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         Volume 3 Number 12      48/39/38            December 1978

                     Newsletter of the SR-52 Users Club
                                published at
                           9459 Taylorsville Road
                              Dayton, OH 45424
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Roots of Polynomials (V3N9p1)
     Dix Fulton (83) and Frank Stallings (389) join Bill in address-
ing this topic.  Some time ago Dix had submitted a Quartic, Cubic,
Quadratic program to PPX-59 (#338005B), and has sent me an improved
version, listed below.  This is shorter than Bill's V3N9 program,
primarily because the same processing is used for all 3 degrees:  All
5 coefficients are entered, regardless of the degree.  For a quadratic,
D and E are zero, and for a cubic. E is zero, since the "quartic"
Ax4+Bx3+Cx2+0x+0=0 can be divided by x2, making it the quadratic
Ax2+Bx+C=0 which it really is, and similarly the "quartic" Ax4+Bx3+
Cx2+Dx+0=0 reduces to a cubic.  But quadratic and cubic processing
are no faster than quartic, and the correct 2 or 3 roots must be
identified among the always output 4.
     Dix spotted a flaw in Bill's V3N9 program:  In same cases pre-
cision is unnecessarily lost following a rounding procedure.  Bill
has corrected that, and reworked quadratic processing to improve
accuracy when the 2 discriminant terms have a large difference in
magnitude (V3N9p2).  So for some cases, Bill's new program, listed
below, is more accurate than Dix's.
     Maurice Swinnen (779) has translated for English-speaking 59/PC
users a Quartic, Cubic, Quadratic program written for German-speaking
users, appearing in Display (V3N4/5S91).  However, it requires 4
card sides of memory, and suffers the quadratic accuracy loss with
critical discriminants.
     Frank Stallings has been experimenting with iterative methods
for finding the roots of higher order polynomials, and wrote the
program listed below following the so-called Bairstow method.  Frank's
program appears to work well on a few sample problems, producing both
real and complex roots for polynomials up to 61st degree, at a rate of
about a minute per degree.  However, there are no known iterative
methods which work for all polynomials, and the Bairstow method (like
the Newton method which it uses) is known to be bad for multiple roots
solutions.  So in cases where convergence is slow, Frank has provided
a flag option allowing the monitoring of successive root approximations.

TI-59/PC Program:  Polynomial Roots                L Frank Stallings (389)

User Instructions:  Press RST, R/S; see the prompter:  DEGRE printed.
Key degree (2-61), press R/S; key the coefficients, beginning with the
highest, and follow each with R/S.  Processing begins following input
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
  The SR-52 Users Club is a non-profit loosely organised group of TI PPC owners/users
  who wish to get more out of their machines by exchanging ideas.  Activity centers
  on a monthly newsletter, 52-NOTES edited and published by Richard C Vanderburgh
  in Dayton, Ohio.  The SR-52 Users Club is neither sponsored nor officially sanctioned
  by Texas Instruments, Inc.  Membership is open to any interested person:  $6.00 ($10
  abroad) includes 6 issues of 52-NOTES; back issues start June 1976 @ $1.00 each.
  ($1.67 abroad).

of the constant.  Monitor convergence (or lack thereof) by setting
flag 0.  With good convergence, roots are printed in pairs about every
2 minutes.

Program Listing:

000:  CMs Adv Adv Op00 1617223517 Op2 CLR Op05 7 Op17 R/S Prt Adv GTO
027:  053 LE EE INV EE rtn LB 1) (R*69 ÷ R0 rtn Prt S*69 1 SUM69 ±
053:  SUM68 R68 CP x≥t 045 LC Adv Adv 3532323736 Op2 Op5 GTO 084 Prt
083:  Adv Dsz69 p51 R*69 x=t 082 2 ± ÷ R69 ÷ R0 X (CP X R1) S62 R2 ÷
112:  (1 ± ÷ R69 = S64 R64 X R68 ÷ R63 X R67 + R*62) (Exc67 S68 R64
143:  X R66 ÷ R63 X R65 + (1 SUM62 INV SUM69 R69 CP x=t 121) INV x≥t
169:  181 R67) Exc65 S66 GTO 121 SUM62) 0 = X S69 R66 ± + R65 x2 =
197:  Exc66 X R67 - 0 Exc68 X R65 = ÷ 0 Exc66 INV x=t 219 B = SUM63
222:  Prd69 Abs + ((0 Exc62 Exc69 ÷ 0 Exc67) ÷ 0 Exc65 INV x=t 247 B)
248:  INV SUM64 Abs = X 1 EE 8 = Op0 INV Ifflg0 266 Pause INV x=t 121
270:  CLR 2 INV SUM69 1/x X (R63 x2 + 4 X R64) INV x≥t 308 (√x + R63)
297:  - E Prt Adv R63 = E GTO 331 ± √x = E x:t 24 Op4 R63 ÷ 2 = E Prt
324:  x:t Op6 Adv x:t Prt x:t ± Op6 Adv R*62 X (CE X 1 SUM62 R62 x:t
347:  R69 INV x≥t 370 R63) SUM*62 1 SUM62 R64 = SUM*62 Dsz62 335 CLR
371:  S62 2 x:t R69 x≥t 121 x:t CLR x=t 395 R1 ÷ R0) ± E Prt Adv RST

Note:  The p51 at step 086 is the pseudo BST which must be created
       synthetically (R51 BST BST Del)
                         - - - - - - - - - -

TI-59/(PC) Program:  Quadratic, Cubic, Quartic Roots  Bill Skillman (710)

User Instructions:  Same as V3N9p2

Program Listing:

000:  GTO 620 LE' SBR623 rtn LD' 1 Exc1 1/x Prd2 Prd3 Prd4 rtn LB' D'
025:  35 Op4 R2 ÷ 3 = S8 x2 ± + R3 ÷ x:t 3 = S10 R4 + R8 X (x2 X 2 -
058:  0 x:t = S11 x2 + R10 X x2 X 4 = INV x≥t 147 √x + x:t R11 = ÷
083:  2 ± + SBR126 x:t = SBR126 + x:t - R8 + Ifflg2 142 E' R8 = ÷ 2
110:  ± + x:t = X 3 √x = x:t - R8 = GTO 609 (S13 Op10 Exc13 Abs INV yx
136:  3 X R13) rtn 0 = S21 rtn R10 ± √x X x:t 2 = S12 R11 ÷ 2 ÷ x:t yx
164:  3 = ± Rad INV Cos ÷ 3 = S9 SBR196 S21 SBR187 S22 2 X 2 X p ÷ 3 +
193:  R9 = Cos X R12 - R8 = S23 Ifflg2 719 GTO E' LA x:t 3 Op17 CMs
219:  x:t S1 x:t 13 RST LB S2 x:t 14 RST LC S3 x:t 15 RST LD s4 x:t
247:  16 RST LE S5 x:t 17 RST LA' D' 35 Op4 R2 ÷ 2 = ± S6 S7 x2 - R3
279:  = CP x≥t 296 ± √x x:t R6 Ifflg4 516 GTO 609 √x X R6 Op10 =
303:  SUM6 R3 ÷ R6 GTO 628 LC' R2 CP x=t 469 D' Prd5 R2 X R4 - 4 X R5
334:  = Exc3 S15 X ± Exc2 S14 4 - R14 x2 = X R5 - R4 x2 = Exc4 S16
363:  Stflg2 20 S0 SBR025 INV Stflg2 CP CLR Op20 R*0 + R14 x2 ÷ 4 -
388:  R15 = SBR581 INV x≥t 376 √x S19 ± + R14 ÷ 2 INV Prd*0 = S2 R14
415:  X R*0 - R16 = Op10 X (R*0 x2 - R5) SBR581 INC x≥t 376 √x = S20
444:  ± + R*0 = S3 SBR261 R19 SUM2 SUM2 R20 SUM3 SUM3 GTO 261 R4 INV
472:  x=t 321 R3 x=t 540 R5 Exc3 S2 Stflg4 GTO 260 SBR496 R7 CP x≥t
498:  511 ± √x x:t SBR616 ± x:t GTO 616 √x E' ± GTO E' x:t INV P/R ÷ 2
521:  = x:t √x x:t P/R x:t S6 SBR 603 x:t 1 ± PRD6 GTO 603 R5 ÷ R1 =
546:  x≥t 569 ± √x √x x:t 35 SBR620 ± E' ± x:t SBR616 ± GTO E' Nop ÷
570:  4 = √x √x X x:t 1 = GTO 527 Nop S18 Fix4 EE INV EE INV Fix INV
592:  x=t 597 CLR rtn R18 rtn Nop Nop Nop 35 Op4 R6 E' 47632024 Stflg3
620:  Op4 x:t Op6 GTO 705 = S7 R6 Ifflg4 491 E' R7 GTO 623
705:  Op8 Ifflg3 717 Op69 INV Stflg3 R/S CE rtn
                           - - - - - - - - -

                              52-NOTES V3N12p2

TI-59 Program:  Quartic, Cubic, Quadratic Solutions  Dix Fulton (83)

User Instructions:  Input coefficients per V3N9p2, following a manual
CMs.  Press E' and see 4 roots printed; or without printer press E',
see 0, press A', see root 1, press B', see root 2, press C', see root
3, press D', see root 4.  For each root, the imaginary part is pause-
displayed, followed by an x:t displaying the real part.

Program Listing:

000:  LA' INV LB' Stflg1 6 GTO STO LC' INV LD' Stflg1 8 LBL STO S9
022:  R*9 ÷ 2 = x2 - Op39 R*9 Op29 = CP x≥t Rcl Abs √x x:t Lbl Rcl
044:  √x Ifflg1 SUM x:t ± x:t ± Lbl SUM - R*9 ÷ 35171327 Op04 2 = Adv
071:  Op6 x:t S09 24301322 Op4 R9 Op6 Pause x:t rtn LE' R4 1/x Prd0
100:  Prd1 Prd2 Prd3 R1 S6 x2 + R3 Prd6 x2 X R0 + R2 ± S7 X (4 X R0)
133:  INV SUM6 = ± S5 R7 ÷ 3 X S7 R6 - R5 = ÷ 2 - R7 X x2 = S8 x2 +
165:  (R6 ÷ 3 - R7 x2) X x2 = CP x≥t Cos ± √x x:t R8 x:t INV P/R
189:  x:t INV yx 3 = x:t ÷ 3 = P/R X 3 √x ÷ 2 = S6 ± S5 x:t INV SUM5
214:  INV SUM6 X 2 = x:t R5 INV x≥t Sin x:t Lbl Sin R6 x≥t Tan x:t
234:  GTO Tan Lbl Cos √x SUM8 ± X 2 + R8 + x:t = op10 X x:t Abs INV
256:  yx 3 + R8 Op10 X R8 Abs INV yx 3 Lbl Tan - R7 = S6 ÷ 2 = S5 +
284:  ((x2 - R0 + Abs) ÷ 2) √x INV SUM5 = S7 R3 ÷ 2 = S8 + (x2 + R6
316:  - R2) √x INV SUM8 = S6 X R7 + R8 X R5 - R1 = Abs x:t R6 X R5 +
348:  R8 X R7 - R1 = Abs x≥t Exc R5 Exc7 S5 Lbl Exc R4 Prd0 Prd1 Prd2
376:  Prd3 0 H8 1 Op4 H18 CP x=t Prd INV Fix A' B' C' D' Adv Adv Adv
398:  Lbl Prd R/S LA S4 x:t 1305 Op4 x:t Op6 R/S LB S3 x:t 1404 Op4
427:  x:t Op6 R/S LC S2 x:t 1303 Op4 x:t Op6 R/S LD S1 x:t 1302 Op4
457:  x:t Op6 R/S LE S0 x:t 1301 Op4 x:t Op6 R/S GTO E'
                          - - - - - - - - -

The Math/Utilities CROM (V3N8p5,6)
     A few pre-production modules have now been made, and TI expects
production to begin mid-December, with first deliveries expected by
early January.  Many of the programs in this new CROM do indeed reflect
better programming quality than has gone into earlier ones, and the
routines themselves will probably turn out to be the most universally
useful.  Fortunately, subroutine callability (from RAM) was one of the
design criteria.  Following is a brief run-down of the 21 programs:
     MU-01 is the usual module check and register-clearer.  (60 steps)
     MU-02 works only with the PC, providing a few common prompting
messages, including multiple-card read-write directions with code
which cleverly interrogates the first card-side read to determine how
many more sides are to be read.  Unfortunately, the prompter refers
to "cards" instead of "sides", which may confuse some users. (329 steps)
     MU-03 is similar to the LE-10 Memopad, but with more symbol-code
keyboard-addressable, and with the words ENTER, PRESS, and PRESS SBR
single-key specifiable.  Users will need to make their own overlays
for finding the alpha prefixes, as TI budget constraints did not allow
for one to be included in the module package.  (474 steps)
     MU-04 is used with MU-03 to format both data and text in any
desired configuration for each line.  It's slow, but can save paper.
(430 steps)
     MU-05 is titled Superplotter, and makes it easy for the user to
generate plots of up to ten different functions, with size and precision
limited only by acceptable execution time and paper.  The result is one
long tape which the user cuts and reassembles with the aid of well-
chosen alignment and coordinate symbols.  This looks like a winner.
(444 steps)

                              52-NOTES V3N12p3

     MU-06 is the fast sorter (V3N8p6) which orders one sequence of
99 random numbers in 9¼ minutes, but which takes 19½ minutes for the
same code executed in RAM.  The V3N2p5 program, which was TI's point
of departure, takes 23 1/3 minutes to order the same sequence.  The
MU-06 interface with the user was well planned, giving the user a
good choice of I/O options for either keyboard or RAM program call.
(132 steps)
     MU-07 is a data array processor, and at 650 steps is the longest
MU program.  Up to 93 (53 for the 58) elements of a 2-dimensional
array can be arithmetically combined or changed by row or column, and
the elements shifted right or left by row.  There are a lot of business
oriented manipulative options, probably best run via a RAM program with
lots of prompting.  MU-07 appears to be an extension of the Business
Decisions Project Planning and Budgeting program (BD-05).
     MU-08 is a general-purpose data packer and unpacker of positive
integers, allowing the user to specify mixed as well as constant data
lengths for 13-digit per register packing into Registers 4 onward to
the end of partitioning.  The 3 functions:  store, recall, and ex-
change each take about 4 seconds to execute, and may be invoked in
any order any number of times.  (226 steps)
     MU-09 is a disappointing factor finder:  It's short (TI claims it
needed to cut down on memory requirements, although MU-09 is only 7
steps shorter than the V3N11p4 program), but slow (doesn't bypass trial
divisors which are multiples of smaller ones already tried (V3N11p4),
and does quite a bit of flag manipulating).  (85 steps)
     MU-10 is a fast, straightforward generator of hyperbolic trig
functions, and their inverses.  (100 steps)
     MU-11 calculates the gamma function and factorials for both inte-
gers and positive reals up to 69 or 70, and the natural logs of these
2 functions for reals up to about 1010.  (147 steps)
     MU-12 is a short random number generator which produces uniformly
distributed numbers in the 0-1 range the same way ML-15 does, but a
bit more efficiently.  It uses MU-13 to generate normally distributed
numbers.  (54 steps)
     MU-13 does normal distribution processing, 2 of the functions
producing the same results as 2 of the Applied Statistics Normal Dist-
ribution program (ST-19); Routines A and C of MU-13 correspond to
routines C and A, respectively, of ST-19.  (260 steps)
     MU-14 mechanizes the Aitken interpolation algorithm, fitting up
to a 26th order polynomial to 27 data points (13th order for 14 data
points for the 58), which is then used to solve for intermediate points.
(193 steps)
     MU-15 is a primitive root-finder, requiring the user to provide
approximations to each (real only) root.  It does provide an option to
specify the max number of iterations for cases where convergence is
expected to be slow or maybe nonexistent.  (99 steps)
     MU-16 finds the max and min values of user-supplied functions in
specifiable intervals by looking for a change in sign of an approxima-
tion to the function's derivative.  This approximation (also used by
MU-15) imposes processing limitations of which the user needs to be
aware.  It would seem that a straight forward sample-and-save-largest-
and-smallest approach (V2N8p4,5) would have been better.  (211 steps)

                              52-NOTES V3N12p4

     MU-17 mechanizes Romberg integration of definite integrals, allow-
ing the user to specify accuracy.  (243 steps)
     MU-18 uses a 4th-order Runge-Kutta method to solve differential
equations of the y'=f(x,y) and y"=f(x,y,y') types.  (293 steps)
     MU-19 performs discrete Fourier series summations on as many
input equally spaced data points as there are registers available
from Reg 16 onward.  If a user's RAM program needs a few registers
from Reg 16 on, he can specify a higher start register for MU-19 to
use for input storage.  (115 steps)
     MU-20 does a good job of determining, recording, and resetting
the status (V3N2p2) of flags, partitioning, angle mode, and Fix.  The
number of open parentheses, and printer connection can be determined,
but not recorded or reset.  (307 steps)
     MU-21 is designed to aid manual (keyboard) calculations by
assigning 5 of the user-defined keys to specified variables and sub-
routines.  I can see where single-key variable recall might help to
speed things up for the non-programmer, but since the user has to
write whatever subroutines he wants, anyway, he might as well call
them directly.  (101 steps)
     I'll comment further in future articles, as I delve into the MU
programs in greater detail, and/or start getting inputs from the
membership.
                      - - - - - - - - - - - - -
Revealed Firmware (V3N10p4, V3N11p1,2)
     Steve Bepko (45), Maurice Swinnen (779), Dave Leising (890), and
John Mickelsen (990) have all found more ROM code past step 487.  Steve
and John got to step 575 by single stepping without the printer;
Maurice and Dave to step 583 with the printer in trace mode.  Appar-
ently no one else had bothered to SST past step 487, since what Steve
and John did works with any of the established ROM-revelation proce-
dures.  For a shortcut confirmation, key at turn-on:  GTO 479 9 Op17
Pgm 12 SBR 444 R/S P/R LRN, and SST to step 575, at which point the
next SST causes a switch to RUN mode.  If LRN is then pressed, the
whole keyboard appears to be locked out with a displayed C.
     Confirmation of Maurice's and Dave's approach can be made with a
59/PC at turn-on with:  Stflg 9 9 Op17 Pgm12 SBR 444 P/R P/R LRN BST
which lists steps 000-583 without mnemonics, and can only be stopped
by turning the machine off.  If instead of keying the Stflg 9 you press
(latch) the TRACE key, printing will stop when the TRACE key is
unlatched.
     Steve found that some regular (with mnemonics) listing could be
made past step 487 by first SSTing past step 487, then keying LRN List.
It turns out that step 489 is the lowest that will work, and in any
case step 503 is as high as it goes, starting over at step 039 set to
code 80 (Grd) this time, and continuing on listing RAM from step 040
to the end of the current partition.
     Bill Skillman (710) identifies steps 384-511 as non-normalized
constants used by the transcendental functions.  (Steps 512-575 repeat
steps 384-447).  Bill finds that if the significant digits in the
equivalent 16 data registers are allowed to be rescaled, they can be
interpreted as:  ln 10, ln 2, ln 1.1, ln 1.01, ln 1.001, ln 1.0001,
ln 1.00001, ln 1.000001, π/4, arctan .1, arctan .01, arctan .001, arc-
tan .0001, π/2, π, and 180/π.  If steps 384-447 are written in RAM,

                              52-NOTES V3N12p5