I'm trying to implement the a block conjugate gradient algorithm that is not subject to breakdown from non invertible residual matrices; But I'm getting nonsensical results (in each iteration, the rank of Rcurrent should be getting smaller, not increasing). It is presented in the paper "A breakdown-free block conjugate gradient method" by Hao Ji and Yaohang Li.
Here is the algorithm:
This is my implementation in Julia:
function orth(M::Matrix)
matrixRank = rank(M)
Ufactor = svdfact(M)[:U]
return Ufactor[:,1:matrixRank]
end
function BFBCG(A::Matrix, Xcurrent::Matrix, M::Matrix, tol::Number, maxit::Number, Rcurrent::Matrix)
# initialization
#Rcurrent = B - A*Xcurrent;
Zcurrent = M*Rcurrent;
Pcurrent = orth(Zcurrent);
Xnext::Matrix = ones(size(Xcurrent))
# iterative method
for i = 0:maxit
Qcurrent = A*Pcurrent
acurrent = (Pcurrent' * Qcurrent)\(Pcurrent'*Rcurrent)
Xnext = Xcurrent+Pcurrent*acurrent
Rnext = Rcurrent-Qcurrent*acurrent
# if Residual norm of columns in Rcurrent < tol, stop
Znext = M*Rnext
bcurrent = -(Pcurrent' * Qcurrent)\ (Qcurrent'*Znext)
Pnext = orth(Znext+Pcurrent*bcurrent)
Xcurrent = Xnext
Zcurrent = Znext
Rcurrent = Rnext
Pcurrent = Pnext
#printf("\nRANK:\t%d",rank(Rcurrent))
#printf("\nNORM column1:\t%1.8f",vecnorm(Rcurrent[:,1]))
#printf("\nNORM column2:\t%1.8f\n=============",vecnorm(Rcurrent[:,2]))
end
return Xnext
end
The results of the paper for those inputs:
A = [15 5 4 3 2 1; 5 35 9 8 7 6; 4 9 46 12 11 10; 3 8 12 50 14 13; 2 7 11 14 19 15; 1 6 10 13 15 45]
M = eye(6)
guess = rand(6,2)
R0 = [1 0.537266261211281;2 0.043775211060964;3 0.964458562037146;4 0.622317517840541;5 0.552735938776748;6 0.023323943544997]
X = BFBCG(A,guess,M,tol,9,R0)
are a rank that reaches zero in the third iteration.
The algorithm works, and the rank goes to zero in the third iteration. The problem is numerical inaccuracies which would leave any matrix fully ranked. To get a better result, use rank(Rcurrent, tol) instead of rank(Rcurrent) which is a version which takes tolerance into account. After which, at least on my machine, the rank drops to zero.
julia> X = BFBCG(A,guess,M,tol,9,R0)
RANK: 2
NORM column1: 1.78951939
NORM column2: 0.41155080
=============
RANK: 2
NORM column1: 0.97949620
NORM column2: 0.16170799
=============
RANK: 0
NORM column1: 0.00000000
NORM column2: 0.00000000
=============
RANK: 0
NORM column1: 0.00000000
NORM column2: 0.00000000
=============
After the you guys helped me out so gracefully last time, here is another tricky array sorter for you.
I have the following array:
a = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
I use it for some visual stuff and render it like this:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Now I want to sort the array to have a "snake" later:
// rearrange the array according to this schema
1 2 3 4
12 13 14 5
11 16 15 6
10 9 8 7
// the original array should look like this
a = [1,2,3,4,12,13,14,5,11,16,15,6,10,9,8,7]
Now I'm looking for a smart formula / smart loop to do that
ticker = 0;
rows = 4; // can be n
cols = 4; // can be n
originalArray = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16];
newArray = [];
while(ticker < originalArray.length)
{
//do the magic here
ticker++;
}
Thanks again for the help.
I was bored, so I made a python version for you with 9 lines of code inside the loop.
ticker = 0
rows = 4
cols = 4
originalArray = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]
newArray = [None] * (rows * cols)
row = 0
col = 0
dir_x = 1
dir_y = 0
taken = {}
while (ticker < len(originalArray)):
newArray[row * cols + col] = originalArray[ticker]
taken[row * cols + col] = True
if col + dir_x >= cols or row + dir_y >= rows or col + dir_x < 0:
dir_x, dir_y = -dir_y, dir_x
elif ((row + dir_y) * cols + col + dir_x) in taken:
dir_x, dir_y = -dir_y, dir_x
row += dir_y
col += dir_x
ticker += 1
print newArray
You can index into the snake coil directly if you recall that
1 + 2 + 3 + ... + n = n*(n+1)/2
m^2 + m - k = 0 => m - (-1+sqrt(1+4*k))/2
and look at the pattern of the coils. (I'll leave it as a hint for the time being--you could also use that n^2 = (n-1)^2 + (2*n+1) with reverse-indexing, or a variety of other things to solve the problem.)
When translating to code, it's not really any shorter than Tuomas' solution if all you want to do is fill the matrix, however.
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
Generate a list of lists (or print, I don't mind) a Pascal's Triangle of size N with the least lines of code possible!
Here goes my attempt (118 characters in python 2.6 using a trick):
c,z,k=locals,[0],'_[1]'
p=lambda n:[len(c()[k])and map(sum,zip(z+c()[k][-1],c()[k][-1]+z))or[1]for _ in range(n)]
Explanation:
the first element of the list comprehension (when the length is 0) is [1]
the next elements are obtained the following way:
take the previous list and make two lists, one padded with a 0 at the beginning and the other at the end.
e.g. for the 2nd step, we take [1] and make [0,1] and [1,0]
sum the two new lists element by element
e.g. we make a new list [(0,1),(1,0)] and map with sum.
repeat n times and that's all.
usage (with pretty printing, actually out of the code-golf xD):
result = p(10)
lines = [" ".join(map(str, x)) for x in result]
for i in lines:
print i.center(max(map(len, lines)))
output:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
K (Wikipedia), 15 characters:
p:{x{+':x,0}\1}
Example output:
p 10
(1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1
1 10 45 120 210 252 210 120 45 10 1)
It's also easily explained:
p:{x {+':x,0} \ 1}
^ ^------^ ^ ^
A B C D
p is a function taking an implicit parameter x.
p unfolds (C) an anonymous function (B) x times (A) starting at 1 (D).
The anonymous function simply takes a list x, appends 0 and returns a result by adding (+) each adjacent pair (':) of values: so e.g. starting with (1 2 1), it'll produce (1 2 1 0), add pairs (1 1+2 2+1 1+0), giving (1 3 3 1).
Update: Adapted to K4, which shaves off another two characters. For reference, here's the original K3 version:
p:{x{+':0,x,0}\1}
J, another language in the APL family, 9 characters:
p=:!/~#i.
This uses J's builtin "combinations" verb.
Output:
p 10
1 1 1 1 1 1 1 1 1 1
0 1 2 3 4 5 6 7 8 9
0 0 1 3 6 10 15 21 28 36
0 0 0 1 4 10 20 35 56 84
0 0 0 0 1 5 15 35 70 126
0 0 0 0 0 1 6 21 56 126
0 0 0 0 0 0 1 7 28 84
0 0 0 0 0 0 0 1 8 36
0 0 0 0 0 0 0 0 1 9
0 0 0 0 0 0 0 0 0 1
Haskell, 58 characters:
r 0=[1]
r(n+1)=zipWith(+)(0:r n)$r n++[0]
p n=map r[0..n]
Output:
*Main> p 5
[[1],[1,1],[1,2,1],[1,3,3,1],[1,4,6,4,1],[1,5,10,10,5,1]]
More readable:
-- # row 0 is just [1]
row 0 = [1]
-- # row (n+1) is calculated from the previous row
row (n+1) = zipWith (+) ([0] ++ row n) (row n ++ [0])
-- # use that for a list of the first n+1 rows
pascal n = map row [0..n]
69C in C:
f(int*t){int*l=t+*t,*p=t,r=*t,j=0;for(*t=1;l<t+r*r;j=*p++)*l++=j+*p;}
Use it like so:
int main()
{
#define N 10
int i, j;
int t[N*N] = {N};
f(t);
for (i = 0; i < N; i++)
{
for (j = 0; j <= i; j++)
printf("%d ", t[i*N + j]);
putchar('\n');
}
return 0;
}
F#: 81 chars
let f=bigint.Factorial
let p x=[for n in 0I..x->[for k in 0I..n->f n/f k/f(n-k)]]
Explanation: I'm too lazy to be as clever as the Haskell and K programmers, so I took the straight forward route: each element in Pascal's triangle can be uniquely identified using a row n and col k, where the value of each element is n!/(k! (n-k)!.
Python: 75 characters
def G(n):R=[[1]];exec"R+=[map(sum,zip(R[-1]+[0],[0]+R[-1]))];"*~-n;return R
Shorter prolog version (112 instead of 164):
n([X],[X]).
n([H,I|T],[A|B]):-n([I|T],B),A is H+I.
p(0,[[1]]):-!.
p(N,[R,S|T]):-O is N-1,p(O,[S|T]),n([0|S],R).
another stab (python):
def pascals_triangle(n):
x=[[1]]
for i in range(n-1):
x.append(list(map(sum,zip([0]+x[-1],x[-1]+[0]))))
return x
Haskell, 164C with formatting:
i l=zipWith(+)(0:l)$l++[0]
fp=map (concatMap$(' ':).show)f$iterate i[1]
c n l=if(length l<n)then c n$' ':l++" "else l
cl l=map(c(length$last l))l
pt n=cl$take n fp
Without formatting, 52C:
i l=zipWith(+)(0:l)$l++[0]
pt n=take n$iterate i[1]
A more readable form of it:
iterateStep row = zipWith (+) (0:row) (row++[0])
pascalsTriangle n = take n $ iterate iterateStep [1]
-- For the formatted version, we reduce the number of rows at the final step:
formatRow r = concatMap (\l -> ' ':(show l)) r
formattedLines = map formatRow $ iterate iterateStep [1]
centerTo width line =
if length line < width
then centerTo width (" " ++ line ++ " ")
else line
centerLines lines = map (centerTo (length $ last lines)) lines
pascalsTriangle n = centerLines $ take n formattedLines
And perl, 111C, no centering:
$n=<>;$p=' 1 ';for(1..$n){print"$p\n";$x=" ";while($p=~s/^(?= ?\d)(\d* ?)(\d* ?)/$2/){$x.=($1+$2)." ";}$p=$x;}
Scheme — compressed version of 100 characters
(define(P h)(define(l i r)(if(> i h)'()(cons r(l(1+ i)(map +(cons 0 r)(append r '(0))))))(l 1 '(1)))
This is it in a more readable form (269 characters):
(define (pascal height)
(define (next-row row)
(map +
(cons 0 row)
(append row '(0))))
(define (iter i row)
(if (> i height)
'()
(cons row
(iter (1+ i)
(next-row row)))))
(iter 1 '(1)))
VBA/VB6 (392 chars w/ formatting)
Public Function PascalsTriangle(ByVal pRows As Integer)
Dim iRow As Integer
Dim iCol As Integer
Dim lValue As Long
Dim sLine As String
For iRow = 1 To pRows
sLine = ""
For iCol = 1 To iRow
If iCol = 1 Then
lValue = 1
Else
lValue = lValue * (iRow - iCol + 1) / (iCol - 1)
End If
sLine = sLine & " " & lValue
Next
Debug.Print sLine
Next
End Function
PHP 100 characters
$v[]=1;while($a<34){echo join(" ",$v)."\n";$a++;for($k=0;$k<=$a;$k++)$t[$k]=$v[$k-1]+$v[$k];$v=$t;}
Ruby, 83c:
def p(n);n>0?(m=p(n-1);k=m.last;m+[([0]+k).zip(k+[0]).map{|x|x[0]+x[1]}]):[[1]];end
test:
irb(main):001:0> def p(n);n>0?(m=p(n-1);k=m.last;m+[([0]+k).zip(k+[0]).map{|x|x[0]+x[1]}]):[[1]];end
=> nil
irb(main):002:0> p(5)
=> [[1], [1, 1], [1, 2, 1], [1, 3, 3, 1], [1, 4, 6, 4, 1], [1, 5, 10, 10, 5, 1]]
irb(main):003:0>
Another python solution, that could be much shorter if the builtin functions had shorter names... 106 characters.
from itertools import*
r=range
p=lambda n:[[len(list(combinations(r(i),j)))for j in r(i+1)]for i in r(n)]
Another try, in prolog (I'm practising xD), not too short, just 164c:
s([],[],[]).
s([H|T],[J|U],[K|V]):-s(T,U,V),K is H+J.
l([1],0).
l(P,N):-M is N-1,l(A,M),append(A,[0],B),s(B,[0|A],P).
p([],-1).
p([H|T],N):-M is N-1,l(H,N),p(T,M).
explanation:
s = sum lists element by element
l = the Nth row of the triangle
p = the whole triangle of size N
VBA, 122 chars:
Sub p(n)
For r = 1 To n
l = "1"
v = 1
For c = 1 To r - 1
v = v / c * (r - c)
l = l & " " & v
Next
Debug.Print l
Next
End Sub
I wrote this C++ version a few years ago:
#include <iostream>
int main(int,char**a){for(int b=0,c=0,d=0,e=0,f=0,g=0,h=0,i=0;b<atoi(a[1]);(d|f|h)>1?e*=d>1?--d:1,g*=f>1?--f:1,i*=h>1?--h:1:((std::cout<<(i*g?e/(i*g):1)<<" "?d=b+=c++==b?c=0,std::cout<<std::endl?1:0:0,h=d-(f=c):0),e=d,g=f,i=h));}
The following is just a Scala function returning a List[List[Int]]. No pretty printing or anything. Any suggested improvements? (I know it's inefficient, but that's not the main challenge now, is it?). 145 C.
def p(n: Int)={def h(n:Int):List[Int]=n match{case 1=>1::Nil;case _=>(0::h(n-1) zipAll(h(n-1),0,0)).map{n=>n._1+n._2}};(1 to n).toList.map(h(_))}
Or perhaps:
def pascal(n: Int) = {
def helper(n: Int): List[Int] = n match {
case 1 => 1 :: List()
case _ => (0 :: helper(n-1) zipAll (helper(n-1),0,0)).map{ n => n._1 + n._2 }
}
(1 to n).toList.map(helper(_))
}
(I'm a Scala noob, so please be nice to me :D )
a Perl version (139 chars w/o shebang)
#p = (1,1);
while ($#p < 20) {
#q =();
$z = 0;
push #p, 0;
foreach (#p) {
push #q, $_+$z;
$z = $_
}
#p = #q;
print "#p\n";
}
output starts from 1 2 1
PHP, 115 chars
$t[][]=1;
for($i=1;$i<$n;++$i){
$t[$i][0]=1;
for($j=1;$j<$i;++$j)$t[$i][$j]=$t[$i-1][$j-1]+$t[$i-1][$j];
$t[$i][$i]=1;}
If you don't care whether print_r() displays the output array in the correct order, you can shave it to 113 chars like
$t[][]=1;
for($i=1;$i<$n;++$i){
$t[$i][0]=$t[$i][$i]=1;
for($j=1;$j<$i;++$j)$t[$i][$j]=$t[$i-1][$j-1]+$t[$i-1][$j];}
Perl, 63 characters:
for(0..9){push#z,1;say"#z";#z=(1,map{$z[$_-1]+$z[$_]}(1..$#z))}
My attempt in C++ (378c). Not anywhere near as good as the rest of the posts.. but I'm proud of myself for coming up with a solution on my own =)
int* pt(int n)
{
int s=n*(n+1)/2;
int* t=new int[s];
for(int i=0;i<n;++i)
for(int j=0;j<=i;++j)
t[i*n+j] = (!j || j==i) ? 1 : t[(i-1)*n+(j-1)] + t[(i-1)*n+j];
return t;
}
int main()
{
int n,*t;
std::cin>>n;
t=pt(n);
for(int i=0;i<n;++i)
{
for(int j=0;j<=i;j++)
std::cout<<t[i*n+j]<<' ';
std::cout<<"\n";
}
}
Old thread, but I wrote this in response to a challenge on another forum today:
def pascals_triangle(n):
x=[[1]]
for i in range(n-1):
x.append([sum(i) for i in zip([0]+x[-1],x[-1]+[0])])
return x
for x in pascals_triangle(5):
print('{0:^16}'.format(x))
[1]
[1, 1]
[1, 2, 1]
[1, 3, 3, 1]
[1, 4, 6, 4, 1]