# Singular          #### D.8.3.1 laguerre_solve

Procedure from library `solve.lib` (see solve_lib).

Usage:
laguerre_solve(f [, m, l, n, s] ); f = polynomial,
m, l, n, s = integers (control parameters of the method)
m: precision of output in digits ( 4 <= m), if basering is not ring of complex numbers;
l: precision of internal computation in decimal digits ( l >=8 ) only if the basering is not complex or complex with smaller precision;
n: control of multiplicity of roots or of splitting of f into squarefree factors
n < 0, no split of f (good, if all roots are simple)
n >= 0, try to split
n = 0, return only different roots
n > 0, find all roots (with multiplicity)
s: s != 0, returns ERROR if | f(root) | > 0.1^m (when computing in the current ring)
( default: m, l, n, s = 8, 30, 1, 0 )

Assume:
f is a univariate polynomial;
basering has characteristic 0 and is either complex or without parameters.

Return:
list of (complex) roots of the polynomial f, depending on n. The entries of the result are of type
string: if the basering is not complex,
number: otherwise.

Note:
If printlevel >0: displays comments ( default = 0 ).
If s != 0 and if the procedure stops with ERROR, try a higher internal precision m.

Example:
 ```LIB "solve.lib"; // Find all roots of an univariate polynomial using Laguerre's method: ring rs1= 0,(x,y),lp; poly f = 15x5 + x3 + x2 - 10; // 10 digits precision laguerre_solve(f,10); ==> : ==> 0.8924637479 ==> : ==> (-0.7392783383+i*0.5355190078) ==> : ==> (-0.7392783383-i*0.5355190078) ==> : ==> (0.2930464644-i*0.9003002396) ==> : ==> (0.2930464644+i*0.9003002396) // Now with complex coefficients, // internal precision is 30 digits (default) printlevel=2; ring rsc= (real,10,i),x,lp; poly f = (15.4+i*5)*x^5 + (25.0e-2+i*2)*x^3 + x2 - 10*i; list l = laguerre_solve(f); ==> //BEGIN laguerre_solve ==> //control: complex ring with precision 30 ==> //working in: ring lagc=(complex,30,30),x,lp; ==> // polynomial has complex coefficients ==> //split in working ring: ==> //split without result ==> //END laguerre_solve l; ==> : ==> (-0.8557376852+i*0.3557664188) ==> : ==> (-0.5462895588-i*0.6796668873) ==> : ==> (0.04588498039+i*0.9133296179) ==> : ==> (0.5037408279-i*0.8058051828) ==> : ==> (0.8524014357+i*0.2163760334) // check result, value of substituted polynomial should be near to zero // remember that l contains a list of strings // in the case of a different ring subst(f,x,l); ==> 0 subst(f,x,l); ==> 0 ```

### Misc 