math - Specialised algorithm to find positive real solutions to quartic equations? -


i'm looking specialised algorithm find positive real solutions quartic equations real coefficients (also know bi-quadratic or polynomial equations of order 4). have form:

a4 x4 + a3 x3 +a2 x2 +a1 x + a0 = 0

with a1, a2,... being real numbers.

it's supposed run on microcontroller, need quite lot of calculations. performance issue. that's why i'm looking specialised algorithm positive solutions. if possible i'd compute exact solutions.

i know there general way compute solution of quartic equation rather involved in terms of computation.

can point me in right direction?

edit:

judging answers: seem have misunderstood me (though pretty clear it). i know of standard ways of solving quartic equations. don't me - neither fit in memory nor sufficiently fast. need high accuracy highly efficient algorithm find real solutions (if helps) quartic equations real coefficients. i'm not sure there such algorithm, thought guys might know. p.s.: downvotes didn't come me.

this 1 of situations easier find roots using complex arithmetic find positive real roots. , since sounds need find multiple roots @ once, recommend using durand-kerner method, refinement of method of weierstrass:

http://en.wikipedia.org/wiki/durand%e2%80%93kerner_method

weierstrass' method in turn refinement of newton's method solves for roots of polynomial in parallel (and has big advantage brain-dead easy code up). converges @ quadratic rate in general, though linearly multiple roots. quartic polynomials, can pretty nail roots in few iterations. if need more general purpose solution, should use instead use jenkins-traub:

http://en.wikipedia.org/wiki/jenkins%e2%80%93traub_method

this faster higher degree polynomials, , works converting problem finding eigenvalues of companion matrix:

http://en.wikipedia.org/wiki/companion_matrix

edit: second suggestion, try using power method on companion matrix. since equation has non-negative coefficients, may find useful apply perron-frobenius theorem companion matrix. @ minimal, certifies there exists @ least 1 non-negative root:

http://en.wikipedia.org/wiki/perron%e2%80%93frobenius_theorem


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