Diophantine linear system solving: Difference between revisions

Line 868:

=={{header|Phix}}==

You can run this online [http://phix.x10.mx/p2js/LLLHnf.htm here].

A mechanical and (no disrespect) somewhat laborious translation of FreeBasic, with a bit of help where needed from the Wren entry.

⚫

⚫

~~(Admittedly made~~ ~~harder~~ by the need to xlate 0 and -1 based idx to 1 based.)

⚫

--

-- demo\rosetta\Diophantine_linear_system_solving.exw

-- ==================================================

--

-- Translation of FreeBasic, with some help from Wren, admittedly made harder

⚫

-- by the need to xlate 0 and -1 based idx to 1 based.

--

⚫

-- Note that for problem 16 (HMM extended gcd (example 7.2)), the signs of

⚫

-- the (20) and (37) rows are flipped, which I'm told is OK.

--

~~without~~ js -- (~~due~~ to file ~~reading~~)

with javascript_semantics -- (using an include instead of file i/o)

⚫

include Diophantine_linear_system_constants.e -- DLS_PROBS/SOLNS

constant echo = false,

intext = ~~get_text~~(~~"Diophantine_linear_system_problems.txt"~~,~~GT_LF_STRIPPED~~),

intext = split(DLS_PROBS,"\n"),

outtxt = ~~get_text~~("~~Diophantine_linear_system_solution.txt~~",~~GT_LF_STRIPPED~~)

outtxt = split(DLS_SOLNS,"\n",false)

integer nxi = 1, nxo = 1

function input(string prompt)

function input(string /*prompt*/)

string in = intext[nxi]

string in_line = intext[nxi]

nxi += 1

return in

return in_line

end function

Line 892:

Line 898:

if out_line != expected then

printf(1,"%s <<=== expected ***ERROR***\n",{expected})

{} = wait_key()

{} = wait_key() -- (nb does nowt in a browser)

end if

nxo += 1

Line 914:

Line 920:

sequence a

procedure ~~InputABC~~(integer pr)

procedure InputAb_or_c(integer pr)

-- input A and b, or a complex constant and compute powers into A

if pr then -- (complex constant)

integer m2 = m1+1

Line 921:

Line 927:

string g = input(" a + bi:")

integer r = find('+',g)

integer plus = find('+',g)

string line = trim(g[1..r-1])

string line = trim(g[1..plus-1])

atom x = to_number(line),

y = iff(r?to_integer(trim(g[r+1..$])):0)

y = iff(plus?to_integer(trim(g[plus+1..$])):0)

if y then line &= sprintf(" + %g*i",y) end if

output(" "&line)

Line 970:

Line 976:

procedure PrntM(integer sw)

integer k, r, s

string g~~, line~~

string g

atom q

Line 994:

Line 1,000:

sw = sw and a[k,s]=0

end for

if sw then g = "" end if -- trivial

if sw then g = "" end if -- trivial

⚫

sequence lensq = repeat("",m+1)

⚫

~~-- Hnf~~ ~~and~~ ~~solution~~

~~for~~ r=m~~+1~~ ~~to k~~ ~~by -~~~~1 do~~

lensq[k] = g

⚫

-- Calculate lengths squared

⚫

~~output~~(~~print_row~~(r,~~fmts~~)&~~<span~~ ~~style="color: #008080;">iif~~(r=k?g:""))

⚫

end for

⚫

-- ~~Null space with~~ lengths squared

for r=1 to k-1 do

⚫

~~line~~ = ~~print_row~~(r,~~fmts~~)

q = 0

for s=1 to m+1 do

q += a[r,s]*a[r,s]

end for

~~output~~(~~line~~&sprintf(" (%d)",q))

lensq[r] &= sprintf(" (%d)",q)

⚫

end for

-- Hnf and solution and null space

sequence order = tagset(k,m+1,-1)&tagset(k-1)

⚫

for r=1 to m+1 do

⚫

integer rr = order[r]

output(print_row(rr,fmts)&lensq[rr])

end for

Line 1,014:

Line 1,024:

for r=1 to m+1 do

printf(1,print_row(r,fmts)~~&"\n"~~)

printf(1,print_row(r,fmts))

-- (not particularly helpful:)

-- printf(1,", L:")

-- for s=1 to m+1 do

-- printf(1," %.20g",lambda[r,s])

⚫

-- end for

⚫

printf(1,"\n")

end for

end if

Line 1,037:

Line 1,053:

-- LLL reduce rows k

procedure Reduce(integer k, t)

⚫

integer s, sx

⚫

~~atom~~ lk, q

col1 = nx

col2 = nx

Line 1,054:

Line 1,068:

end if

end for

q = 0

atom q = 0

if col1<nx then

if a[t,col1+1]<0 then Minus(t) end if

q = floor(a[k,col1+1]/a[t,col1+1])

else

lk = lambda[k,t]

atom lk = lambda[k,t]

if 2*abs(lk)>d[t+1] then

-- 2|lambda_kt| > d_t

Line 1,068:

Line 1,082:

if q then

sx = iif(col1=nx?m:mn)

integer sx = iif(col1=nx?m:mn)

-- reduce row k

for s=1 to sx+1 do

Line 1,082:

Line 1,096:

-- exchange rows k and k-1

procedure Swop(integer k)

integer r, s, t = k-1

integer t = k-1, tm1 = t-1

⚫

~~atom~~ db, lk, <span~~ style~~="color~~: ~~#000000;">lr

{a[k], a[t]} = {a[t], a[k]}

{~~<span~~ ~~style="color: #000000;">lambda[~~k~~<span~~ ~~style="color: #0000FF;">][1~~~~..t-1],~~ lambda[t][1..t-1]} = {lambda[t][1..t-1], lambda[k][1..t-1]}

object tmp = lambda[t][1..tm1]

⚫

lambda[t][1..tm1] = lambda[k][1..tm1]

lambda[k][1..tm1] = tmp; tmp = 0

-- update Gram coefficients

-- columns k, k-1 for r > k

lk = lambda[k,t]

atom lk = lambda[k,t],

db = (d[t]*d[k+1]+lk*lk)/d[~~t+1~~]

db = (d[t]*d[k+1]+lk*lk)/d[k]

for r=k+1 to m+1 do

lr = lambda[r,k]

atom lr = lambda[r,k],

~~<span~~ ~~style="color:~~ ~~#000000;">lambda<span~~ ~~style="color:~~ ~~#0000FF;">[<span~~ ~~style="color: #000000;">r,~~~~k]~~ = ~~(~~d[k+1]*lambda[r,t]-lk*lr)/d[t+1]

dk1 = d[k+1]

lambda[r,t] = (db*lr+lk*~~lambda~~[r,k])/d[k~~+1~~]

lambda[r,k] = (dk1*lambda[r,t]-lk*lr)/d[k]

lambda[r,t] = (db*lr+lk*lambda[r,k])/dk1

end for

d[~~t+1~~] = db

d[k] = db

end procedure

Line 1,105:

Line 1,121:

problem_no += 1

printf(1,"problem #%d\n",problem_no)

⚫

~~integer~~ i, k, t, tl = 0

⚫

~~<span~~ ~~style="color:~~ ~~#004080;">atom~~ db, lk

~~InputABC~~(sw)

InputAb_or_c(sw)

-- augment Ab~ with column e_m

Line 1,119:

Line 1,133:

if echo then PrntM(0) end if

k = 1

integer k = 1,

⚫

tl = 0

while k<=m do

t = k-1

integer t = k-1

-- partial size reduction

Reduce(k+1, t+1)

Line 1,128:

Line 1,143:

if sw then

-- zero rows k-1, k

lk = lambda[k+1,t+1]

atom lk = lambda[k+1,t+1]

-- Lovasz condition

-- Bk >= (alpha - mu_kt^2) * Bt

db = d[t+1]*d[k+2]+lk*lk

atom db = d[t+1]*d[k+2]+lk*lk

-- not satisfied

sw = db*ald<d[t+2]*d[t+2]*aln

Line 1,202:

Line 1,217:

end if

end while

?"done"

{} = wait_key()

⚫

Note that for problem 16 (HMM extended gcd (example 7.2)), the signs of

⚫

the (20) and (37) rows ~~in the null space~~ are flipped, ~~somehow,~~ ~~otherwise~~

output matches that of FreeBasic (and is auto-checked). (I suspect the

issue is FreeBasic distinguishing 0 and -0 in a way that Phix does not)

Eagle eyed viewers may have spotted my blase use of null space there as

if I knew what I was talking about (tee hee).

=={{header|Wren}}==

@@ Line 868: / Line 868: @@
 =={{header|Phix}}==
+You can run this online [http://phix.x10.mx/p2js/LLLHnf.htm here].
-A mechanical and (no disrespect) somewhat laborious translation of FreeBasic, with a bit of help where needed from the Wren entry.<br>
+<!--<lang Phix>(phixonline)-->
-(Admittedly made harder by the need to xlate 0 and -1 based idx to 1 based.)
-<!--<lang Phix>(notonline)-->
  <span style="color: #000080;font-style:italic;">--
  -- demo\rosetta\Diophantine_linear_system_solving.exw
  -- ==================================================
+ --
+ --  Translation of FreeBasic, with some help from Wren, admittedly made harder
+ --  by the need to xlate 0 and -1 based idx to 1 based.
+ --
+ --  Note that for problem 16 (HMM extended gcd (example 7.2)), the signs of
+ --  the (20) and (37) rows are flipped, which I'm told is OK.
  --</span>
- <span style="color: #008080;">without</span> <span style="color: #008080;">js</span> <span style="color: #000080;font-style:italic;">-- (due to file reading)</span>
+ <span style="color: #008080;">with</span> <span style="color: #008080;">javascript_semantics</span> <span style="color: #000080;font-style:italic;">-- (using an include instead of file i/o)</span>
+ <span style="color: #008080;">include</span> <span style="color: #000000;">Diophantine_linear_system_constants</span><span style="color: #0000FF;">.</span><span style="color: #000000;">e</span> <span style="color: #000080;font-style:italic;">-- DLS_PROBS/SOLNS</span>
  <span style="color: #008080;">constant</span> <span style="color: #000000;">echo</span> <span style="color: #0000FF;">=</span> <span style="color: #004600;">false</span><span style="color: #0000FF;">,</span>
- <span style="color: #000000;">intext</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_text</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"Diophantine_linear_system_problems.txt"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">GT_LF_STRIPPED</span><span style="color: #0000FF;">),</span>
+ <span style="color: #000000;">intext</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">split</span><span style="color: #0000FF;">(</span><span style="color: #000000;">DLS_PROBS</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">),</span>
- <span style="color: #000000;">outtxt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">get_text</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"Diophantine_linear_system_solution.txt"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">GT_LF_STRIPPED</span><span style="color: #0000FF;">)</span>
+ <span style="color: #000000;">outtxt</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">split</span><span style="color: #0000FF;">(</span><span style="color: #000000;">DLS_SOLNS</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">,</span><span style="color: #004600;">false</span><span style="color: #0000FF;">)</span>
  <span style="color: #004080;">integer</span> <span style="color: #000000;">nxi</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">nxo</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
- <span style="color: #008080;">function</span> <span style="color: #000000;">input</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000000;">prompt</span><span style="color: #0000FF;">)</span>
+ <span style="color: #008080;">function</span> <span style="color: #000000;">input</span><span style="color: #0000FF;">(</span><span style="color: #004080;">string</span> <span style="color: #000080;font-style:italic;">/*prompt*/</span><span style="color: #0000FF;">)</span>
-     <span style="color: #004080;">string</span> <span style="color: #7060A8;">in</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">intext</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nxi</span><span style="color: #0000FF;">]</span>
+     <span style="color: #004080;">string</span> <span style="color: #000000;">in_line</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">intext</span><span style="color: #0000FF;">[</span><span style="color: #000000;">nxi</span><span style="color: #0000FF;">]</span>
      <span style="color: #000000;">nxi</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
-     <span style="color: #008080;">return</span> <span style="color: #7060A8;">in</span>
+     <span style="color: #008080;">return</span> <span style="color: #000000;">in_line</span>
  <span style="color: #008080;">end</span> <span style="color: #008080;">function</span>
@@ Line 892: / Line 898: @@
      <span style="color: #008080;">if</span> <span style="color: #000000;">out_line</span> <span style="color: #0000FF;">!=</span> <span style="color: #000000;">expected</span> <span style="color: #008080;">then</span>
          <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"%s &lt;&lt;=== expected ***ERROR***\n"</span><span style="color: #0000FF;">,{</span><span style="color: #000000;">expected</span><span style="color: #0000FF;">})</span>
-         <span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span>
+         <span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span> <span style="color: #000080;font-style:italic;">-- (nb does nowt in a browser)</span>
      <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
      <span style="color: #000000;">nxo</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
@@ Line 914: / Line 920: @@
  <span style="color: #004080;">sequence</span> <span style="color: #000000;">a</span>
- <span style="color: #008080;">procedure</span> <span style="color: #000000;">InputABC</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">pr</span><span style="color: #0000FF;">)</span>
+ <span style="color: #008080;">procedure</span> <span style="color: #000000;">InputAb_or_c</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">pr</span><span style="color: #0000FF;">)</span>
- <span style="color: #000080;font-style:italic;">-- input A and b, or a complex constant and compute powers into A</span>
+     <span style="color: #000080;font-style:italic;">-- input A and b, or a complex constant and compute powers into A</span>
      <span style="color: #008080;">if</span> <span style="color: #000000;">pr</span> <span style="color: #008080;">then</span> <span style="color: #000080;font-style:italic;">-- (complex constant)</span>
          <span style="color: #004080;">integer</span> <span style="color: #000000;">m2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">m1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span>
@@ Line 921: / Line 927: @@
          <span style="color: #004080;">string</span> <span style="color: #000000;">g</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">input</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" a + bi:"</span><span style="color: #0000FF;">)</span>
-         <span style="color: #004080;">integer</span> <span style="color: #000000;">r</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'+'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">g</span><span style="color: #0000FF;">)</span>
+         <span style="color: #004080;">integer</span> <span style="color: #000000;">plus</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">find</span><span style="color: #0000FF;">(</span><span style="color: #008000;">'+'</span><span style="color: #0000FF;">,</span><span style="color: #000000;">g</span><span style="color: #0000FF;">)</span>
-         <span style="color: #004080;">string</span> <span style="color: #000000;">line</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">r</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
+         <span style="color: #004080;">string</span> <span style="color: #000000;">line</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">[</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">plus</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
          <span style="color: #004080;">atom</span> <span style="color: #000000;">x</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">to_number</span><span style="color: #0000FF;">(</span><span style="color: #000000;">line</span><span style="color: #0000FF;">),</span>
-              <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">?</span><span style="color: #7060A8;">to_integer</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$])):</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
+              <span style="color: #000000;">y</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iff</span><span style="color: #0000FF;">(</span><span style="color: #000000;">plus</span><span style="color: #0000FF;">?</span><span style="color: #7060A8;">to_integer</span><span style="color: #0000FF;">(</span><span style="color: #7060A8;">trim</span><span style="color: #0000FF;">(</span><span style="color: #000000;">g</span><span style="color: #0000FF;">[</span><span style="color: #000000;">plus</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..$])):</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span>
          <span style="color: #008080;">if</span> <span style="color: #000000;">y</span> <span style="color: #008080;">then</span> <span style="color: #000000;">line</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" + %g*i"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">y</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
          <span style="color: #000000;">output</span><span style="color: #0000FF;">(</span><span style="color: #008000;">" "</span><span style="color: #0000FF;">&</span><span style="color: #000000;">line</span><span style="color: #0000FF;">)</span>
@@ Line 970: / Line 976: @@
  <span style="color: #008080;">procedure</span> <span style="color: #000000;">PrntM</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">sw</span><span style="color: #0000FF;">)</span>
      <span style="color: #004080;">integer</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span>
-     <span style="color: #004080;">string</span> <span style="color: #000000;">g</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">line</span>
+     <span style="color: #004080;">string</span> <span style="color: #000000;">g</span>
      <span style="color: #004080;">atom</span> <span style="color: #000000;">q</span>
@@ Line 994: / Line 1,000: @@
              <span style="color: #000000;">sw</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">sw</span> <span style="color: #008080;">and</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s</span><span style="color: #0000FF;">]=</span><span style="color: #000000;">0</span>
          <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
-         <span style="color: #008080;">if</span> <span style="color: #000000;">sw</span> <span style="color: #008080;">then</span> <span style="color: #000000;">g</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- trivial
+         <span style="color: #008080;">if</span> <span style="color: #000000;">sw</span> <span style="color: #008080;">then</span> <span style="color: #000000;">g</span> <span style="color: #0000FF;">=</span> <span style="color: #008000;">""</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span> <span style="color: #000080;font-style:italic;">-- trivial</span>
+         <span style="color: #004080;">sequence</span> <span style="color: #000000;">lensq</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">repeat</span><span style="color: #0000FF;">(</span><span style="color: #008000;">""</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
-         -- Hnf and solution</span>
-         <span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">k</span> <span style="color: #008080;">by</span> <span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
+         <span style="color: #000000;">lensq</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">g</span>
+         <span style="color: #000080;font-style:italic;">-- Calculate lengths squared</span>
-             <span style="color: #000000;">output</span><span style="color: #0000FF;">(</span><span style="color: #000000;">print_row</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmts</span><span style="color: #0000FF;">)&</span><span style="color: #008080;">iif</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">k</span><span style="color: #0000FF;">?</span><span style="color: #000000;">g</span><span style="color: #0000FF;">:</span><span style="color: #008000;">""</span><span style="color: #0000FF;">))</span>
-         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
-         <span style="color: #000080;font-style:italic;">-- Null space with lengths squared</span>
          <span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
-             <span style="color: #000000;">line</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">print_row</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmts</span><span style="color: #0000FF;">)</span>
              <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
              <span style="color: #008080;">for</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
                  <span style="color: #000000;">q</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">s</span><span style="color: #0000FF;">]</span>
              <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
-             <span style="color: #000000;">output</span><span style="color: #0000FF;">(</span><span style="color: #000000;">line</span><span style="color: #0000FF;">&</span><span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"   (%d)"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">))</span>
+             <span style="color: #000000;">lensq</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">&=</span> <span style="color: #7060A8;">sprintf</span><span style="color: #0000FF;">(</span><span style="color: #008000;">"   (%d)"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">q</span><span style="color: #0000FF;">)</span>
+         <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
+         <span style="color: #000080;font-style:italic;">-- Hnf and solution and null space</span>
+         <span style="color: #004080;">sequence</span> <span style="color: #000000;">order</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)&</span><span style="color: #7060A8;">tagset</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
+         <span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
+             <span style="color: #004080;">integer</span> <span style="color: #000000;">rr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">order</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">]</span>
+             <span style="color: #000000;">output</span><span style="color: #0000FF;">(</span><span style="color: #000000;">print_row</span><span style="color: #0000FF;">(</span><span style="color: #000000;">rr</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmts</span><span style="color: #0000FF;">)&</span><span style="color: #000000;">lensq</span><span style="color: #0000FF;">[</span><span style="color: #000000;">rr</span><span style="color: #0000FF;">])</span>
          <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
@@ Line 1,014: / Line 1,024: @@
          <span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
-             <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">print_row</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmts</span><span style="color: #0000FF;">)&</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
+             <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">print_row</span><span style="color: #0000FF;">(</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">fmts</span><span style="color: #0000FF;">))</span>
+ <span style="color: #000080;font-style:italic;">-- (not particularly helpful:)
+ --          printf(1,", L:")
+ --          for s=1 to m+1 do
+ --              printf(1," %.20g",lambda[r,s])
+ --          end for</span>
+             <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"\n"</span><span style="color: #0000FF;">)</span>
          <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
      <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
@@ Line 1,037: / Line 1,053: @@
  <span style="color: #000080;font-style:italic;">-- LLL reduce rows k</span>
  <span style="color: #008080;">procedure</span> <span style="color: #000000;">Reduce</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">)</span>
-     <span style="color: #004080;">integer</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">sx</span>
-     <span style="color: #004080;">atom</span> <span style="color: #000000;">lk</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">q</span>
      <span style="color: #000000;">col1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">nx</span>
      <span style="color: #000000;">col2</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">nx</span>
@@ Line 1,054: / Line 1,068: @@
          <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
      <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
-     <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
+     <span style="color: #004080;">atom</span> <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
      <span style="color: #008080;">if</span> <span style="color: #000000;">col1</span><span style="color: #0000FF;"><</span><span style="color: #000000;">nx</span> <span style="color: #008080;">then</span>
          <span style="color: #008080;">if</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">col1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]<</span><span style="color: #000000;">0</span> <span style="color: #008080;">then</span> <span style="color: #000000;">Minus</span><span style="color: #0000FF;">(</span><span style="color: #000000;">t</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
          <span style="color: #000000;">q</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">floor</span><span style="color: #0000FF;">(</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">col1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]/</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">,</span><span style="color: #000000;">col1</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">])</span>
      <span style="color: #008080;">else</span>
-         <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span>
+         <span style="color: #004080;">atom</span> <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span>
          <span style="color: #008080;">if</span> <span style="color: #000000;">2</span><span style="color: #0000FF;">*</span><span style="color: #7060A8;">abs</span><span style="color: #0000FF;">(</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">)></span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #008080;">then</span>
              <span style="color: #000080;font-style:italic;">-- 2|lambda_kt| &gt; d_t
@@ Line 1,068: / Line 1,082: @@
      <span style="color: #008080;">if</span> <span style="color: #000000;">q</span> <span style="color: #008080;">then</span>
-         <span style="color: #000000;">sx</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iif</span><span style="color: #0000FF;">(</span><span style="color: #000000;">col1</span><span style="color: #0000FF;">=</span><span style="color: #000000;">nx</span><span style="color: #0000FF;">?</span><span style="color: #000000;">m</span><span style="color: #0000FF;">:</span><span style="color: #000000;">mn</span><span style="color: #0000FF;">)</span>
+         <span style="color: #004080;">integer</span> <span style="color: #000000;">sx</span> <span style="color: #0000FF;">=</span> <span style="color: #008080;">iif</span><span style="color: #0000FF;">(</span><span style="color: #000000;">col1</span><span style="color: #0000FF;">=</span><span style="color: #000000;">nx</span><span style="color: #0000FF;">?</span><span style="color: #000000;">m</span><span style="color: #0000FF;">:</span><span style="color: #000000;">mn</span><span style="color: #0000FF;">)</span>
          <span style="color: #000080;font-style:italic;">-- reduce row k</span>
          <span style="color: #008080;">for</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">=</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">sx</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
@@ Line 1,082: / Line 1,096: @@
  <span style="color: #000080;font-style:italic;">-- exchange rows k and k-1</span>
  <span style="color: #008080;">procedure</span> <span style="color: #000000;">Swop</span><span style="color: #0000FF;">(</span><span style="color: #004080;">integer</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">)</span>
-     <span style="color: #004080;">integer</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">s</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
+     <span style="color: #004080;">integer</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tm1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
-     <span style="color: #004080;">atom</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">lk</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">lr</span>
      <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">a</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]}</span>
-     <span style="color: #0000FF;">{</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]}</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">{</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">],</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">t</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]}</span>
+     <span style="color: #004080;">object</span> <span style="color: #000000;">tmp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">tm1</span><span style="color: #0000FF;">]</span>
+     <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">tm1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">tm1</span><span style="color: #0000FF;">]</span>
+     <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">][</span><span style="color: #000000;">1</span><span style="color: #0000FF;">..</span><span style="color: #000000;">tm1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">tmp</span><span style="color: #0000FF;">;</span> <span style="color: #000000;">tmp</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
     <span style="color: #000080;font-style:italic;">-- update Gram coefficients
     -- columns k, k-1 for r &gt; k</span>
-     <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span>
+     <span style="color: #004080;">atom</span> <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">],</span>
-     <span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
+          <span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
      <span style="color: #008080;">for</span> <span style="color: #000000;">r</span><span style="color: #0000FF;">=</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">to</span> <span style="color: #000000;">m</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span> <span style="color: #008080;">do</span>
-         <span style="color: #000000;">lr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
+         <span style="color: #004080;">atom</span> <span style="color: #000000;">lr</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">],</span>
-         <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lr</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
+              <span style="color: #000000;">dk1</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
-         <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">db</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lr</span><span style="color: #0000FF;">+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">])/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
+         <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">dk1</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]-</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lr</span><span style="color: #0000FF;">)/</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span>
+         <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #0000FF;">(</span><span style="color: #000000;">db</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lr</span><span style="color: #0000FF;">+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">r</span><span style="color: #0000FF;">,</span><span style="color: #000000;">k</span><span style="color: #0000FF;">])/</span><span style="color: #000000;">dk1</span>
      <span style="color: #008080;">end</span> <span style="color: #008080;">for</span>
-     <span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">db</span>
+     <span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">]</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">db</span>
  <span style="color: #008080;">end</span> <span style="color: #008080;">procedure</span>
@@ Line 1,105: / Line 1,121: @@
      <span style="color: #000000;">problem_no</span> <span style="color: #0000FF;">+=</span> <span style="color: #000000;">1</span>
      <span style="color: #7060A8;">printf</span><span style="color: #0000FF;">(</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #008000;">"problem #%d\n"</span><span style="color: #0000FF;">,</span><span style="color: #000000;">problem_no</span><span style="color: #0000FF;">)</span>
-     <span style="color: #004080;">integer</span> <span style="color: #000000;">i</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">tl</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
-     <span style="color: #004080;">atom</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">lk</span>
-     <span style="color: #000000;">InputABC</span><span style="color: #0000FF;">(</span><span style="color: #000000;">sw</span><span style="color: #0000FF;">)</span>
+     <span style="color: #000000;">InputAb_or_c</span><span style="color: #0000FF;">(</span><span style="color: #000000;">sw</span><span style="color: #0000FF;">)</span>
      <span style="color: #000080;font-style:italic;">-- augment Ab~ with column e_m</span>
@@ Line 1,119: / Line 1,133: @@
      <span style="color: #008080;">if</span> <span style="color: #000000;">echo</span> <span style="color: #008080;">then</span> <span style="color: #000000;">PrntM</span><span style="color: #0000FF;">(</span><span style="color: #000000;">0</span><span style="color: #0000FF;">)</span> <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
-     <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span>
+     <span style="color: #004080;">integer</span> <span style="color: #000000;">k</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">1</span><span style="color: #0000FF;">,</span>
+             <span style="color: #000000;">tl</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">0</span>
      <span style="color: #008080;">while</span> <span style="color: #000000;">k</span><span style="color: #0000FF;"><=</span><span style="color: #000000;">m</span> <span style="color: #008080;">do</span>
-         <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
+         <span style="color: #004080;">integer</span> <span style="color: #000000;">t</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">k</span><span style="color: #0000FF;">-</span><span style="color: #000000;">1</span>
          <span style="color: #000080;font-style:italic;">-- partial size reduction</span>
          <span style="color: #000000;">Reduce</span><span style="color: #0000FF;">(</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span> <span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">)</span>
@@ Line 1,128: / Line 1,143: @@
          <span style="color: #008080;">if</span> <span style="color: #000000;">sw</span> <span style="color: #008080;">then</span>
              <span style="color: #000080;font-style:italic;">-- zero rows k-1, k</span>
-             <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
+             <span style="color: #004080;">atom</span> <span style="color: #000000;">lk</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">lambda</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">,</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]</span>
              <span style="color: #000080;font-style:italic;">-- Lovasz condition
              -- Bk &gt;= (alpha - mu_kt^2) * Bt</span>
-             <span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lk</span>
+             <span style="color: #004080;">atom</span> <span style="color: #000000;">db</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">1</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">k</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]+</span><span style="color: #000000;">lk</span><span style="color: #0000FF;">*</span><span style="color: #000000;">lk</span>
              <span style="color: #000080;font-style:italic;">-- not satisfied</span>
              <span style="color: #000000;">sw</span> <span style="color: #0000FF;">=</span> <span style="color: #000000;">db</span><span style="color: #0000FF;">*</span><span style="color: #000000;">ald</span><span style="color: #0000FF;"><</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">d</span><span style="color: #0000FF;">[</span><span style="color: #000000;">t</span><span style="color: #0000FF;">+</span><span style="color: #000000;">2</span><span style="color: #0000FF;">]*</span><span style="color: #000000;">aln</span>
@@ Line 1,202: / Line 1,217: @@
      <span style="color: #008080;">end</span> <span style="color: #008080;">if</span>
  <span style="color: #008080;">end</span> <span style="color: #008080;">while</span>
+ <span style="color: #0000FF;">?</span><span style="color: #008000;">"done"</span>
+ <span style="color: #0000FF;">{}</span> <span style="color: #0000FF;">=</span> <span style="color: #7060A8;">wait_key</span><span style="color: #0000FF;">()</span>
 <!--</lang>-->
-Note that for problem 16 (HMM extended gcd (example 7.2)), the signs of
-the (20) and (37) rows in the null space are flipped, somehow, otherwise
-output matches that of FreeBasic (and is auto-checked). (I suspect the
-issue is FreeBasic distinguishing 0 and -0 in a way that Phix does not)
-Eagle eyed viewers may have spotted my blase use of null space there as
-if I knew what I was talking about (tee hee).
 =={{header|Wren}}==