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# Recursive Functions

 P: n/a I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? Thanks! Nov 13 '05 #1
64 Replies

 P: n/a On Mon, 27 Oct 2003 09:54:05 -0800, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This question is off topic here. I would suggest you try posting in comp.programming, but I can't actually figure out what you want. You've described the algorithm completely, what more do you want? -Sheldon p.s. I'm hoping that what you want isn't for someone to simply do your homework for you. Nov 13 '05 #2

 P: n/a On 2003-10-27, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This is not a C question... but the C answer is use pow() (unless you are purposefully avoiding floating point, in which case a table lookup is in order). Rewrite your word problem into a recurrence relationship. Use induction to prove this recurrence correctly calculates x to the nth power. It should be obvious at that point what your function should look like and what your base case is. -- James Nov 13 '05 #3

 P: n/a dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This seems a fairly clear description of the procedure, assuming `n' is a non-negative integer. (If `n' can be negative the problem is harder; if `n' can be a non-integer problem is much harder.) What problem are you having with it? Perhaps if you'd show us what you've written thus far, someone will be able to help. -- Er*********@sun.com Nov 13 '05 #4

 P: n/a dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? /* untested, unreadable, unambitious, unenthusiastic */ #define k \ unsigned char k Power(k x,k n){k r=1;if(n ){if(1==n){r= x;}else{k t = Power(x,n>>1) ;if(1&n){r=x; }r *= t*t; }} return r ;} -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton Nov 13 '05 #5

 P: n/a "Sheldon Simms" wrote in message news:pa****************************@yahoo.com... On Mon, 27 Oct 2003 09:54:05 -0800, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This question is off topic here. I would suggest you try posting in comp.programming, but I can't actually figure out what you want. You've described the algorithm completely, what more do you want? Well, to help make it on topic, I sometimes notice C's lack of an integer power function, such as that described. (Though I believe that it works better as an iterative function than a recursive function.) Sometimes I use an SQ macro, which squares an expression through mutliplication. If the expression is more than a single variable, I hope that the compiler does common subexpression elimination. -- glen Nov 13 '05 #6

 P: n/a dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? Thanks! Here is tested C code for raising an integer to an integer power, recursively. (I hope this isn't a HW assignment but a legitimate query.) // Recursive raise integer to integer power // // If j is even, return (i^2)^(j/2), else return i * (i^2)^(j/2) // // ---------------------------------------------------- // (c) Copyright 2003 Julian V. Noble. // // Permission is granted by the author to // // use this software for any application pro- // // vided this copyright notice is preserved. // // ---------------------------------------------------- #include #include int power( int i, int j ); // prototype int main( void ) { int a; int b; printf("What are m and n? "); scanf(" %d", &a); scanf(" %d", &b); printf( " %d\n", power(a,b) ) ; return 0; } int power( int i, int j ) { int k; if (j==0) // i^0 = 1 { return 1; } if (j & 1) // odd? k = i; else // even? k = 1; return k * power( i*i, j/2); } -- Julian V. Noble Professor Emeritus of Physics jv*@spamfree.virginia.edu ^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ "Science knows only one commandment: contribute to science." -- Bertolt Brecht, "Galileo". Nov 13 '05 #7

 P: n/a Glen Herrmannsfeldt wrote: "Sheldon Simms" wrote in message news:pa****************************@yahoo.com... On Mon, 27 Oct 2003 09:54:05 -0800, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This question is off topic here. I would suggest you try posting in comp.programming, but I can't actually figure out what you want. You've described the algorithm completely, what more do you want? Well, to help make it on topic, I sometimes notice C's lack of an integer power function, such as that described. (Though I believe that it works better as an iterative function than a recursive function.) Sometimes I use an SQ macro, which squares an expression through mutliplication. If the expression is more than a single variable, I hope that the compiler does common subexpression elimination. -- glen You are correct in saying that the iterative is probably better than the recursive version. If anyone is interested I'll post my iterative version. -- Julian V. Noble Professor Emeritus of Physics jv*@spamfree.virginia.edu ^^^^^^^^ http://galileo.phys.virginia.edu/~jvn/ "Science knows only one commandment: contribute to science." -- Bertolt Brecht, "Galileo". Nov 13 '05 #8

 P: n/a dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? double Power(double x, unsigned int n) { return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; } Nov 13 '05 #9

 P: n/a "E. Robert Tisdale" wrote: dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? double Power(double x, unsigned int n) { return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; } I confess that I considered suggesting this method, but then decided that wanton cruelty was not (yet) justified. -- Er*********@sun.com Nov 13 '05 #10

 P: n/a On Mon, 27 Oct 2003, Eric Sosman wrote: "E. Robert Tisdale" wrote: dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? double Power(double x, unsigned int n) { return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; } I confess that I considered suggesting this method, but then decided that wanton cruelty was not (yet) justified. Well, either that or the interpretation of "suitable base case to stop the recursion" might need some clarification. Nov 13 '05 #11

 P: n/a On Tue, 28 Oct 2003, Jarno A Wuolijoki wrote: On Mon, 27 Oct 2003, Eric Sosman wrote: "E. Robert Tisdale" wrote: double Power(double x, unsigned int n) { return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; } I confess that I considered suggesting this method, but then decided that wanton cruelty was not (yet) justified. Well, either that or the interpretation of "suitable base case to stop the recursion" might need some clarification. In particular, where is it guaranteed that the base case (n <= 0) is ever reached? I admit I'm not conversant in the details of C floating point, but it seems like an unwarrantedly dubious assumption to be making. What's to say that 1e-100/2 != 1e-100 ? -Arthur Nov 13 '05 #12

 P: n/a "Arthur J. O'Dwyer" wrote: On Tue, 28 Oct 2003, Jarno A Wuolijoki wrote: On Mon, 27 Oct 2003, Eric Sosman wrote: > "E. Robert Tisdale" wrote: > > > > double Power(double x, unsigned int n) { > > return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; > > } > > I confess that I considered suggesting this method, but > then decided that wanton cruelty was not (yet) justified. Well, either that or the interpretation of "suitable base case to stop the recursion" might need some clarification.In particular, where is it guaranteed that the base case (n <= 0)is ever reached? I admit I'm not conversant in the details ofC floating point, but it seems like an unwarrantedly dubiousassumption to be making. What's to say that 1e-100/2 != 1e-100 ? Err, n is of type unsigned int, thus all divisions are integer divisions... Regards -- Irrwahn (ir*******@freenet.de) Nov 13 '05 #13

 P: n/a On Tue, 28 Oct 2003, Irrwahn Grausewitz wrote: "Arthur J. O'Dwyer" wrote:> > "E. Robert Tisdale" wrote: > > > > double Power(double x, unsigned int n) { > > return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; > > }In particular, where is it guaranteed that the base case (n <= 0)is ever reached? I admit I'm not conversant in the details ofC floating point, but it seems like an unwarrantedly dubiousassumption to be making. What's to say that 1e-100/2 != 1e-100 ? Err, n is of type unsigned int, thus all divisions are integer divisions... Oops. I was thrown by the use of 'double' everywhere else. Seems silly to use 'double' for the first operand if you're only going to allow positive integers for the second operand... plus, in this case Tisdale's solution does exponentially more work than it really has to. There should be only one call to Power() within the function itself. (Proper solution left as an exercise for the OP.) -Arthur Nov 13 '05 #14

 P: n/a In article , "Arthur J. O'Dwyer" wrote: On Tue, 28 Oct 2003, Irrwahn Grausewitz wrote: "Arthur J. O'Dwyer" wrote: >> > "E. Robert Tisdale" wrote:> > >> > > double Power(double x, unsigned int n) {> > > return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0;> > > }In particular, where is it guaranteed that the base case (n <= 0)is ever reached? I admit I'm not conversant in the details ofC floating point, but it seems like an unwarrantedly dubiousassumption to be making. What's to say that 1e-100/2 != 1e-100 ? Err, n is of type unsigned int, thus all divisions are integer divisions... Oops. I was thrown by the use of 'double' everywhere else. Seems silly to use 'double' for the first operand if you're only going to allow positive integers for the second operand... plus, in this case Tisdale's solution does exponentially more work than it really has to. There should be only one call to Power() within the function itself. I think it does more than exponentially more work than needed... What is n - n/2 if n is equal to 1? Nov 13 '05 #15

 P: n/a Irrwahn Grausewitz wrote in message news:. .. "Arthur J. O'Dwyer" wrote:On Tue, 28 Oct 2003, Jarno A Wuolijoki wrote: On Mon, 27 Oct 2003, Eric Sosman wrote: > "E. Robert Tisdale" wrote: > > > > double Power(double x, unsigned int n) { > > return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; > > } > > I confess that I considered suggesting this method, but > then decided that wanton cruelty was not (yet) justified. Well, either that or the interpretation of "suitable base case to stop the recursion" might need some clarification.In particular, where is it guaranteed that the base case (n <= 0)is ever reached? I admit I'm not conversant in the details ofC floating point, but it seems like an unwarrantedly dubiousassumption to be making. What's to say that 1e-100/2 != 1e-100 ? Err, n is of type unsigned int, thus all divisions are integer divisions... The floating point reference is a red herring. Think about what the function will do when n is 1. -- Peter Nov 13 '05 #16

 P: n/a "James Hu" wrote in message news:x9********************@comcast.com... On 2003-10-27, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd, and to find a suitable base case to stop the recursion. Can someone give me an example of this? This is not a C question... but the C answer is use pow() (unless you are purposefully avoiding floating point, in which case a table lookup is in order). The algorithm described, in non-recursive form, is commonly used in languages that supply an integer exponentiation operator. A table dimensioned INT_MAX-INT_MIN+1 will take up a lot of memory. Rewrite your word problem into a recurrence relationship. Use induction to prove this recurrence correctly calculates x to the nth power. It should be obvious at that point what your function should look like and what your base case is. Yes, he should do that. -- glen Nov 13 '05 #17

 P: n/a On 2003-10-28, Glen Herrmannsfeldt wrote: "James Hu" wrote in message news:x9********************@comcast.com... On 2003-10-27, dmattis wrote: > I am trying to write a recursive version of Power(x,n) that works by > breaking n down into halves(where half of n=n/2), squaring > Power(x,n/2), and multiplying by x again if n was odd, and to find a > suitable base case to stop the recursion. Can someone give me an > example of this? This is not a C question... but the C answer is use pow() (unless you are purposefully avoiding floating point, in which case a table lookup is in order). The algorithm described, in non-recursive form, is commonly used in languages that supply an integer exponentiation operator. Yes, but those typically take a floating point type for the x argument, and an int type for the n argument. A table dimensioned INT_MAX-INT_MIN+1 will take up a lot of memory. That is an imaginative approach, but not what I had in mind. #include static int8_t hbit = {-1 ,0 ,1,1 ,2,2,2,2 ,3,3,3,3,3,3,3,3 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5 ,5,5,5,5,5,5,5 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 }; int int_pow(int x, uint8_t n) { int t = 1; if (n == 0) return 1; if (x == 0) return 0; switch (hbit[n]) { case 7: if (n & 1) t *= x; n >>= 1; x *= x; case 6: if (n & 1) t *= x; n >>= 1; x *= x; case 5: if (n & 1) t *= x; n >>= 1; x *= x; case 4: if (n & 1) t *= x; n >>= 1; x *= x; case 3: if (n & 1) t *= x; n >>= 1; x *= x; case 2: if (n & 1) t *= x; n >>= 1; x *= x; case 1: if (n & 1) t *= x; n >>= 1; x *= x; default: if (n & 1) t *= x; } return t; } -- James Nov 13 '05 #18

 P: n/a Christian Bau wrote: "Arthur J. O'Dwyer" wrote: > >> > "E. Robert Tisdale" wrote: > >> > > > >> > > double Power(double x, unsigned int n) { > >> > > return (0 < n)? Power(x, n/2)*Power(x, n - n/2): 1.0; > >> > > } plus, in this case Tisdale's solution does exponentially more work than it really has to. There should be only one call to Power() within the function itself.I think it does more than exponentially more work than needed...What is n - n/2 if n is equal to 1? Arrgh, Trollsdale did it again! And on first glance I thought he actually posted valid code; stupid me. -- Irrwahn (ir*******@freenet.de) Nov 13 '05 #19

 P: n/a Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Alea iacta "Julian V. Noble" Here is tested C code for raising an integer to an integer power, recursively. (I hope this isn't a HW assignment but a legitimate query.) Nov 13 '05 #20

 P: n/a Marcus Lessard wrote: Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. If computing positive integer powers is of so little interest, it's hard to see why Knuth gives the topic an entire section of its own in "The Art of Computer Programming, Volume II: Seminumerical Algorithms." The method outlined is the first serious power-computing algorithm Knuth introduces in that section. He writes that some authors have declared the method optimal, but he is kind enough to spare them embarrassment by omitting their names; clearly they forgot to consider cases like n==15. -- Er*********@sun.com Nov 13 '05 #21

 P: n/a "James Hu" wrote in message news:Nv********************@comcast.com... On 2003-10-28, Glen Herrmannsfeldt wrote: "James Hu" wrote in message news:x9********************@comcast.com... On 2003-10-27, dmattis wrote: > I am trying to write a recursive version of Power(x,n) that works by > breaking n down into halves(where half of n=n/2), squaring > Power(x,n/2), and multiplying by x again if n was odd, and to find a > suitable base case to stop the recursion. Can someone give me an > example of this? This is not a C question... but the C answer is use pow() (unless you are purposefully avoiding floating point, in which case a table lookup is in order). The algorithm described, in non-recursive form, is commonly used in languages that supply an integer exponentiation operator. Yes, but those typically take a floating point type for the x argument, and an int type for the n argument. The languages I know of supply routines for integer n, and integer, float, double, complex, and complex double x. The OP didn't supply the type of x, but the algorithm works for any type where mulitply is defined. A table dimensioned INT_MAX-INT_MIN+1 will take up a lot of memory. That is an imaginative approach, but not what I had in mind. #include static int8_t hbit = {-1 ,0 ,1,1 ,2,2,2,2 ,3,3,3,3,3,3,3,3 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5 ,5,5,5,5,5,5,5 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7,7 ,7,7,7,7,7,7,7 }; int int_pow(int x, uint8_t n) { int t = 1; if (n == 0) return 1; if (x == 0) return 0; switch (hbit[n]) { case 7: if (n & 1) t *= x; n >>= 1; x *= x; case 6: if (n & 1) t *= x; n >>= 1; x *= x; case 5: if (n & 1) t *= x; n >>= 1; x *= x; case 4: if (n & 1) t *= x; n >>= 1; x *= x; case 3: if (n & 1) t *= x; n >>= 1; x *= x; case 2: if (n & 1) t *= x; n >>= 1; x *= x; case 1: if (n & 1) t *= x; n >>= 1; x *= x; default: if (n & 1) t *= x; } return t; } The algorithm also works for n > 255. -- glen Nov 13 '05 #22

 P: n/a "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Computing integer powers of numbers is fairly common in scientific programming, and is one of the relatively few things missing from C used in scientific programming. However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. -- glen Nov 13 '05 #23

 P: n/a On Tue, 28 Oct 2003, Eric Sosman wrote: If computing positive integer powers is of so little interest, it's hard to see why Knuth gives the topic an entire section of its own in "The Art of Computer Programming, Volume II: Seminumerical Algorithms." The method outlined is the first serious power-computing algorithm Knuth introduces in that section. He writes that some authors have declared the method optimal, but he is kind enough to spare them embarrassment by omitting their names; clearly they forgot to consider cases like n==15. What's wrong with n==15? The algorithm given is still O(lg N) in such cases. No, probably Knuth was thinking of *better* algorithms that could compute powers more quickly -- I don't know of any, but perhaps a couple of exp() and log() lookup tables, plus iteration, could do things faster than straight recursion. -Arthur Nov 13 '05 #24

 P: n/a On 2003-10-28, Glen Herrmannsfeldt wrote: "James Hu" wrote in message news:Nv********************@comcast.com... On 2003-10-28, Glen Herrmannsfeldt wrote: > "James Hu" wrote in message > news:x9********************@comcast.com... >> On 2003-10-27, dmattis wrote: >> > I am trying to write a recursive version of Power(x,n) that works by >> > breaking n down into halves(where half of n=n/2), squaring >> > Power(x,n/2), and multiplying by x again if n was odd, and to find a >> > suitable base case to stop the recursion. Can someone give me an >> > example of this? >> >> This is not a C question... but the C answer is use pow() (unless you >> are purposefully avoiding floating point, in which case a table lookup >> is in order). > > The algorithm described, in non-recursive form, is commonly used in > languages that supply an integer exponentiation operator. Yes, but those typically take a floating point type for the x argument, and an int type for the n argument. The languages I know of supply routines for integer n, and integer, float, double, complex, and complex double x. Well, lets just call most of those floating point, except for integer and complex x (which I assume you mean A+Bi with A and B integers, but C has no such type). The OP didn't supply the type of x, but the algorithm works for any type where mulitply is defined. Since the proposed algorithm does not deal with negative powers, it is safe to assume the the type of n is unsigned. > A table dimensioned INT_MAX-INT_MIN+1 will take up a lot of memory. That is an imaginative approach, but not what I had in mind. [.. snip .. ] The algorithm also works for n > 255. But not without overflowing the int type. My function does not prevent overflow from happening, but the caller can check to make sure overflow will not before invoking the function. And the caller can then discover the degenerate unity cases and dispatch it without incurring the function call overhead. However, here is an alternate implementation (now, with two table lookups): #include #include static int32_t xpow = {0,INT_MAX,46341,1291,216,74,36,22,15,11,9,8,6,6,5 ,5,4,4,4,4,3,3 ,3,3,3,3,3,3,3,3,3,2 }; static int8_t hbit = {-1 ,0 ,1,1 ,2,2,2,2 ,3,3,3,3,3,3,3,3 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 }; int32_t int_pow(int32_t x, unsigned n) { int32_t t = 1; int32_t ax; if (n == 0) return 1; if (x == 0) return 0; if (n == 1) return x; if (x == INT_MIN) return 0; if (((x<0)?-x:x) >= xpow[(n<32)?n:31]) return (x<0)?((n&1)?INT_MIN:INT_MAX):INT_MAX; if (x == 1 || x == -1) n &= 1; switch (hbit[n]) { case 4: if (n & 1) t *= x; n >>= 1; x *= x; case 3: if (n & 1) t *= x; n >>= 1; x *= x; case 2: if (n & 1) t *= x; n >>= 1; x *= x; case 1: if (n & 1) t *= x; n >>= 1; x *= x; default: t *= x; } return t; } -- James Nov 13 '05 #25

 P: n/a "Arthur J. O'Dwyer" wrote: On Tue, 28 Oct 2003, Eric Sosman wrote: If computing positive integer powers is of so little interest, it's hard to see why Knuth gives the topic an entire section of its own in "The Art of Computer Programming, Volume II: Seminumerical Algorithms." The method outlined is the first serious power-computing algorithm Knuth introduces in that section. He writes that some authors have declared the method optimal, but he is kind enough to spare them embarrassment by omitting their names; clearly they forgot to consider cases like n==15. What's wrong with n==15? The algorithm given is still O(lg N) in such cases. No, probably Knuth was thinking of *better* algorithms that could compute powers more quickly -- I don't know of any, but perhaps a couple of exp() and log() lookup tables, plus iteration, could do things faster than straight recursion. The binary method starts with x and then computes x^2, x^3, x^6, x^7, x^14, x^15 -- six multiplications. The factor method starts with x and then computes x^2 and x^3 with two multiplications. Letting y = x^3 it then computes y^2, y^4, y^5 (= x^15) in three more multiplications, for five in all. Perhaps not a big deal when `x' is something simple like a floating-point number -- but worth paying heed to if `x' is, say, a large matrix. -- Er*********@sun.com Nov 13 '05 #26

 P: n/a On 2003-10-28, James Hu wrote: [ ... ] {0,INT_MAX,46341,1291,216,74,36,22,15,11,9,8,6,6,5 ,5,4,4,4,4,3,3 [ ... ] if (x == INT_MIN) return 0; [ ... ] return (x<0)?((n&1)?INT_MIN:INT_MAX):INT_MAX; [ ... ] References to INT_MAX and INT_MIN should be changed to INT32_MAX and INT32_MIN respectively. -- James Nov 13 '05 #27

 P: n/a My take double Power(double x, unsigned int n) { if(n==0) return 1.; else return (n&1) ? x*Power(x*x,(n-1)/2) : Power(x*x,n/2); } Nov 13 '05 #28

 P: n/a Glen Herrmannsfeldt wrote: "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Computing integer powers of numbers is fairly common in scientific programming, and is one of the relatively few things missing from C used in scientific programming. However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton Nov 13 '05 #29

 P: n/a In article , "Arthur J. O'Dwyer" wrote: What's wrong with n==15? The algorithm given is still O(lg N) in such cases. No, probably Knuth was thinking of *better* algorithms that could compute powers more quickly -- I don't know of any, but perhaps a couple of exp() and log() lookup tables, plus iteration, could do things faster than straight recursion. Knuth was thinking of the fact that this algorithm uses six multiplications to calculate x^15, but it can be done using five multiplications. Nov 13 '05 #30

 P: n/a In article , Richard Heathfield wrote: Glen Herrmannsfeldt wrote: "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Computing integer powers of numbers is fairly common in scientific programming, and is one of the relatively few things missing from C used in scientific programming. However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? double power (double x, unsigned long n) { if (n == 0) { return 1.0; } else { double result = x; unsigned long k = 1; while (2*k <= n) k *= 2; while (k > 1) { k /= 2; result *= (n & k) ? result * x : result; } return result; } } But your example is much more difficult because the execution time of a single multiplication depends on the size of the numbers involved, so it is not the number of operations that counts, but their cost. Nov 13 '05 #31

 P: n/a On Tue, 28 Oct 2003, Richard Heathfield wrote: Glen Herrmannsfeldt wrote: However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) Really. I guess you must have used 'long double', huh? :) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? #include double pow3(double x, unsigned int n) { do { return exp(n*log(x)); } while (1); } -Arthur Nov 13 '05 #32

 P: n/a Arthur J. O'Dwyer wrote: On Tue, 28 Oct 2003, Richard Heathfield wrote: Glen Herrmannsfeldt wrote: > > However, requiring it as a recursive function does scream of homework. > A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) Really. I guess you must have used 'long double', huh? :) No, I used bignum routines written in ISO C. The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? #include double pow3(double x, unsigned int n) { do { return exp(n*log(x)); } while (1); } Ah, I don't think you were taking me entirely seriously. That's a shame, because I was in fact asking a serious question. -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton Nov 13 '05 #33

 P: n/a Arthur J. O'Dwyer wrote: On Tue, 28 Oct 2003, Eric Sosman wrote: If computing positive integer powers is of so littleinterest, it's hard to see why Knuth gives the topic anentire section of its own in "The Art of Computer Programming,Volume II: Seminumerical Algorithms." The method outlined is the first serious power-computingalgorithm Knuth introduces in that section. He writes thatsome authors have declared the method optimal, but he iskind enough to spare them embarrassment by omitting theirnames; clearly they forgot to consider cases like n==15. What's wrong with n==15? The algorithm given is still O(lg N) in such cases. No, probably Knuth was thinking of *better* algorithms that could compute powers more quickly -- I don't know of any, but perhaps a couple of exp() and log() lookup tables, plus iteration, could do things faster than straight recursion. (x << 4) - x There. O(1) /me runs for cover ^_^ Nov 13 '05 #34

 P: n/a On 2003-10-27, James Hu wrote: On 2003-10-27, dmattis wrote: I am trying to write a recursive version of Power(x,n) that works by [...] ... the C answer is use pow() (unless you are purposefully avoiding floating point, in which case a table lookup is in order). Ok, my previous versions contained minor snafus. This one is tested, and profiled: #include #include static int32_t xpow = {0,INT32_MAX,46340,1290,215,73,35,21,14,10,8,7 ,5,5,4,4,3,3,3,3,2,2,2,2,2,2,2,2,2,2,2,1 }; static int8_t hbit = {-1,0,1,1,2,2,2,2,3,3,3,3,3,3,3,3 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 }; int32_t int_pow(int32_t x, unsigned n) { int32_t t = 1; if (n == 0) return 1; if (x == 0) return 0; if (n == 1) return x; if (x == INT32_MIN) return (n&1)?x:INT32_MAX; if (((x<0)?-x:x) > xpow[(n>31)?31:n]) return (x<0)?((n&1)?INT32_MIN:INT32_MAX):INT32_MAX; if (n>31) return (n&1)?x:1; switch (hbit[n]) { case 4: if (n & 1) t *= x; n >>= 1; x *= x; case 3: if (n & 1) t *= x; n >>= 1; x *= x; case 2: if (n & 1) t *= x; n >>= 1; x *= x; case 1: if (n & 1) t *= x; n >>= 1; x *= x; default: t *= x; } return t; } -- James Nov 13 '05 #35

 P: n/a On 2003-10-29, Richard Heathfield wrote: Arthur J. O'Dwyer wrote: On Tue, 28 Oct 2003, Richard Heathfield wrote: The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? #include double pow3(double x, unsigned int n) { do { return exp(n*log(x)); } while (1); } Ah, I don't think you were taking me entirely seriously. That's a shame, because I was in fact asking a serious question. Richard, could you post your iterative version? Maybe we can help you tweak it. -- James Nov 13 '05 #36

 P: n/a Then maybe it is just me, but I didn't explain myself clearly. The objective of calculating powers is all well and good and no doubt of use but what confuses me is the highly specified nature of the algorithm: (from OP): "...breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd..." Will this be more efficient? Or is it just the solution demanded by the professor? ML "Eric Sosman" If computing positive integer powers is of so little interest, it's hard to see why Knuth gives the topic an entire section of its own in "The Art of Computer Programming, Volume II: Seminumerical Algorithms." Nov 13 '05 #37

 P: n/a Richard Heathfield wrote in message news:... Glen Herrmannsfeldt wrote: However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? I'm surprised there is much if any difference: long mpow(long x, long e) { long r=1; while(e) { if(e&1) r*=x; x*=x; e>>=1; } return r; } Its fairly trivial to remove one multiplication at the cost of n if tests (where n is the number of bits in e) and one multiplication can become an assignment but other than that I don't think fewer multiplications are possible using the square and multiply idiom. IIRC Knuth goes into this in some detail and mentions some exponents where this idiom does not use the minimum possible number of multiplications. (e=15 being the smallest which can be evaluated as mpow(mpow(x,3),5)) (Getting very OT now. squaring is particularly fast when very big number are involved where FFT techniques become "efficient") Tim. Nov 13 '05 #38

 P: n/a Richard Heathfield wrote: Glen Herrmannsfeldt wrote: "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Computing integer powers of numbers is fairly common in scientific programming, and is one of the relatively few things missing from C used in scientific programming. However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton 6573 (!?) Nov 13 '05 #39

 P: n/a On Wed, 29 Oct 2003, Richard Heathfield wrote: Arthur J. O'Dwyer wrote: On Tue, 28 Oct 2003, Richard Heathfield wrote: Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) Really. I guess you must have used 'long double', huh? :) No, I used bignum routines written in ISO C. [I did figure as much; still, I don't think the original discussion was meant to generalize to bignums. I thought we'd been talking about 'foo(double, unsigned int)'.] The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? #include double pow3(double x, unsigned int n) { do { return exp(n*log(x)); } while (1); } Ah, I don't think you were taking me entirely seriously. That's a shame, because I was in fact asking a serious question. :) Well, suppose someone goes and looks up ISO C implementations of 'exp' and 'log', and plugs them into the above function in the right places, then. I'd be willing to bet that there's at least one efficient iterative method of computing exp(n) and log(n), although maybe not to the appropriate precision for bignum work. Certainly a bignum solution would need lots of support code. For a serious solution, you could simulate the recursive solution by looking at the individual bits of the exponent; but since your own iterative solution didn't take too long, I'm guessing that that's what you did already. -Arthur Nov 13 '05 #40

 P: n/a "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Then maybe it is just me, but I didn't explain myself clearly. The objective of calculating powers is all well and good and no doubt of use but what confuses me is the highly specified nature of the algorithm: (from OP): "...breaking n down into halves(where half of n=n/2), squaring Power(x,n/2), and multiplying by x again if n was odd..." Will this be more efficient? Or is it just the solution demanded by the professor? Except for some special cases, it is the most efficient set of multiplies to generate a given power. In the iterative form, it is the algorithm used by languages that I know of that supply such an operation. If n is a power of two, such as 2**m, it will square the number m times. I suppose as an example for recursive algorithms it is a little better than factorial. -- glen Nov 13 '05 #41

 P: n/a Richard Heathfield wrote: Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? SomeType my_power(SomeType x, unsigned int n) { /* This is the right-to-left binary method used * by the O.P., expressed iteratively rather * than recursively. A similar left-to-right * method would use one less multiplication * (note that the first multiplication in this * method is always by unity, and thus wasted), * but needs extra code to find the high-order * 1-bit of `n'. Other methods are faster for * certain values of `n' (e.g., 15), but are * also more complicated. See Knuth, "The * Art of Computer Programming, Volume II: * Seminumerical Algorithms" Section 4.6.3; * this is Algorithm A from that section. */ SomeType y = 1; /* Optional: Make an explicit test for x==0 here. * As things stand, my_power(0,0) -> 1. */ for (;;) { if (n & 1) y *= x; if ((n >>= 1) == 0) return y; x *= x; } } -- Er*********@sun.com Nov 13 '05 #42

 P: n/a "Glen Herrmannsfeldt" .... If n is a power of two, such as 2**m, it will square the number m times. I suppose as an example for recursive algorithms it is a little better than factorial. -- glen From a mathematical point of view there is no difference in breaking up the computation into a set of squares and then multiplying. What is going on in the underlying processing that makes using the squares more efficient than just multiplying x by itself n times? ML Nov 13 '05 #43

 P: n/a Marcus Lessard wrote: "Glen Herrmannsfeldt" ... If n is a power of two, such as 2**m, it will square the number m times. I suppose as an example for recursive algorithms it is a little better than factorial. -- glen From a mathematical point of view there is no difference in breaking up the computation into a set of squares and then multiplying. What is going on in the underlying processing that makes using the squares more efficient than just multiplying x by itself n times? Count the multiplications: x*x*x*...*x*x*x = x^128 x*x = x^2, x^2*x^2 = x^4, ..., x^64*x^64 = x^128 Which do you think will be faster: 127 multiplications or 7? -- Er*********@sun.com Nov 13 '05 #44

 P: n/a James Hu wrote: Richard, could you post your iterative version? Maybe we can help you tweak it. Well, I'm not really ready to release the underlying library code yet. But here's the high-level stuff: while(MF_Compare(a2, zero) != 0) { MF_Multiply(t, a3, a1); MF_Copy(a3, t); MF_Subtract(t, a2, one); MF_Copy(a2, t); } As you can see, this does involve two copy operations, which I concede is a bit lame, but eliminating them is not going to make a huge difference. (The profile suggests that they take up zero time, which can't be quite true, of course, but you get the general idea.) The profile says that the recursive version spends 0.21 seconds doing 18 multiplications, whilst the iterative version spends 0.96 seconds doing 7777 multiplications. -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton Nov 13 '05 #45

 P: n/a Wolfgang Riedel wrote: Richard Heathfield wrote: Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? 6573 (!?) Yes, exactly right. :-) In case you want a quick check against your own code, the first few digits are 21254808227343471907345591571884156862... and it finishes ...65911020435734015379207. -- Richard Heathfield : bi****@eton.powernet.co.uk "Usenet is a strange place." - Dennis M Ritchie, 29 July 1999. C FAQ: http://www.eskimo.com/~scs/C-faq/top.html K&R answers, C books, etc: http://users.powernet.co.uk/eton Nov 13 '05 #46

 P: n/a In article , Richard Heathfield wrote:James Hu wrote: Richard, could you post your iterative version? Maybe we can help you tweak it.Well, I'm not really ready to release the underlying library code yet. Buthere's the high-level stuff: while(MF_Compare(a2, zero) != 0) { MF_Multiply(t, a3, a1); MF_Copy(a3, t); MF_Subtract(t, a2, one); MF_Copy(a2, t); }As you can see, this does involve two copy operations, which I concede is abit lame, but eliminating them is not going to make a huge difference. (Theprofile suggests that they take up zero time, which can't be quite true, ofcourse, but you get the general idea.) If you have bitwise operators (and-with-1 for "is odd" and shift-right for "integer divide by two"), you might want to try an iterative square-and-multiply, something like this: -------- /*Raise a1 to the power of a2, storing result in a3*/ if(MF_Compare(a2,zero) == 0) /*Assumes return 0 -> equal*/ { /*Special case, zeroth power is always 1*/ MF_Copy(a3,one); } else { MF_Copy(a3,a1); while(MF_Compare(a2,one) > 0) /*assumes return > 0 -> arg1>arg2*/ { MF_Multiply(t,a3,a3); /*MF_IS_ODD can be implemented using a bitwise and with 1*/ if(MF_IS_ODD(a2)) { /*As a side effect of the multiplication, we can put the result back in the variable we want it to end up in, and avoid a separate copy operation */ MF_Multiply(a3,t,a1); } else { MF_Copy(a3,t); } /*MF_DIVIDE_BY_TWO can be implemented by a bitwise right shift, avoiding having to do a long-division operation */ MF_DIVIDE_BY_TWO(a2); } } /*a3 now contains the desired result*/ -------- This was translated without additional checking from code I wrote that passed a brief sanity check doing the same thing with unsigned longs. Any bugs are a result of either translation error or incomplete testing of the original. The profile says that the recursive version spends 0.21 seconds doing 18multiplications, whilst the iterative version spends 0.96 seconds doing7777 multiplications. Ahh. So what you were really comparing was a recursive square-and- multiply(?) and an iterative counted-multiply. The iterative square-and-multiply should compare somewhat more favorably with the recursive one. dave -- Dave Vandervies dj******@csclub.uwaterloo.ca[O]ur recruitment drive isn't until June, so we're short-handed as it is. Damn! My copy of the Gay Agenda must be out of date. --Graham Reed and Zebee Johnstone in the scary devil monastery Nov 13 '05 #47

 P: n/a Richard Heathfield writes: Glen Herrmannsfeldt wrote: "Marcus Lessard" wrote in message news:bn**********@pyrite.mv.net... Maybe it's just me but doesn't the contrived nature of the function scream out "Homework Assignment?" Maybe I'm missing something but I just can't see why you'd ever want to compute this value.. Computing integer powers of numbers is fairly common in scientific programming, and is one of the relatively few things missing from C used in scientific programming. However, requiring it as a recursive function does scream of homework. A simple for loop should be easier and faster. Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since the result occupies over 6500 decimal digits, I won't display it here!) The recursive calculation took 0.22 seconds, and the iterative version 1.06 seconds - almost five times slower. Perhaps you could demonstrate an iterative version that can hold a candle to the recursive technique? Richard, are you serious? How about "emulating" the recursion via stack and iteration? There are several advantages to this, IMHO: - You have precise control the size of the stack - If the stack runs out, you can exit gracefully: or even grow the stack. Try doing that when your implementation's stack runs out. -- Micah J. Cowan mi***@cowan.name Nov 13 '05 #48

 P: n/a "Arthur J. O'Dwyer" wrote in message news:Pi**********************************@unix49.a ndrew.cmu.edu... On Wed, 29 Oct 2003, Richard Heathfield wrote: Arthur J. O'Dwyer wrote: On Tue, 28 Oct 2003, Richard Heathfield wrote:>> Hmmm. I raised 7 to the power 7777 using recursion and iteration. (Since> the result occupies over 6500 decimal digits, I won't display it here!) Really. I guess you must have used 'long double', huh? :) No, I used bignum routines written in ISO C. [I did figure as much; still, I don't think the original discussion was meant to generalize to bignums. I thought we'd been talking about 'foo(double, unsigned int)'.] Well, mostly the suggestion was int, float, and double, but the basic algorithm should work for other types. Though by iterative algorithm I did mean an iterative implementation of the square and multiply algorithm. Here is the loop from the OS/360 (not copyrighted) Fortran library routine for double base to integer power. Not included is the check for a negative power, and later reciprocal of the result. They use a two register shift, instead of AND to test the low bit after the shift, which might save one or two instructions. PLUS LD FACTOR,ONE LOAD FACTOR OF ONE IN FACTOR REG LOOP SRDL EXPN,1 SHIFT LOW BIT EXPN REG INTO ADDR REG LTR ADDR,ADDR TEST SIGN POS ADDR REG FOR MINUS BIT BC 10,JUMP IF SIGN BIT NOT MINUS,BRANCH TO JUMP MDR FACTOR,BASE MULTIPLY FACTOR REG BY BASE NO REG JUMP LTR EXPN,EXPN CHECK IF EXPONENT PLUS,MINUS,OR ZERO BC 8,NEXT IF EXPONENT NOW ZERO, BRANCH TO NEXT MDR BASE,BASE MULTIPLY BASE NO BY DOUBLING ITSELF BC 15,LOOP BRANCH TO LOOP TO TEST NEXT EXPN BIT I post this as proof that the algorithm has actually been used in production software. With any function call overhead, I would expect it to be slower in the recursive version. Though for bignum versions, the difference may be small, as the multiply will be somewhat slower. Also, I don't know which bignum libraries can do a square in place (without copying it first). -- glen -- glen Nov 13 '05 #49

 P: n/a On Tue, 28 Oct 2003 12:41:13 -0500 (EST), Arthur J. O'Dwyer wrote: What's wrong with n==15? The algorithm given is still O(lg N) in such cases. No, probably Knuth was thinking of *better* algorithms that could compute powers more quickly -- I don't know of any, but perhaps a couple of exp() and log() lookup tables, plus iteration, could do things faster than straight recursion. x^3 can be done with two multiplications and x^5 can be done with 3 using the square and multiply algorithm x2 =x * x x2 = x * x y =x * x2 z = x * x2 x4 = x2 * x2 y2 = y * y z = z * x4 y4 = y2 * y2 x8 = x4 * x4 z = y * y4 z = z * x8 So x^15 can be done with 5 multiplications by calling the square and multiply algorithm twice. But the square and multiply algorithm requires 6 multiplications for x^15. A general algorithm would require the ability to factor n which is not known to be easy ;-) (Factoring n isn't guaranteed to give a better solution than square and multiply, 33 being the smallest counter example. Knuth has about 20 pages on finding a general optimal solution) Tim. -- God said, "div D = rho, div B = 0, curl E = - @B/@t, curl H = J + @D/@t," and there was light. http://tjw.hn.org/ http://www.locofungus.btinternet.co.uk/ Nov 13 '05 #50

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