PC-1211, PC-1250, etc. TVM

[Albert Chan]

To make the post more clearly, I quoted the code used here

(06-05-2024 01:05 AM)Albert Chan Wrote:  10 INPUT "B,N,P,M,F? ";B,N,P,M,F @ B=B*M
20 DEF FNF(I) @ S=-EXPM1(LOGP1(I)*-N)
25 D=(P+F)/S*(1-N*(1-S)/(S+S/I))+(B-F)
30 FNF=(P+F)*I/S+(B-F)*I+M @ END DEF ! FNF=NPMT
40 X=(ABS(P)<ABS(F))*(P+F)-B @ I=M/(X-P) @ Y=FNF(I)
50 H=-Y/D @ I=I+H @ Y=FNF(I) ! Newton's method
60 DISP 100*I,Y @ IF I+H*H<>I THEN 50

Since Sharp does not have EXPM1, LOGP1, lets try without them.
For good measure, we also report the next iteration.

>20 DEF FNF(I) @ S=1-(1+I)^(-N)
>70 DISP (I-Y/D)*100
>run
B,N,P,M,F? 0,360,0,-1000,1e6
.313028391431      504.47069099
.471270710728      62.96905505
.497374290313      1.51995026
.498036167469      .00095736
.498036584886      -.0000025
.498036583796

Final I is an improvement ... but it is really luck. Garbage in, Garbage out.
More Newton iterations suggested we only have 8 to 9 digits accuracy.

.498036583796
.498036583626
.498036583308
.49803658459
.498036583243
...

---

One way is to correct for S (1+I) error.

(1+I)^(-n) = (1+I+ε)^(-n) * (1 + ε/(1+I)) ^n ≈ (1+I+ε)^(-n) * (1 + n*ε/(1+I))

>20 DEF FNF(I) @ X=1+I @ S=X^(-N) @ S=1-S-N*(X-1-I)/X*S
>run
B,N,P,M,F? 0,360,0,-1000,1e6
.313028391431      504.47068755
.471270710304      62.96905734
.497374290775      1.51994785
.498036166884      .00095668
.498036584005      .00000001
.498036584009

---

Much better, but S calculation is messy. A novel way is to set ε=0 !!!

>20 DEF FNF(I) @ S=1-(1+I)^(-N)
>50 H=-Y/D @ I=I+H+1-1 @ Y=FNF(I) ! Newton's method
>run
B,N,P,M,F? 0,360,0,-1000,1e6
.313028391           504.47068892
.47127071             62.96905807
.497374291           1.51994733
.498036167           .00095642
.498036584           .00000002
.498036584009