In Sage notebook, I want to print the standard mathematical notation for the limit of a function.
I expect something like this:
f = x+1
latex(limit(f, x=0)) + '=' + lim(f, x=0)
to print a properly formatted 'lim x->0 x+1 = 1'
naturally, I figured it out shortly after posting the question, always seems to work that way. here's the code from my notes-book:
show("NOTE: how to print lim")
# raw latex
f = (x^2-1)/(x-1)
Lf = LatexExpr(r'\lim_{x\to\infty}') + latex(f)
show(Lf)
# dummy limit
from sage.calculus.calculus import dummy_limit as lim
Lf = lim(f, x, 0)
out = latex(Lf) + "=" + latex(limit(f, x=0))
show(out)
Not very elegant, really wish the limit had a better way to represent itself in unevaluated form.
Related
I see several posts about making a string in to a lua table, but my problem is a little different [I think] because there is an additional dimension to the table.
I have a table of tables saved as a file [i have no issue reading the file to a string].
let's say we start from this point:
local tot = "{{1,2,3}, {4,5,6}}"
When I try the answers from other users I end up with:
local OneDtable = {"{1,2,3}, {4,5,6}"}
This is not what i want.
how can i properly create a table, that contains those tables as entries?
Desired result:
TwoDtable = {{1,2,3}, {4,5,6}}
Thanks in advance
You can use the load function to read the content of your string as Lua code.
local myArray = "{{1,2,3}, {4,5,6}}"
local convert = "myTable = " .. myArray
local convertFunction = load(convert)
convertFunction()
print(myTable[1][1])
Now, myTable has the values in a 2-dimensional array.
For a quick solution I suggest going with the load hack, but be aware that this only works if your code happens to be formatted as a Lua table already. Otherwise, you'd have to parse the string yourself.
For example, you could try using lpeg to build a recursive parser. I built something very similar a while ago:
local lpeg = require 'lpeg'
local name = lpeg.R('az')^1 / '\0'
local space = lpeg.S('\t ')^1
local function compile_tuple(...)
return string.char(select('#', ...)) .. table.concat{...}
end
local expression = lpeg.P {
'e';
e = name + lpeg.V 't';
t = '(' * ((lpeg.V 'e' * ',' * space)^0 * lpeg.V 'e') / compile_tuple * ')';
}
local compiled = expression:match '(foo, (a, b), bar)'
print(compiled:byte(1, -1))
Its purpose is to parse things in quotes like the example string (foo, (a, b), bar) and turn it into a binary string describing the structure; most of that happens in the compile_tuple function though, so it should be easy to modify it to do what you want.
What you'd have to adapt:
change name for number (and change the pattern accordingly to lpeg.R('09')^1, without the / '\0')
change the compile_tuple function to a build_table function (local function build_tanle(...) return {...} end should do the trick)
Try it out and see if something else needs to be changed; I might have missed something.
You can read the lpeg manual here if you're curious about how this stuff works.
So I would like to print polynomials in one variable (s) with one parameter (a), say
a·s^3 − s^2 - a^2·s − a + 1.
Sage always displays it with decreasing degree, and I would like to get something like
1 - a - a^2·s - s^2 + a·s^3
to export it to LaTeX. I can't figure out how to do this... Thanks in advance.
As an alternative to string manipulation, one can use the series expansion.
F = a*s^3 - s^2 - a^2*s - a + 1
F.series(s, F.degree(s)+1)
returns
(-a + 1) + (-a^2)*s + (-1)*s^2 + (a)*s^3
which appears to be what you wanted, save for some redundant parentheses.
This works because (a) a power series is ordered from lowest to highest coefficients; (b) making the order of remainder greater than the degree of the polynomial ensures that the series is just the polynomial itself.
This is not easy, because the sort order is defined in Pynac, a fork of Ginac, which Sage uses for its basic symbolic manipulation. However, depending on what you need, it is possible programmatically:
sage: F = 1 + x + x^2
sage: "+".join(map(str,sorted([f for f in F.operands()],key=lambda exp:exp.degree(x))))
'1+x+x^2'
I don't know whether this sort of thing is powerful enough for your needs, though. You may have to traverse the "expression tree" quite a bit but at least your sort of example seems to work.
sage: F = a + a^2*x + x^2 - a*x^2
sage: "+".join(map(str,sorted([f for f in F.operands()],key=lambda exp:exp.degree(x))))
'a+a^2*x+-a*x^2+x^2'
Doing this in a short statement requires a number of Python tricks like this, which are very well worth learning if you are going to use Sage (or Numpy, or pandas, or ...) a fair amount.
I'm writing a LSL to Lua translator, and I'm having all sorts of trouble implementing incrementing and decrementing operators. LSL has such things using the usual C like syntax (x++, x--, ++x, --x), but Lua does not. Just to avoid massive amounts of typing, I refer to these sorts of operators as "crements". In the below code, I'll use "..." to represent other parts of the expression.
... x += 1 ...
Wont work, coz Lua only has simple assignment.
... x = x + 1 ...
Wont work coz that's a statement, and Lua can't use statements in expressions. LSL can use crements in expressions.
function preIncrement(x) x = x + 1; return x; end
... preIncrement(x) ...
While it does provide the correct value in the expression, Lua is pass by value for numbers, so the original variable is not changed. If I could get this to actually change the variable, then all is good. Messing with the environment might not be such a good idea, dunno what scope x is. I think I'll investigate that next. The translator could output scope details.
Assuming the above function exists -
... x = preIncrement(x) ...
Wont work for the "it's a statement" reason.
Other solutions start to get really messy.
x = preIncrement(x)
... x ...
Works fine, except when the original LSL code is something like this -
while (doOneThing(x++))
{
doOtherThing(x);
}
Which becomes a whole can of worms. Using tables in the function -
function preIncrement(x) x[1] = x[1] + 1; return x[1]; end
temp = {x}
... preincrement(temp) ...
x = temp[1]
Is even messier, and has the same problems.
Starting to look like I might have to actually analyse the surrounding code instead of just doing simple translations to sort out what the correct way to implement any given crement will be. Anybody got any simple ideas?
I think to really do this properly you're going to have to do some more detailed analysis, and splitting of some expressions into multiple statements, although many can probably be translated pretty straight-forwardly.
Note that at least in C, you can delay post-increments/decrements to the next "sequence point", and put pre-increments/decrements before the previous sequence point; sequence points are only located in a few places: between statements, at "short-circuit operators" (&& and ||), etc. (more info here)
So it's fine to replace x = *y++ + z * f (); with { x = *y + z * f(); y = y + 1; }—the user isn't allowed to assume that y will be incremented before anything else in the statement, only that the value used in *y will be y before it's incremented. Similarly, x = *--y + z * f(); can be replaced with { y = y - 1; x = *y + z * f (); }
Lua is designed to be pretty much impervious to implementations of this sort of thing. It may be done as kind of a compiler/interpreter issue, since the interpreter can know that variables only change when a statement is executed.
There's no way to implement this kind of thing in Lua. Not in the general case. You could do it for global variables by passing a string to the increment function. But obviously it wouldn't work for locals, or for variables that are in a table that is itself global.
Lua doesn't want you to do it; it's best to find a way to work within the restriction. And that means code analysis.
Your proposed solution only will work when your Lua variables are all global. Unless this is something LSL also does, you will get trouble translating LSL programs that use variables called the same way in different places.
Lua is only able of modifying one lvalue per statement - tables being passed to functions are the only exception to this rule. You could use a local table to store all locals, and that would help you out with the pre-...-crements; they can be evaluated before the expression they are contained in is evauated. But the post-...-crements have to be evaluated later on, which is simply not possible in lua - at least not without some ugly code involving anonymous functions.
So you have one options: you must accept that some LSL statements will get translated to several Lua statements.
Say you have a LSL statement with increments like this:
f(integer x) {
integer y = x + x++;
return (y + ++y)
}
You can translate this to a Lua statement like this:
function f(x) {
local post_incremented_x = x + 1 -- extra statement 1 for post increment
local y = x + post_incremented_x
x = post_incremented_x -- extra statement 2 for post increment
local pre_incremented_y = y + 1
return y + pre_incremented_y
y = pre_incremented_y -- this line will never be executed
}
So you basically will have to add two statements per ..-crement used in your statements. For complex structures, that will mean calculating the order in which the expressions are evaluated.
For what is worth, I like with having post decrements and predecrements as individual statements in languages. But I consider it a flaw of the language when they can also be used as expressions. The syntactic sugar quickly becomes semantic diabetes.
After some research and thinking I've come up with an idea that might work.
For globals -
function preIncrement(x)
_G[x] = _G[x] + 1
return _G[x]
end
... preIncrement("x") ...
For locals and function parameters (which are locals to) I know at the time I'm parsing the crement that it is local, I can store four flags to tell me which of the four crements is being used in the variables AST structure. Then when it comes time to output the variables definition, I can output something like this -
local x;
function preIncrement_x() x = x + 1; return x; end
function postDecrement_x() local y = x; x = x - 1; return y; end
... preIncrement_x() ...
In most of your assessment of configurability to the code. You are trying to hard pass data types from one into another. And call it a 'translator'. And in all of this you miss regex and other pattern match capacities. Which are far more present in LUA than LSL. And since the LSL code is being passed into LUA. Try using them, along with other functions. Which would define the work more as a translator, than a hard pass.
Yes I know this was asked a while ago. Though, for other viewers of this topic. Never forget the environments you are working in. EVER. Use what they give you to the best ability you can.
I need a base converter function for Lua. I need to convert from base 10 to base 2,3,4,5,6,7,8,9,10,11...36 how can i to this?
In the string to number direction, the function tonumber() takes an optional second argument that specifies the base to use, which may range from 2 to 36 with the obvious meaning for digits in bases greater than 10.
In the number to string direction, this can be done slightly more efficiently than Nikolaus's answer by something like this:
local floor,insert = math.floor, table.insert
function basen(n,b)
n = floor(n)
if not b or b == 10 then return tostring(n) end
local digits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"
local t = {}
local sign = ""
if n < 0 then
sign = "-"
n = -n
end
repeat
local d = (n % b) + 1
n = floor(n / b)
insert(t, 1, digits:sub(d,d))
until n == 0
return sign .. table.concat(t,"")
end
This creates fewer garbage strings to collect by using table.concat() instead of repeated calls to the string concatenation operator ... Although it makes little practical difference for strings this small, this idiom should be learned because otherwise building a buffer in a loop with the concatenation operator will actually tend to O(n2) performance while table.concat() has been designed to do substantially better.
There is an unanswered question as to whether it is more efficient to push the digits on a stack in the table t with calls to table.insert(t,1,digit), or to append them to the end with t[#t+1]=digit, followed by a call to string.reverse() to put the digits in the right order. I'll leave the benchmarking to the student. Note that although the code I pasted here does run and appears to get correct answers, there may other opportunities to tune it further.
For example, the common case of base 10 is culled off and handled with the built in tostring() function. But similar culls can be done for bases 8 and 16 which have conversion specifiers for string.format() ("%o" and "%x", respectively).
Also, neither Nikolaus's solution nor mine handle non-integers particularly well. I emphasize that here by forcing the value n to an integer with math.floor() at the beginning.
Correctly converting a general floating point value to any base (even base 10) is fraught with subtleties, which I leave as an exercise to the reader.
you can use a loop to convert an integer into a string containting the required base. for bases below 10 use the following code, if you need a base larger than that you need to add a line that mapps the result of x % base to a character (usign an array for example)
x = 1234
r = ""
base = 8
while x > 0 do
r = "" .. (x % base ) .. r
x = math.floor(x / base)
end
print( r );
I have a multiple sequence alignment (Clustal) file and I want to read this file and arrange sequences in such a way that it looks more clear and precise in order.
I am doing this from Biopython using an AlignIO object:
alignment = AlignIO.read("opuntia.aln", "clustal")
print "Number of rows: %i" % len(align)
for record in alignment:
print "%s - %s" % (record.id, record.seq)
My output looks messy and long scrolling. What I want to do is print only 50 sequences in each line and continue until the end of the alignment file.
I wish to have output like this, from http://www.ebi.ac.uk/Tools/clustalw2/.
Br,
I don't have biopython on this computer, so this isn't tested, but it should work:
chunk_size = 50
for i in range(0, alignment.get_alignment_length(), chunk_size):
print ""
for record in alignment:
print "%s\t%s %i" % (record.name, record.seq[i:i + chunk_size], i + chunk_size)
Does the same trick as Eli's one - using range to set up an index to slice from then iterating over the record in the alignment for each slice.
Do you require something more complex than simply breaking record.seq into chunks of 50 characters, or am I missing something?
You can use Python sequence slicing to achieve that very easily. seq[N:N+50] accesses the 50 sequence elements starting with N:
In [24]: seq = ''.join(str(random.randint(1, 4)) for i in range(200))
In [25]: seq
Out[25]: '13313211211434211213343311221443122234343421132111223234141322124442112343143112411321431412322123214232414331224144142222323421121312441313314342434231131212124312344112144434314122312143242221323123'
In [26]: for n in range(0, len(seq), 50):
....: print seq[n:n+50]
....:
....:
13313211211434211213343311221443122234343421132111
22323414132212444211234314311241132143141232212321
42324143312241441422223234211213124413133143424342
31131212124312344112144434314122312143242221323123