Singular Manual: laguerre_solve

| #### D.8.4.1 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);
    ==> [1]:
    ==>    0.8924637479
    ==> [2]:
    ==>    (-0.7392783383+i*0.5355190078)
    ==> [3]:
    ==>    (-0.7392783383-i*0.5355190078)
    ==> [4]:
    ==>    (0.2930464644-i*0.9003002396)
    ==> [5]:
    ==>    (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:  complex,10,10,i
    ==> //         polynomial has complex coefficients
    ==> //split in working ring:
    ==> //split without result
    ==> //END laguerre_solve
    l;
    ==> [1]:
    ==> (-0.8557376852+i*0.3557664188)
    ==> [2]:
    ==> (-0.5462895588-i*0.6796668873)
    ==> [3]:
    ==> (0.04588498039+i*0.9133296179)
    ==> [4]:
    ==> (0.5037408279-i*0.8058051828)
    ==> [5]:
    ==> (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[1]);
    ==> 0
    subst(f,x,l[2]);
    ==> 0

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