About thermocouples

• x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x) + (a7 * x * x * x * x * x * x * x) ;

Yeah, but doesn't C have an exponentiation operator? It's been so long since I've done any real programming, I've forgotten most of what I learned. Isn't there a construct like x**y or so? (which is shorthand to write, but still needs all those multiplies)...

Thanks, Rich

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

No, unfortunately. Just a standard library function-- pow(x,y) => x^y

Anyway, you shouldn't ever use exponentiation to evaluate a polynomial.

• x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x) + (a7 * x * x * x * x * x * x * x) ;

???? Topic was in algebra, not boolean logic...

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

But i am..multiplies can be further reduced.

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

Oh, i waited, ad read the article, was just bitching that this pageload BS is becoming more common.

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

X^y does not need y multiplies..

On a sunny day (Tue, 21 Sep 2010 11:15:56 -0400) it happened Spehro Pefhany wrote in :

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) +

Well, really. Your example was not working, so anyways I decided to write the reverse of v2t, t2v (means voltage to temperature) for all thermocouples one is likely to use. In the same time as people are discussing when to multiply (pun intended), I have a nice working program using the NIST polinomials you pointed out. I tested some randopm values, and the range warnings, all seems OK. Note that this is a bit trickier then it may seem at first sight, as different polinomials are used for different temperature ranges, and also additional processing is needed, for example in typ K: if temp > 0, than E = sum(i=0 to n) c_i t^i + a0 exp(a1 (t - a2)^2) if temp 400.0) return 0;

// E = sum(i=0 to n) c_i t^i

// E = sum(i=0 to n) c_i t^i

// E = sum(i=0 to n) c_i t^i

// if > 0, than E = sum(i=0 to n) c_i t^i + a0 exp(a1 (t - a2)^2) // if 1768.00) return 0;

// E = sum(i=0 to n) c_i t^i

return 1; } /* end function temperature_to_voltage_type_S */

void print_usage() { fprintf(stderr, "Panteltje (c) t2v-0.1 thermocouple temperature (in °C) to voltage calculator\n"); fprintf(stderr, "Usage: t2v type[E|J|K|R|S|T] temperature\n"); fprintf(stderr, "Example:\n\ t2v T -270\n"); } /* end function print_usage */

int main(int argc, char **argv) { double temperature; double voltage;

if(argc != 3) { print_usage();

exit(1); }

/* get temperature */ temperature = atof(argv[2]);

/* get thermocouple type */ switch(argv[1][0]) { case 'E': if(! temperature_to_voltage_type_E(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); }

break; case 'J': if(! temperature_to_voltage_type_J(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); }

break; case 'K': if(! temperature_to_voltage_type_K(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); }

break; case 'R': if(! temperature_to_voltage_type_R(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); } break; case 'S': if(! temperature_to_voltage_type_S(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); } break; case 'T': if(! temperature_to_voltage_type_T(temperature, &voltage) ) { fprintf(stderr, "temperature out of range, aborting.\n");

exit(1); }

break; default: print_usage();

exit(1); break; } /* end switch thermocouple type */

/* display to user in C, K, and F */ fprintf(stdout, "voltage is %.6f V\n", voltage / 1000.0);

exit(0); } /* end function main */

You mean, less than the order of the polynomial? You can get to, say, the fifth power of some number with fewer than five multiplies?

OK, I'm teachable. Can you show me?

Thanks, Rich

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

The point was that the caret operator in C doesn't mean "x to the y", it means "x XOR y".

I guess part of the point is that C isn't English. ;-)

Cheers! Rich

Log rule?

VLV

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

The question was about _implementation_, no?

The answer is that no, it's a really, really bad way to implement a polynomial evaluation.

Whether pow(x,n) is better or worse than than x*x*..*x (for floating point x and positive integer n) is probably dependent on the value of n, as well as the compiler and the target.

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

What algorithm are you assuming? And what value(s) of y? Ln and exp are not cheap unless you have a suitable FPU.

W=(x*x), Y=(W*W), ANS=Y*x (three multiplies) I leave the rest to the student.

Nope; just a bit of factoring and use of the rules of math. See my previous answer.

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

Symple . . . only multiplies . . . three for y^5.

(a4 * x * x * x * x ) + (a5 * x * x * x * x * x) + (a6 * x * x * x * x * x * x)

• (a7 * x * x * x * x * x * x * x) ;

It's simple, but inferior to using Horner's rule when you look at total performance, which has n multiplies total vs. (O) n * log2(n), right?

Best regards, Spehro Pefhany

The polynomial is going to be used to correct a _thermocouple_. How often is that going to need to be evaluated? 10 times per second at most?

If it really matters, there are also representations more compact than Horner--you don't need to do N multiplies to get x**N, it's more like log(N) multiplies. There are even FFT tricks to evaluate polynomials in a compact way.

The issue with polynomials represented as sums of coefficients times powers of x**N, using Horner or any of the trickier algorithms, is that the basis set is very nearly linearly dependent. That is, X**N looks a whole lot like X**N-2. (There's a discussion of this in the Wikipedia article on the Hilbert matrix,

That makes the evaluation of the coefficients horribly ill conditioned, which loses accuracy quite independently of execution speed, compiler details, and everything else.

Using Chebyshev polynomials makes the whole thing very well conditioned, because Chebyshevs are just cosines with a distorted horizontal axis, so they're all nicely orthogonal and everything.

If you don't know about Clenshaw's rule, it's worth looking it up, because it's both useful and pretty. Wikipedia isn't that useful because it's just the math--

Cheers

Phil Hobbs

It was actually more of an operational slip-up. I studied it at the time.