 SIGN
75 /
76 ABS
77 E-8
78 X<Y?

79 GTO 03
80 CLX
81 STO 02
82 LBL 03
83 RCL 02
84 SQRT
85 RTN
86 LBL 02
87 CF 02
88 2
89 /
90 CHS
91 ENTER
92 X^2
93 RCL Z
94 -
95 X<0?
96 GTO 02
97 SQRT
98 RCL Y
99 SIGN
100 *
101 +
102 X#0?
103 ST/ Y
104 RTN

105 LBL 02
106 CHS
107 SQRT
108 X<>Y
109 SF 02
110 RTN
111 LBL 00
112 -
113 SQRT
114 RCL 05
115 ABS
116 +
117 2
118 /
119 3
120 1/X
121 Y^X
122 RCL 05
123 SIGN
124 CHS
125 *
126 RCL 00
127 RCL Y
128 /
129 +

130 LBL 01
131 RCL 02
132 +
133 3
134 /
135 STO 00
136 RCL 02
137 -
138 RCL 01
139 2
140 ST/ 00
141 ST/ 01
142 ST/ 03
143 /
144 X^2
145 +
146 XEQ 03
147 STO 05
148 RCL 00
149 X^2
150 RCL 04
151 -
152 XEQ 03
153 RCL 00
154 RCL 01

155 *
156 RCL 03
157 -
158 SIGN
159 *
160 RCL 00
161 X<>Y
162 ST- 00
163 +
164 RCL 01
165 RCL 05
166 ST- 01
167 +
168 XEQ 02
169 FS? 02
170 SF 01
171 STO 03
172 X<>Y
173 STO 04
174 RCL 00
175 RCL 01
176 XEQ 02
177 RCL 04
178 RCL 03
179 END

( 217 bytes / SIZE 006 )


STACK	INPUTS	OUTPUTS
T	a	t
Z	b	z
Y	c	y
X	d	x

Example1: Solve x⁴-2.x³-35.x²+36.x+180 = 0

       -2   ENTER^
      -35 ENTER^
       36   ENTER^
      180 XEQ "P4" >>>>   -5 RDN   -2 RDN   6 RDN   3    Flags F01 & F02 are clear , the 4 solutions are   -5 ; -2 ; 6 ; 3

Example2: Solve x⁴-5.x³+11.x²-189.x+522 = 0

        -5   ENTER^
        11   ENTER^
     -189   ENTER^
       522      R/S     >>>>   -2 RDN    5   RDN   6 RDN 3 Flag F01 is set & F02 is clear , the 4 solutions are -2+5.i ; -2-5.i ; 6 ; 3

Example3: Solve x⁴-8.x³+26.x²-168.x+1305 = 0

       -8   ENTER^
       26   ENTER^
     -168 ENTER^
     1305    R/S      >>>>   -2 RDN    5   RDN   6   RDN   3    Flags F01 & F02 are set , the 4 solutions are -2+5.i ; -2-5.i ; 6+3.i ; 6-3.i

Example4: Solve x⁴ + 2.x³+ 5.x² + 4.x+ 4 = 0

2 ENTER^
5 ENTER^
4 ENTER^ R/S >>>> -0.5 RDN 1.322875656 RDN -0.5 RDN 1.322875656 Flags F01 & F02 are set , the 4 solutions are -0.5 +/- 1.322875656 i , -0.5 +/- 1.322875656 i

Notes:

-It's better to replace lines 119-120-121 by the M-Code routine CBRT ( cf "A few M-Code routines for the HP41" )
-LBL 03 avoids a small negative number ( instead of 0 ) that would produce a DATAERROR with SQRT

-We can also delete LBL 03 ( lines 70 to 85 ) after replacing lines 152 & 146 by 0? SQRT where 0? is the following M-Code routine:

0BF "?"
030 "0"
2A0 SETDEC
0F8 C=X
10E A=C ALL
138 C=L
2EE ?C#0
01F JC+03
0AE A<>C ALL
01B JNC+03
261 C=
060 A/C
05E C=|C|
10E A=C ALL
04E C
2BE
35C
050 =
21C
250
250
090 -1 10^(-8)
01D C=
060 A+C
2FE ?C<0?
3A0 ?NC RTN
04E C=0
0E8 X=C
3E0 RTN

3°) Palindromic Polynomials

a) Degree = 4

-Palindromic polynomials are p(x) = a_n.xⁿ+a_n-1.x^n-1+ ... + a₁.x+a₀ with a_n = a₀ , a_n-1 = a₁ , a_n-2 = a₂ and so on ......

"PAL4" solves x⁴ + a x³ + b x² + a x + 1 = 0 we use x + 1/x = y it becomes a quadratic equation.

Data Registers: R00: unused R01 thru R04 = roots
Flags: F01 F02
Subroutines: /

01 LBL "PAL4"
02 CF 01
03 CF 02
04 2
05 ST/ Z
06 -
07 X<>Y
08 CHS
09 ENTER
10 X^2
11 RCL Z
12 -
13 X<0?
14 GTO 02
15 SQRT
16 RCL Y
17 SIGN
18 *
19 +
20 X#0?
21 ST/ Y
22 2

23 ST/ Z
24 /
25 STO 03
26 X<>Y
27 STO 01
28 X^2
29 1
30 -
31 X<0?
32 GTO 00
33 SQRT
34 RCL 01
35 SIGN
36 *
37 RCL 01
38 +
39 STO 01
40 1/X
41 STO 02
42 GTO 01
43 LBL 00
44 CHS

45 SQRT
46 STO 02
47 SF 01
48 LBL 01
49 RCL 03
50 X^2
51 1
52 -
53 X<0?
54 GTO 00
55 SQRT
56 RCL 03
57 SIGN
58 *
59 RCL 03
60 +
61 STO 03
62 1/X
63 STO 04
64 GTO 03
65 LBL 00
66 CHS

67 SQRT
68 STO 04
69 SF 02
70 GTO 03
71 LBL 02
72 CHS
73 SQRT
74 2
75 ST/ Z
76 /
77 STO 02
78 STO 04
79 X<>Y
80 STO 01
81 STO 03
82 R-P
83 X^2
84 X<>Y
85 ST+ X
86 X<>Y
87 P-R
88 1

89 -
90 R-P
91 SQRT
92 X<>Y
93 2
94 /
95 X<>Y
96 P-R
97 ST+ 01
98 ST- 03
99 X<>Y
100 ST+ 02
101 ST- 04
102 SF 01
103 SF 02
104 LBL 03
105 RCL 04
106 RCL 03
107 RCL 02
108 RCL 01
109 END

( 141 bytes / SIZE 005 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	a	y
X	b	x

-If CF 01 & CF 02 the 4 solutions are    x ; y ; z ; t
-If SF 01 & CF 02  the 4 solutions are    x+i.y ; x-i.y ; z ; t
-If CF 01 & SF 02 the 4 solutions are x ; y ; z+i.t ; z-i.t
-If SF 01 & SF 02 the 4 solutions are    x+i.y ; x-i.y ; z+i.t ; z-i.t

Example: x⁴ + 4 x³ - 7 x² + 4 x + 1 = 0

4 ENTER^
-7 XEQ "PAL4" >>>> 0.802775638 = R01
RDN 0.596281205 = R02
RDN -5.421086406 = R03
RDN -0.184464870 = R04

-F01 is set & F02 is clear, so the solutions are 0.802775638 +/- 0.596281205 i ; -5.421086406 ; -0.184464870

b) Degree = 5

"PAL5" solves p(x) = x⁵ + a x⁴ + b x³ + b x² + a x + 1 = 0

(-1) is always a root, so we can divide p by (x+1) thus p(x) = (x+1) q(x) where q(x) is a palindromic polynomial of degree 4

Data Registers: R00: unused    R01 thru R04 = roots R05 = -1
Flags:   F01 F02
Subroutine: "PAL4"

01 LBL "PAL5"
02 RCL Y
03 -
04 1
05 CHS
06 STO 05
07 ST+ Z
08 -
09 XEQ "PAL4"
10 END

( 26 bytes / SIZE 006 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	a	y
X	b	x

-If CF 01 & CF 02 the 4 solutions are    x ; y ; z ; t
-If SF 01 & CF 02  the 4 solutions are    x+i.y ; x-i.y ; z ; t and the 5th root = -1 = R05
-If CF 01 & SF 02 the 4 solutions are x ; y ; z+i.t ; z-i.t
-If SF 01 & SF 02 the 4 solutions are    x+i.y ; x-i.y ; z+i.t ; z-i.t

Example: x⁵ + 4 x⁴ -7 x³ - 7 x² + 4 x + 1 = 0

4 ENTER^
-7 XEQ "PAL5" >>>> 1.679502968 = R01
RDN    0.595414250 = R02
RDN -5.077988862 = R03
RDN -0.196928356 = R04

-Since F01 & F02 are clear, these numbers are the first 4 roots and the 5th root = -1 is stored in R05

c) Degree = 6

"PAL6" solves p(x) = x⁶ + a x⁵ + b x⁴ + c x³ + b x² + a x + 1 = 0 with x + 1/x = y , it becomes a cubic equation.

Data Registers: R00: temp R01 thru R06 = roots
Flags: F01 F02 F03
Subroutine: "P3"

01 LBL "PAL6"
02 RCL Z
03 ST+ X
04 -
05 X<>Y
06 3
07 -
08 X<>Y
09 1
10 RDN
11 XEQ "P3"
12 CF 02
13 CF 03
14 2
15 ST/ T
16 ST/ Z
17 /
18 STO 05
19 RDN
20 FS?C 01
21 GTO 02
22 STO 03
23 X<>Y
24 STO 01
25 X^2

26 1
27 -
28 X<0?
29 GTO 00
30 SQRT
31 RCL 01
32 SIGN
33 *
34 RCL 01
35 +
36 STO 01
37 1/X
38 STO 02
39 GTO 01
40 LBL 00
41 CHS
42 SQRT
43 STO 02
44 SF 01
45 LBL 01
46 RCL 03
47 X^2
48 1
49 -
50 X<0?

51 GTO 00
52 SQRT
53 RCL 03
54 SIGN
55 *
56 RCL 03
57 +
58 STO 03
59 1/X
60 STO 04
61 GTO 03
62 LBL 00
63 CHS
64 SQRT
65 STO 04
66 SF 02
67 GTO 03
68 LBL 02
69 STO 01
70 STO 03
71 X<>Y
72 STO 02
73 STO 04
74 X<>Y
75 R-P

76 X^2
77 X<>Y
78 ST+ X
79 X<>Y
80 P-R
81 1
82 -
83 R-P
84 SQRT
85 X<>Y
86 2
87 /
88 X<>Y
89 P-R
90 ST+ 01
91 ST- 03
92 X<>Y
93 ST+ 02
94 ST- 04
95 SF 01
96 SF 02
97 LBL 03
98 RCL 05
99 X^2
100 1

101 -
102 X<0?
103 GTO 00
104 SQRT
105 RCL 05
106 SIGN
107 *
108 RCL 05
109 +
110 STO 05
111 1/X
112 STO 06
113 GTO 01
114 LBL 00
115 CHS
116 SQRT
117 STO 06
118 SF 03
119 LBL 01
120 RCL 04
121 RCL 03
122 RCL 02
123 RCL 01
124 END

( 161 bytes / SIZE 007 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	a	z
Y	b	y
X	c	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t
-If CF 03 the last 2 roots are u ; v u , v = R05 R06
-If SF 03 the last 2 roots are u+i.v ; u-i.v

Example:     x⁶ + 2 x⁵ - 4 x⁴ + 3 x³ - 4 x² + 2 x + 1 = 0

2 ENTER^
-4 ENTER^
3 XEQ "PAL6" >>>> -3.495175097 = R01
RDN -0.286108699 = R02
RDN -0.068902601 = R03
RDN 0.997623392 = R04

and R05 = 0.959544499 & R06 = 0.281557020 we have CF 01 SF 02 & SF 03

-So the 6 roots are -3.495175097 ; -0.286108699 ; -0.068902601 +/- 0.997623392 i ; 0.959544499 +/- 0.281557020 i

d) Degree = 7

-Like all the palindromic polynomials with an odd degree, -1 is a root
-So, there is a ploynomial q such that (x) = ( x + 1 ) q(x) and deg q = -1 + deg p
-Thus "PAL7" calls "PAL6" as a subroutine

Data Registers: R00: temp R01 thru R06 = roots R07 = -1
Flags:   F01 F02 F03
Subroutine: "PAL6"

01 LBL "PAL7"
02 RCL Z
03 1
04 CHS
05 STO 07
06 +
07 R^
08 RCL Y
09 -
10 R^
11 RCL Y
12 -
13 XEQ "PAL6"
14 END

( 31 bytes / SIZE 008 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	a	z
Y	b	y
X	c	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t
-If CF 03 the 3rd 2 roots are u ; v u , v = R05 R06 and R07 = -1
-If SF 03 the 3rd 2 roots are u+i.v ; u-i.v

Example:     x⁷ + 2 x⁶ - 4 x⁵ + 3 x⁴ + 3 x³ - 4 x² + 2 x + 1 = 0

2 ENTER^
-4 ENTER^
3 XEQ "PAL7" >>>> -3.346973965 = R01
RDN -0.298777347 = R02
RDN    0.500000000 = R03
RDN 0.866025404 = R04

and R05 = 0.822875656 & R06 = 0.568221485 & R07 = -1 we have CF 01 SF 02 & SF 03

-So the 7 roots are -3.346973965 ; -0.298777347 ; 0.5 +/- 0.866025404 i ; 0.822875656 +/- 0.568221485 i ; -1

e) Degree = 8

"PAL8" solves p(x) = x⁸ + a x⁷ + b x⁶ + c x⁵ + d x⁴ + c x³ + b x² + a x + 1 = 0 with x + 1/x = y , it becomes a quartic equation.

Data Registers: R00: temp R01 thru R08: roots
Flags: F01 F02 F03 F04
Subroutine: "P4"

01 LBL "PAL8"
02 CF 03
03 CF 04
04 STO 00
05 RDN
06 STO 01
07 CLX
08 SIGN
09 -
10 ST- 00
11 ST- 00
12 3
13 -
14 RCL Y
15 LASTX
16 *
17 CHS
18 RCL 01
19 +
20 RCL 00
21 XEQ "P4"
22 STO 01
23 RDN
24 STO 03
25 CLX

26 2
27 ST/ 01
28 ST/ 03
29 ST/ Z
30 /
31 FS?C 02
32 GTO 02
33 STO 07
34 X<>Y
35 STO 05
36 X^2
37 1
38 -
39 X<0?
40 GTO 00
41 SQRT
42 RCL 05
43 SIGN
44 *
45 RCL 05
46 +
47 STO 05
48 1/X
49 STO 06
50 GTO 01

51 LBL 04
52 R-P
53 X^2
54 X<>Y
55 ST+ X
56 X<>Y
57 P-R
58 1
59 -
60 R-P
61 SQRT
62 X<>Y
63 2
64 /
65 X<>Y
66 P-R
67 RTN
68 LBL 00
69 CHS
70 SQRT
71 STO 06
72 SF 03
73 LBL 01
74 RCL 07
75 X^2

76 1
77 -
78 X<0?
79 GTO 00
80 SQRT
81 RCL 07
82 SIGN
83 *
84 RCL 07
85 +
86 STO 07
87 1/X
88 STO 08
89 GTO 03
90 LBL 00
91 CHS
92 SQRT
93 STO 08
94 SF 04
95 GTO 03
96 LBL 02
97 STO 05
98 STO 07
99 X<>Y
100 STO 06

101 STO 08
102 X<>Y
103 XEQ 04
104 ST+ 05
105 ST- 07
106 X<>Y
107 ST+ 06
108 ST- 08
109 SF 03
110 SF 04
111 LBL 03
112 FS?C 01
113 GTO 02
114 RCL 01
115 X^2
116 1
117 -
118 X<0?
119 GTO 00
120 SQRT
121 RCL 01
122 SIGN
123 *
124 RCL 01
125 +

126 STO 01
127 1/X
128 STO 02
129 GTO 01
130 LBL 00
131 CHS
132 SQRT
133 STO 02
134 SF 01
135 LBL 01
136 RCL 03
137 X^2
138 1
139 -
140 X<0?
141 GTO 00
142 SQRT
143 RCL 03
144 SIGN
145 *
146 RCL 03
147 +
148 STO 03
149 1/X
150 STO 04
151 GTO 03

152 LBL 00
153 CHS
154 SQRT
155 STO 04
156 SF 02
157 GTO 03
158 LBL 02
159 RCL 03
160 STO 02
161 STO 04
162 RCL 01
163 STO 03
164 XEQ 04
165 ST+ 01
166 ST- 03
167 X<>Y
168 ST+ 02
169 ST- 04
170 SF 01
171 SF 02
172 LBL 03
173 RCL 04
174 RCL 03
175 RCL 02
176 RCL 01
177 END

( 232 bytes / SIZE 009 )


STACK	INPUTS	OUTPUTS
T	a	t
Z	b	z
Y	c	y
X	d	x

-If CF 01 the first roots are    x ; y
-If SF 01 the first roots are x+i.y ; x-i.y
-If CF 02 the 2nd roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd roots are z+i.t ; z-i.t
-If CF 03 the 3rd roots are u ; v u , v = R05 R06 w , r = R07 R08
-If SF 03 the 3rd roots are u+i.v ; u-i.v
-If CF 04 the last roots are w ; r
-If SF 04 the last roots are w+i.r ; w-i.r

Example:     x⁸ + 2 x⁷ - 4 x⁶ + 3 x⁵ + 7 x⁴ + 3 x³ - 4 x² + 2 x + 1 = 0

2 ENTER^
-4 ENTER^
3 ENTER^
7 XEQ "PAL8" >>>> -3.321130852 = R01
RDN -0.301102258 = R02
RDN    -0.792609764 = R03
RDN 0.609729254 = R04

and R05 = 1.134460522 R06 = 1.063259900 R07 = 0.469265796 R08 = -0.439813897

-Since we have CF 01 SF 02 SF 03 SF 04 the 8 roots are:

-3.321130852 ; -0.301102258 ; -0.792609764 +/- 0.609729254 i ; 1.134460522 +/- 1.063259900 i ; 0.469265796 +/- 0.439813897 i

f) Degree = 9

-1 is always a root of p(x) = x⁹ + a x⁸ + b x⁷ + c x⁶ + d x⁵ + d x⁴ + c x³ + b x² + a x + 1 = 0 so p(x) = ( x + 1 ) q(x) with deg q = 8 & q palindromic too.

"PAL9" calls "PAL8" as a subroutine.

Data Registers: R00: temp R01 thru R08 = roots R09 = -1
Flags:   F01 F02 F03 F04
Subroutine: "PAL8"

01 LBL "PAL9"
02 STO 00
03 RDN
04 STO 01
05 CLX
06 RCL Z
07 -
08 ST- 01
09 RCL 01
10 ST- 00
11 1
12 CHS
13 STO 09
14 ST+ T
15 ST- Z
16 ST+ Y
17 ST- 00
18 X<> 00
19 XEQ "PAL8"
20 END

( 42 bytes / SIZE 010 )


STACK	INPUTS	OUTPUTS
T	a	t
Z	b	z
Y	c	y
X	d	x

-If CF 01 the first roots are    x ; y
-If SF 01 the first roots are x+i.y ; x-i.y
-If CF 02 the 2nd roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd roots are z+i.t ; z-i.t
-If CF 03 the 3rd roots are u ; v u , v = R05 R06 w , r = R07 R08 R09 = -1
-If SF 03 the 3rd roots are u+i.v ; u-i.v
-If CF 04 the last roots are w ; r
-If SF 04 the last roots are w+i.r ; w-i.r

Example:     x⁹ + 2 x⁸ - 4 x⁷ + 3 x⁶ + 7 x⁵ + 7 x⁴ + 3 x³ - 4 x² + 2 x + 1 = 0

2 ENTER^
-4 ENTER^
3 ENTER^
7 XEQ "PAL9" >>>> -3.328907873 = R01
RDN -0.300398821 = R02
RDN    -0.428871616 = R03
RDN 0.903365450 = R04

and R05 = 1.289363998 R06 = 1.084685542 R07 = 0.454160965 R08 = -0.382065757 R09 = -1

-Since we have CF 01 SF 02 SF 03 SF 04 the 9 roots are:

-3.328907873 ; -0.300398821 ; -0.428871616 +/- 0.903365450 i ; 1.289363998 +/- 1.084685542 i ; 0.454160965 +/- 0.382065757 i ; -1

4°) Integer Roots

a) Real Roots

-We assume the polynomial p(x) = a_n.xⁿ+a_n-1.x^n-1+ ... + a₁.x+a₀ ( a₀ # 0 ) has integer coefficients and integer roots

Data Registers: R00 .... R04: temp ( Registers Rbb thru Ree are to be initialized before executing "IROOT" )

• Rbb = a_n • Rbb+1 = a_n-1 , ........ , • Ree = a₀ ( bb > 04 )

Flag: F10
Subroutines: /

01 LBL "IROOT"
02 CF 10
03 STO 00
04 STO 04
05 FRC
06 E3
07 *
08 RCL 00
09 INT
10 -
11 1
12 STO 01
13 -
14 STO 03
15 GTO 03
16 LBL 00

17 RCL 00
18 RCL 01
19 0
20 LBL 01
21 RCL Y
22 *
23 RCL IND Z
24 +
25 ISG Z
26 GTO 01
27 X#0?
28 RTN
29 E-3
30 ST- 00
31 RCL 00
32 STO 02

33 CLX
34 DSE 03
35 LBL 02
36 RCL 01
37 *
38 ST+ IND 02
39 RCL IND 02
40 ISG 02
41 GTO 02
42 RCL 01
43 STO IND 02
44 SF 10
45 RTN
46 LBL 03
47 RCL 03
48 X=0?

49 GTO 05
50 XEQ 00
51 FS?C 10
52 GTO 03
53 RCL 01
54 CHS
55 STO 01
56 LBL 04
57 RCL 03
58 X=0?
59 GTO 05
60 XEQ 00
61 FS?C 10
62 GTO 04
63 1
64 RCL 01

65 -
66 STO 01
67 GTO 03
68 LBL 05
69 SIGN
70 RCL 00
71 +
72 RCL IND X
73 RCL IND L
74 /
75 CHS
76 STO IND Y
77 RCL 04
78 ISG X
79 END

( 121 bytes / SIZE ??? )

STACK	INPUTS	OUTPUTS
Y	/	Rbb+1
X	bbb.eee	1+bbb.eee

( with bbb > 004 )

Example: p(x) = x⁵ - 64 x³ - 66 x² + 927 x + 1890 = 0

-If you choose bbb = 011 1 STO 11 0 STO 12 -64 STO 13 -66 STO 14 927 STO 15 1890 STO 16

11.016 XEQ "IROOT" >>>> 12.016 ---Execution time = 26s---
X<>Y    7 = R12

R12 = 7 R13 = -6 R14 = 5 R15 = -3 R16 = -3

-Thus, p(x) = ( x - 7 ) ( x+6 ) ( x - 5 ) ( x + 3 )²

Notes:

-In fact, the last root ( stored in Rbb+1 ) might be non-integer.
-The precision is exact, provided the calculations do not involve numbers > 10¹⁰

-Of course, this program is slow if the roots are very large...

b) Complex Roots

-The same method is used by "IRZ" to comput the complex roots ( Gaussian integers ) of a polynomial

p(z) = c_n zⁿ + c_n-1 z^n-1 + .................. + c₁ z + c₀          with         c_k = a_k + i.b_k where a_k & b_k are integers ( c₀ # 0 )

Data Registers: R00 .... R09: temp ( Registers Rbb thru Ree are to be initialized before executing "IRZ" )

   • Rbb = a_n • Rbb+1 = b_n , ........ , • Ree-1 = a₀ • Ree= b₀    ( bb > 09 )

Flag: F10
Subroutines: /

01 LBL "IRZ"
02 CF 10
03 STO 00
04 STO 03
05 FRC
06 3
07 10^X
08 *
09 RCL 00
10 INT
11 -
12 3
13 -
14 2
15 STO 08
16 /
17 STO 04
18 GTO 03
19 LBL 00
20 RCL 00
21 STO 05
22 CLST
23 LBL 01
24 STO 06
25 RCL 02
26 *
27 X<>Y
28 STO 07

29 RCL 01
30 *
31 +
32 RCL 01
33 RCL 06
34 *
35 RCL 02
36 RCL 07
37 *
38 -
39 RCL IND 05
40 +
41 X<>Y
42 ISG 05
43 RCL IND 05
44 +
45 X<>Y
46 ISG 05
47 GTO 01
48 X=0?
49 X#Y?
50 RTN
51 2 E-3
52 ST- 00
53 RCL 00
54 STO 05
55 CLST
56 DSE 04

57 LBL 02
58 STO 06
59 RCL 01
60 *
61 X<>Y
62 STO 07
63 RCL 02
64 *
65 -
66 RCL IND 05
67 +
68 STO IND 05
69 RCL 02
70 RCL 06
71 *
72 RCL 01
73 RCL 07
74 *
75 +
76 ISG 05
77 RCL IND 05
78 +
79 STO IND 05
80 X<>Y
81 ISG 05
82 GTO 02
83 RCL 01
84 STO IND 05

85 ISG 05
86 CLX
87 RCL 02
88 STO IND 05
89 SF 10
90 RTN
91 LBL 03
92 RCL 08
93 STO 09
94 DSE X
95 STO 01
96 CLX
97 STO 02
98 LBL 04
99 RCL 04
100 X=0?
101 GTO 09
102 XEQ 00
103 FS?C 10
104 GTO 04
105 RCL 02
106 X=0?
107 GTO 08
108 CHS
109 STO 02
110 LBL 05
111 RCL 04
112 X=0?

113 GTO 09
114 XEQ 00
115 FS?C 10
116 GTO 05
117 LBL 08
118 RCL 01
119 X=0?
120 GTO 08
121 CHS
122 STO 01
123 LBL 06
124 RCL 04
125 X=0?
126 GTO 09
127 XEQ 00
128 FS?C 10
129 GTO 06
130 RCL 02
131 X=0?
132 GTO 08
133 CHS
134 STO 02
135 LBL 07
136 RCL 04
137 X=0?
138 GTO 09
139 XEQ 00
140 FS?C 10

141 GTO 07
142 LBL 08
143 RCL 01
144 ABS
145 STO 01
146 SIGN
147 ST- 01
148 ST+ 02
149 DSE 09
150 GTO 04
151 ST+ 08
152 GTO 03
153 LBL 09
154 RCL 03
155 ISG X
156 RCL IND X
157 STO 04
158 ISG Y
159 RCL IND Y
160 STO 05
161 *
162 ISG Y
163 RCL IND Y
164 STO 06
165 RCL IND 03
166 STO 07
167 *
168 -

169 RCL 05
170 RCL 07
171 *
172 RCL 04
173 RCL 06
174 *
175 +
176 CHS
177 RCL 04
178 X^2
179 RCL 07
180 X^2
181 +
182 ST/ Z
183 /
184 2
185 ST+ 03
186 RDN
187 STO IND 03
188 ISG 03
189 X<>Y
190 STO IND 03
191 X<>Y
192 RCL 03
193 1
194 -
195 END

( 268 bytes / SIZE ??? )

STACK	INPUTS	OUTPUTS
Z	/	Rbb+3
Y	/	Rbb+2
X	bbb.eee	2+bbb.eee

( with bbb > 009 )

Example: p(x) = (2+3.i) z⁵ + (34-27.i) z⁴ + (-122-66.i) z³ + (738-466.i) z² + (-4080-3065.i) z + (-5100+6325.i) = 0

-If you choose bbb = 011

2 STO 11 34 STO 13 -122 STO 15 738 STO 17 -4080 STO 19 -5100 STO 21
3 STO 12 -27 STO 14 -66 STO 16 -466 STO 18 -3065 STO 20 6325 STO 22

11.022 XEQ "IRZ" >>>> 13.022 ---Execution time = 6m45s---
RDN -4
RDN 7

-And we have R13 = -4 R14 = 7 R15 = 3 R16 = 4 R17 = 3 R18 = 4 R19 = 0 R20 = -5 R20 = -1 R21 = 2

-Thus, p(z) = ( 2 + 3.i ) ( z + 4 - 7.i ) ( z - 3 - 4.i )² ( z + 5.i ) ( z + 1 - 2.i )

Notes:

-In fact, the last root ( stored in Rbb+2 & Rbb+3 ) might be non-integer.
-The precision is exact, provided the calculations do not involve numbers > 10¹⁰

-Of course, this program is very slow if the roots are very large...

5°) A Few Quintic Equations

a) Program#1

-We have Sinh 5y = 16 Sinh⁵ y + 20 Sinh³ y + 5 Sinh y
and Sin 5y = 16 Sin⁵ y - 20 Sin³ y + 5 Sin y

-So, we can use these formulae to solve x⁵ + 5 k x³ + 5 k² x + A = 0

Data Registers: R00 thru R06: temp R05 = 5th root
Flags: F01 F02
Subroutine: "P4"

01 LBL "P5"
02 CF 01
03 CF 02
04 CHS
05 X<>Y
06 STO 01
07 X<0?
08 GTO 00
09 X#0?
10 GTO 01
11 X<>Y
12 SIGN
13 LASTX
14 ABS
15 5
16 1/X
17 Y^X
18 *
19 GTO 03
20 LBL 00
21 CHS
22 X^2
23 LASTX
24 SQRT
25 ST+ X
26 STO 00
27 *
28 /
29 ABS

30 1
31 X<Y?
32 GTO 02
33 DEG
34 LASTX
35 ASIN
36 5
37 /
38 SIN
39 STO 05
40 LASTX
41 72
42 STO 01
43 +
44 SIN
45 STO 04
46 LASTX
47 RCL 01
48 +
49 SIN
50 STO 03
51 LASTX
52 RCL 01
53 +
54 SIN
55 STO 02
56 LASTX
57 RCL 01
58 +

59 SIN
60 RCL 00
61 ST* 02
62 ST* 03
63 ST* 04
64 ST* 05
65 *
66 STO 01
67 RCL 04
68 RCL 03
69 RCL 02
70 RCL 01
71 GTO 04
72 LBL 01
73 X^2
74 LASTX
75 SQRT
76 STO T
77 ST+ X
78 *
79 /
80 ENTER
81 X^2
82 1
83 +
84 SQRT
85 +
86 LN
87 5

88 /
89 E^X-1
90 LASTX
91 CHS
92 E^X-1
93 -
94 *
95 GTO 03
96 LBL 02
97 X<> L
98 SIGN
99 X<>Y
100 LASTX
101 X^2
102 1
103 -
104 SQRT
105 +
106 LN
107 5
108 /
109 E^X
110 ENTER
111 1/X
112 +
113 *
114 RCL 01
115 CHS
116 SQRT

117 *
118 LBL 03
119 STO 06
120 RCL 01
121 X^2
122 5
123 *
124 STO 05
125 X<>Y
126 X^2
127 RCL 01
128 5
129 *
130 +
131 STO Y
132 RCL 06
133 *
134 RCL X
135 LASTX
136 *
137 RCL 05
138 +
139 RCL 06
140 RDN
141 XEQ "P4"
142 STO 05
143 X<> 06
144 X<> 05
145 LBL 04
146 END

( 177 bytes / SIZE 007 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	k	y
X	A	x

-If CF 01 & CF 02 the 4 solutions are    x ; y ; z ; t
-If SF 01 & CF 02  the 4 solutions are    x+i.y ; x-i.y ; z ; t and the 5th root R05 = u
-If CF 01 & SF 02 the 4 solutions are x ; y ; z+i.t ; z-i.t
-If SF 01 & SF 02 the 4 solutions are    x+i.y ; x-i.y ; z+i.t ; z-i.t

Example1: x⁵ - 15 x³ + 45 x + 4 = 0 ( k = -3 , A = 4 )

-3 ENTER^
4 XEQ "P5" >>>> -3.321006904
RDN -1.963370398
RDN 2.107577266
RDN 3.265924783 and R05 = -0.089124743

-Since we have CF 01 & CF 02, these 5 numbers are the 5 roots of this quintic equation

Example2: x⁵ + 15 x³ + 45 x + 100 = 0 ( k = 3 , A = 100 )

3 ENTER^
100 XEQ "P5" >>>> -0.412527941
   RDN    3.530731294
   RDN 1.080012167
   RDN 2.182111936 and R05 = -1.334968451

-We have SF 01 & SF 02 , so the 5 roots are: -0.412527941 +/- 3.530731294 i ; 1.080012167 +/- 2.182111936 i ; -1.334968451

b) Program#2

-The following program solves x⁵ + A x³ + a x² + b x + c = 0 IF A = c / a + a b / c

Data Registers: R00 thru R07: temp
Flags:   F00 F01
Subroutine: "P3"

01 LBL "P5"
02 CF 00
03 RCL Z
04 /
05 STO 06
06 STO 07
07 /
08 ST+ 06
09 0
10 X<> Z
11 1
12 RDN
13 XEQ "P3"
14 RCL 07
15 X>0?
16 SF 00
17 ABS
18 SQRT
19 END

( 34 bytes / SIZE 008 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	a	z
Y	b	y
X	c	x

-If CF 00 & CF 01 the 5 solutions are    x ; -x ; y ; z ; t
-If SF 00 & CF 01 the 5 solutions are    i.x ; -i.x ; y ; z ; t
-If CF 00 & SF 01 the 5 solutions are x ; -x ; y ; z+i.t ; z-i.t
-If SF 00 & SF 01 the 5 solutions are    i.x ; -i.x ; y ; z+i.t ; z-i.t

Example: x⁵ + A x³ + 3 x² + 4 x - 9 = 0 with A = - 13/3

3 ENTER^
4 ENTER^
-9 XEQ "P5" >>>> 1.732050808
RDN -1.746676973
RDN 0.873338487
RDN -0.977152498

- CF 00 & SF 01 so the 5 roots are 1.732050808 ; -1.732050808 ; -1.746676973 ; 0.873338487 +/- 0.977152498 i

Notes:

A is calculated and stored in R06. In this example, R06 = A = -4.333333333

-We have solved ( x² + c/a ) ( x³ + (ab/c) x + a ) = 0

c) Program#3

-The following program solves x⁵ + A x⁴ + B x³ + C x² + D x + E = 0 after rewriting it ( x² + a x + b ) ( x³ + a x² + b x + c ) = 0

-You choose B , C , D and "P5" solves an equivalent quartic equation

If the quartic equation has no real root, no result.
If the quartic equation has 2 real roots, , there are 2 solutions.
If the quartic equation has 4 real roots , there are 4 solutions.

-When there is no more solutions, the HP41 displays " END"

Data Registers: R00 thru R17: temp R01 thru R05 = 5 roots R06 = A , R07 = B , R08 = C , R09 = D , R10 = E
Flags:   F00 F01 F02
Subroutines: "P2" "P3" "P4"

01 LBL "P5"
02 STO 09
03 RDN
04 STO 08
05 X<>Y
06 STO 07
07 6
08 *
09 CHS
10 X<>Y
11 4
12 *
13 RCL 09
14 LASTX
15 *
16 CHS
17 RCL 07
18 X^2
19 +
20 5

21 ST/ T
22 ST/ Z
23 /
24 0
25 RDN
26 XEQ "P4"
27 STO 11
28 RDN
29 STO 12
30 RDN
31 STO 13
32 X<>Y
33 STO 14
34 FS? 01
35 GTO 00
36 RCL 11
37 XEQ 01
38 RCL 12
39 XEQ 01
40 LBL 00

41 FS?C 02
42 GTO 00
43 RCL 13
44 XEQ 01
45 RCL 14
46 XEQ 01
47 GTO 00
48 LBL 01
49 STO 06
50 X^2
51 CHS
52 RCL 07
53 +
54 2
55 /
56 STO 15
57 1
58 X<>Y
59 RCL 06
60 X<>Y

61 XEQ "P2"
62 STO 16
63 X<>Y
64 STO 17
65 RCL 08
66 RCL 07
67 RCL 06
68 *
69 -
70 RCL 06
71 3
72 Y^X
73 +
74 STO 10
75 1
76 RCL 06
77 RCL 15
78 R^
79 XEQ "P3"
80 STO 05
81 RDN

82 STO 03
83 X<>Y
84 STO 04
85 RCL 15
86 ST* 10
87 RCL 06
88 ST+ 06
89 RCL 17
90 STO 02
91 X<> 16
92 STO 01
93 RCL 04
94 RCL 03
95 RCL 02
96 RCL 01
97 STOP
98 RTN
99 LBL 00
100 " END"
101 AVIEW
102 END

( 143 bytes / SIZE 018 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	B	z
Y	C	y
X	D	x

-If CF 00 & CF 01 the 4 roots are    x ; y ; z ; t
-If SF 00 & CF 01 the 4 roots are    x+i.y ; x-i.y ; z ; t and the 5th root R05 = u
-If CF 00 & SF 01 the 4 roots are x ; y ; z+i.t ; z-i.t
-If SF 00 & SF 01 the 4 roots are    x+i.y ; x-i.y ; z+i.t ; z-i.t

Example: x⁵ + A x⁴ + 3 x³ + 7 x² + 6 x + E = 0

3 ENTER^
7 ENTER^
6 XEQ "P5" >>>> 3.052611654
RDN   -0.541473338
RDN -0.326247858
RDN 0.552188522 and R05 = 3.163634031

CF 00 & SF 01 so the roots are 3.052611654 ; -0.541473338 ; -0.326247858 +/- 0.552188522 i ; 3.163634031

-The equation is x⁵ - 5.022276632 x⁴ + 3 x³ + 7 x² + 6 x + 2.151028651 = 0 because R06 = A = -5.022276632 & R10 = E = 2.151028651

R/S >>>> -0.500000000
RDN   0.866025405
RDN 0.440619700
RDN -1.569610322 and R05 = -1.881239400

SF 00 & SF 01 so the roots are -0.5 +/- 0.866025405 i ; 0.440619700 +/- 1.569610322 i ; -1.881239400

-The equation is x⁵ + 2 x⁴ + 3 x³ + 7 x² + 6 x + 5 = 0 because R06 = A = 1.999999999 & R10 = E = 5.000000005

R/S >>>> " END" thus, there is nomore solution.

6°) A Few Sextic Equations

"P6" solves   x⁶ + A x⁴ + B x³ + C x² + D x + E = 0 if E = B²/4 - ( -C + (D/B)² )² / 4 / ( -A + 2D/B )

Data Registers: R00 thru R15: temp R01 thru R06 = 6 roots R11= A , R12 = B , R13 = C , R14 = D , R15 = E
Flags:   F01 F02
Subroutine: "P3"

01 LBL "P6"
02 CF 02
03 STO 14
04 RDN
05 STO 13
06 RDN
07 STO 12
08 X<>Y
09 STO 11
10 RCL 14
11 RCL 12
12 /
13 STO 09
14 ST+ X

15 -
16 STO 08
17 RCL 09
18 X^2
19 RCL 13
20 -
21 2
22 /
23 STO 07
24 X^2
25 X<>Y
26 /
27 RCL 12
28 2

29 /
30 STO 10
31 X^2
32 +
33 STO 15
34 RCL 08
35 CHS
36 SQRT
37 ST/ 07
38 STO 08
39 CHS
40 RCL 09
41 RCL 10
42 RCL 07

43 -
44 1
45 RDN
46 XEQ "P3"
47 FS? 01
48 SF 02
49 STO 06
50 RDN
51 X<> 10
52 X<>Y
53 X<> 07
54 +
55 1
56 RCL 08

57 RCL 09
58 R^
59 XEQ "P3"
60 STO 05
61 RDN
62 STO 01
63 X<>Y
64 STO 02
65 RCL 07
66 STO 04
67 RCL 10
68 STO 03
69 RCL 02
70 RCL 01
71 END

( 91 bytes / SIZE 016 )


STACK	INPUTS	OUTPUTS
T	A	t
Z	B	z
Y	C	y
X	D	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t

-The last 2 roots are always real u ; v u , v = R05 R06

Example:   x⁶ + x⁴ - 4 x³ - x² - 4 x + E = 0

1 ENTER^
-4 ENTER^
-1 ENTER^
-4 XEQ "P6" >>>> -0.771844506
RDN 1.115142508
RDN -0.287371537
RDN 1.349996398 and R05 = 0.543689012 R06 = 1.574743073

SF 01 & SF 02 thus the 6 roots are -0.771844506 +/- 1.115142508 i ; -0.287371537 +/- 1.349996398 i ; 0.543689012 ; 1.574743073

-R15 = E = 3.000000000 so the equation is x⁶ + x⁴ - 4 x³ - x² - 4 x + 3= 0

7°) A Few Septic Equations

a) Program#1

-We have: Sinh 7y = 64 Sinh⁷ y + 112 Sinh⁵ y + 56 Sinh³ y + 7 Sinh y and Sin 7y = -64 Sin⁷ y + 112 Sin⁵ y - 56 Sin³ y + 7 Sin y

-So we can use these formulae to solve x⁷ + 7 k x⁵ + 14 k² x³ + 7 k³ x + A = 0

Data Registers: R00: temp R01 thru R07 = roots R08: temp
Flags:   F01 F02 F03
Subroutine: /

-Lines 19 & 49 are three-byte GTOs

01 LBL "P7"
02 RAD
03 PI
04 ST+ X
05 7
06 /
07 STO 08
08 RDN
09 CF 00
10 CF 01
11 CF 02
12 CF 03
13 CHS
14 X<>Y
15 STO 01
16 X<0?
17 GTO 01
18 X#0?
19 GTO 03
20 X<>Y
21 SIGN
22 LASTX
23 ABS
24 7
25 1/X
26 Y^X
27 *
28 STO 07
29 RCL 08
30 X<>Y
31 P-R
32 STO 01
33 X<>Y

34 STO 02
35 RCL 08
36 ST+ X
37 ST+ 08
38 RCL 07
39 P-R
40 STO 03
41 X<>Y
42 STO 04
43 RCL 08
44 RCL 07
45 P-R
46 STO 05
47 X<>Y
48 STO 06
49 GTO 05
50 LBL 00
51 RCL 06
52 RCL Y
53 RCL 06
54 E^X
55 P-R
56 R^
57 CHS
58 R^
59 CHS
60 E^X
61 P-R
62 CHS
63 X<>Y
64 CHS
65 ST+ T
66 RDN

67 +
68 FS? 00
69 X<>Y
70 RTN
71 LBL 01
72 CHS
73 ENTER
74 X^2
75 LASTX
76 SQRT
77 ST+ X
78 STO 00
79 *
80 *
81 /
82 ABS
83 1
84 X<Y?
85 GTO 02
86 LASTX
87 ASIN
88 7
89 /
90 SIN
91 STO 07
92 LASTX
93 RCL 08
94 +
95 SIN
96 STO 01
97 LASTX
98 RCL 08
99 +

100 SIN
101 STO 02
102 LASTX
103 RCL 08
104 +
105 SIN
106 STO 03
107 LASTX
108 RCL 08
109 +
110 SIN
111 STO 04
112 LASTX
113 RCL 08
114 +
115 SIN
116 STO 05
117 LASTX
118 RCL 08
119 +
120 SIN
121 RCL 00
122 CHS
123 ST* 01
124 ST* 02
125 ST* 03
126 ST* 04
127 ST* 05
128 ST* 07
129 *
130 STO 06
131 GTO 06
132 LBL 02

133 SF 00
134 X<> L
135 SIGN
136 CHS
137 ST* 00
138 X<>Y
139 ENTER
140 X^2
141 2
142 ST/ 00
143 SIGN
144 -
145 SQRT
146 +
147 PI
148 CHS
149 2
150 /
151 STO 04
152 RDN
153 LN
154 7
155 /
156 STO 06
157 E^X
158 ENTER
159 1/X
160 +
161 CHS
162 GTO 04
163 LBL 03
164 X^2
165 LASTX

166 ST* Y
167 SQRT
168 STO 00
169 ST+ X
170 *
171 /
172 ENTER
173 X^2
174 0
175 STO 04
176 SIGN
177 +
178 SQRT
179 +
180 LN
181 7
182 /
183 E^X-1
184 LASTX
185 STO 06
186 CHS
187 E^X-1
188 -
189 LBL 04
190 STO 07
191 RCL 04
192 RCL 08
193 +
194 XEQ 00
195 STO 01
196 X<>Y
197 STO 02
198 RCL 08

199 ST+ X
200 ST+ 08
201 RCL 04
202 +
203 XEQ 00
204 STO 03
205 X<>Y
206 X<> 04
207 RCL 08
208 +
209 XEQ 00
210 STO 05
211 X<>Y
212 RCL 00
213 ST* 01
214 ST* 02
215 ST* 03
216 ST* 04
217 ST* 05
218 ST* 07
219 *
220 STO 06
221 LBL 05
222 SF 01
223 SF 02
224 SF 03
225 LBL 06
226 DEG
227 CF 00
228 RCL 04
229 RCL 03
230 RCL 02
231 RCL 01
232 END

( 289 bytes / SIZE 009 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	k	y
X	A	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t
-If CF 03 the 3rd 2 roots are u ; v u , v = R05 R06 and R07 = w = the 7th root ( always real )
-If SF 03 the 3rd 2 roots are u+i.v ; u-i.v

Example1:     x⁷ + 49 x⁵ + 686 x³ + 2401 x - 100 = 0 ( k = 7 , A = -100 )

   7 ENTER^
-100 XEQ "P7" >>>> 0.025955070
RDN 4.137191362
RDN -0.009263257
RDN 5.158993245 and R05 = -0.037506163 R06 = 2.295967991 R07 = 0.041628699

   SF 01 & SF 02 & SF 03 so the 7 roots are:

0.025955070 +/- 4.137191362 i ; -0.009263257 +/- 5.158993245 i ; -0.037506163 +/- 2.295967991 i ; 0.041628699

Example2:     x⁷ - 49 x⁵ + 686 x³ - 2401 x + 4 = 0 ( k = -7 , A = 4 )

   -7 ENTER^
4 XEQ "P7" >>>> -4.136024417
RDN -5.159204061
RDN -2.297397818
RDN 2.294395841 and R05 = 5.158462634 R06 = 4.138101851 R07 = 0.001665974

-Since we have CF 01 & CF 02 & CF 03 these 7 real numbers are the 7 roots.

b) Program#2

-The following routine solves the equation x⁷ + A x⁶ + B x⁵ + C x⁴ + D x³ + E x² + F x + G = 0 if it may be re-written: ( x⁴ + a x³ + b x + c ) ( x³ + d x + e ) = 0

-You choose A , B , C , D , E

Data Registers: R00 thru R17: temp    ( Registers R11 thru R15 are to be initialized before executing "P7" )

R01 thru R07 = 7 roots    • R11= A , • R12 = B , • R13 = C , • R14 = D , • R15 = E and when the program stops: R16 = F R17 = G

Flags:   F01 F02 F03
Subroutines: "P3" "P4"

01 LBL "P7"
02 CF 03
03 RCL 12
04 RCL 13
05 RCL 11
06 RCL 12
07 *
08 -
09 RCL 15
10 RCL 12
11 /
12 STO 16

13 -
14 STO 10
15 ST* 16
16 STO 17
17 1
18 0
19 RDN
20 RDN
21 XEQ "P3"
22 FS? 01
23 SF 03
24 STO 07

25 RDN
26 STO 08
27 X<>Y
28 STO 09
29 RCL 11
30 RCL 14
31 RCL 10
32 RCL 11
33 *
34 -
35 STO 00
36 ST* 17

37 RCL 15
38 RCL 12
39 ST* 00
40 /
41 RCL 00
42 ST+ 16
43 CLX
44 X<> Z
45 XEQ "P4"
46 STO 01
47 RDN
48 STO 02

49 RDN
50 STO 03
51 X<>Y
52 STO 04
53 RCL 08
54 STO 05
55 RCL 09
56 STO 06
57 RCL 04
58 RCL 03
59 RCL 02
60 RCL 01
61 END

( 85 bytes / SIZE 018 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	/	z
Y	/	y
X	/	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t
-If CF 03 the 3rd 2 roots are u ; v u , v = R05 R06 and R07 = w = the 7th root ( always real )
-If SF 03 the 3rd 2 roots are u+i.v ; u-i.v

Example:     x⁷ + 3 x⁶ + 2 x⁵ - 7 x⁴ + 5 x³ + 4 x² + F x + G = 0 ( A = 3 , B = 2 , C = -7 , D = 5 , E = 4 )

3 STO 11 -7 STO 13 4 STO 15
2 STO 12 5 STO 14

    XEQ "P7" >>>> -2.825275998
   RDN 1.404195934
   RDN 1.325275998
   RDN 1.807422116 and R05 = -1.098545526 R06 = 2.370739719 R07 = 2.197091051

   SF 01 & SF 02 & SF 03 so the 7 roots are:

   -2.825275998 +/- 1.404195934 i ; 1.325275998 +/- 1.807422116 i ; -1.098545526 +/- 2.370739719 i ; 2.197091051

-And we have R16 = 70 & R17 = -750

-So, we have found the 7 roots of x⁷ + 3 x⁶ + 2 x⁵ - 7 x⁴ + 5 x³ + 4 x² + 70 x - 750 = 0

8°) A Few Octic Equations

-The following routine solves the equation x⁸ + A x⁴ + B x² + C x + D = 0 if it may be re-written: ( x⁴ + a x + b ) ( x⁴ - a x + c ) = 0

-You choose A , B < 0 & C

Data Registers: R00 thru R15: temp    R11= A , R12 = B , R13 = C , R14 = D

R01 thru R08 = 8 roots

Flags:   F01 F02 F03 F04
Subroutine: "P4"

01 LBL "P8"
02 CF 03
03 CF 04
04 STO 13
05 RDN
06 STO 12
07 CHS
08 SQRT
09 STO 14
10 X<>Y
11 STO 11
12 *
13 STO 15

14 RCL 13
15 ST+ 15
16 -
17 RCL 14
18 ST+ X
19 ST/ 15
20 /
21 STO 08
22 0
23 ENTER
24 ENTER
25 RCL 14
26 R^

27 XEQ "P4"
28 FS? 01
29 SF 03
30 FS? 02
31 SF 04
32 STO 09
33 RDN
34 STO 10
35 RDN
36 STO 07
37 X<>Y
38 X<> 08
39 RCL 15

40 *
41 X<> 14
42 CHS
43 LASTX
44 0
45 ENTER
46 R^
47 R^
48 XEQ "P4"
49 STO 01
50 RDN
51 STO 02
52 RDN

53 STO 03
54 X<>Y
55 STO 04
56 RCL 10
57 STO 06
58 RCL 09
59 STO 05
60 RCL 04
61 RCL 03
62 RCL 02
63 RCL 01
64 END

( 88 bytes / SIZE 016 )


STACK	INPUTS	OUTPUTS
T	/	t
Z	A	z
Y	B < 0	y
X	C	x

-If CF 01 the first 2 roots are    x ; y
-If SF 01 the first 2 roots are x+i.y ; x-i.y
-If CF 02 the 2nd 2 roots are z ; t x , y , z , t = R01 R02 R03 R04
-If SF 02 the 2nd 2 roots are z+i.t ; z-i.t
-If CF 03 the 3rd 2 roots are u ; v
-If SF 03 the 3rd 2 roots are u+i.v ; u-i.v u , v , w , p = R05 R06 R07 R08
-If CF 04 the last 2 roots are w ; p
-If SF 04 the last 2 roots are w+i.p ; w-i.p

Example: x⁸ + 4x⁴ - 9 x² + 6 x + D = 0

4 ENTER^
-9 ENTER^
6 XEQ "P8" >>>> -0.973561484 = R01
RDN 1.310797213 = R02
RDN 0.973561484 = R03
RDN    0.421253594 = R04

and R05 = -1.307486100 R06 = -0.337666766 R07 = 0.822576433 R08 = 1.260317961

-We have SF 01 SF 02 CF 03 SF 04 thus, the roots are:

-0.973561484 +/- 1.310797213 i ; 0.973561484 +/- 0.421253594 i ; -1.307486100 ; -0.337666766 ; 0.822576433 +/- 1.260317961 i    and R14 = 3

-So these numbers are the 8 roots of the equation:    x⁸ + 4x⁴ - 9 x² + 6 x + 3 = 0

References:

[1] Abramowitz and Stegun - "Handbook of Mathematical Functions" - Dover Publications - ISBN 0-486-61272-4
[2] David J. Wolters - "Tutorial on Analytic Algorithms to Solve Cubic and Quartic Equations"
[3] Raghavendra G. Kulkarni - "Solving certain quintics"
[4] Raghavendra G. Kulkarni - "Solving Sextic Equations"

hp41programs

Polynomials(III) for the HP-41