This problem comes from the brain of Valentin Albillo on the HP Calculator Museum.

Given a polynomial Ax^3 + Bx^2 + Cx + D = 0 find the coefficients A B C D that give a root at x = pi.

A is in the range 1 to 9
B, C, D are in the range -9 to 9
Use a range [3.1, 3.2] for x

Here’s a program for the HP-41cx (Run-time is about 80 minutes) :

Registers:


01 A    hardcoded to start at 1

02 B    -M to M

03 C    ditto

04 D    ditto

05 x1   3.1

06 x2   3.2

07 f(x1)

08 f(x2)

09 M    9


Labels:


LBL SM start of program

LBL 01 main loop

LBL 02 compute f(x1)

LBL 03 compute f(x2)

LBL 04 success!


The program:


01 LBL SM  set up variables

02 1

03 STO 01

04 RCL 09

05 CHS

06 STO 02

07 STO 03

08 STO 04

09 3.1

10 STO 05

11 3.2

12 STO 06

13 LBL 01  main loop

14 XEQ 02  calc f(x1)

15 XEQ 03  calc f(x2)

16 RCL 07

17 RCL 08

18 /       divide them and look for -ve

19 0

20 X<>Y

21 X<=Y?   -ve?

22 GTO 04  success !! x1 and x2 straddle the root

23 RCL 04

24 1

25 +

26 STO 04  D=D+1

27 RCL 09

28 X<>Y

29 X<=Y?   <=M ?

30 GTO 01  loop

31 X<>Y

32 CHS

33 STO 04  else D=-M

34 1

35 RCL 03

36 +

37 STO 03  C=C+1

38 RCL 09

39 X<>Y

40 X<=Y?   C<=M?

41 GTO 01  loop

42 X<>Y

43 CHS

44 STO 03  else C=-M

45 1

46 RCL 02

47 +

48 STO 02  B=B+1

49 RCL 09

50 X<>Y

51 X<=Y?   B<=M?

52 GTO 01  loop

53 X<>Y

54 CHS

55 STO 02  B=-M

56 1

57 RCL 01

58 +

59 STO 01  A=A+1

60 GTO 01  loop — should be a check here

61 "FAILED"
62 AVIEW

63 LBL 04  we come here if we find a root in [x1,x2]

64 "FOUND IT"

65 AVIEW

66 RTN

67 END

68 LBL 02  Calculate f(x1)

69 RCL 05

70 3

71 Y^X

72 RCL 01

73 *

74 RCL 05

75 X^2

76 RCL 02

77 *

78 +

79 RCL 05

80 RCL 03

81 *

82 +

83 RCL 04

84 +

85 STO 07

86 RTN

87 LBL 03  Calculate f(x2)

88 RCL 06

89 3

90 Y^X

91 RCL 06

92 X^2

93 RCL 02

94 *

95 +

96 RCL 06

97 RCL 03

98 *

99 +

100 RCL 04

101 +

102 STO 08

103 RTN

104 END