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In the Newton method, divide the `df/dx` argument by the
multiplicity of the desired root in order to preserve quadratic
convergence.

- Function:
**newton:find-integer-root***f df/dx x0* Given integer valued procedure

`f`, its derivative (with respect to its argument)`df/dx`, and initial integer value`x0`for which`df/dx`(`x0`) is non-zero, returns an integer`x`for which`f`(`x`) is closer to zero than either of the integers adjacent to`x`; or returns`#f`

if such an integer can’t be found.To find the closest integer to a given integer’s square root:

(define (integer-sqrt y) (newton:find-integer-root (lambda (x) (- (* x x) y)) (lambda (x) (* 2 x)) (ash 1 (quotient (integer-length y) 2)))) (integer-sqrt 15) ⇒ 4

- Function:
**newton:find-root***f df/dx x0 prec* Given real valued procedures

`f`,`df/dx`of one (real) argument, initial real value`x0`for which`df/dx`(`x0`) is non-zero, and positive real number`prec`, returns a real`x`for which`abs`

(`f`(`x`)) is less than`prec`; or returns`#f`

if such a real can’t be found.If

`prec`is instead a negative integer,`newton:find-root`

returns the result of -`prec`iterations.

H. J. Orchard, The Laguerre Method for Finding the Zeros of Polynomials, IEEE Transactions on Circuits and Systems, Vol. 36, No. 11, November 1989, pp 1377-1381.

There are 2 errors in Orchard’s Table II. Line k=2 for starting value of 1000+j0 should have Z_k of 1.0475 + j4.1036 and line k=2 for starting value of 0+j1000 should have Z_k of 1.0988 + j4.0833.

- Function:
**laguerre:find-root***f df/dz ddf/dz^2 z0 prec* Given complex valued procedure

`f`of one (complex) argument, its derivative (with respect to its argument)`df/dx`, its second derivative`ddf/dz^2`, initial complex value`z0`, and positive real number`prec`, returns a complex number`z`for which`magnitude`

(`f`(`z`)) is less than`prec`; or returns`#f`

if such a number can’t be found.If

`prec`is instead a negative integer,`laguerre:find-root`

returns the result of -`prec`iterations.

- Function:
**laguerre:find-polynomial-root***deg f df/dz ddf/dz^2 z0 prec* Given polynomial procedure

`f`of integer degree`deg`of one argument, its derivative (with respect to its argument)`df/dx`, its second derivative`ddf/dz^2`, initial complex value`z0`, and positive real number`prec`, returns a complex number`z`for which`magnitude`

(`f`(`z`)) is less than`prec`; or returns`#f`

if such a number can’t be found.If

`prec`is instead a negative integer,`laguerre:find-polynomial-root`

returns the result of -`prec`iterations.

- Function:
**secant:find-root***f x0 x1 prec* - Function:
**secant:find-bracketed-root***f x0 x1 prec* Given a real valued procedure

`f`and two real valued starting points`x0`and`x1`, returns a real`x`for which`(abs (f x))`

is less than`prec`; or returns`#f`

if such a real can’t be found.If

`x0`and`x1`are chosen such that they bracket a root, that is(or (< (f x0) 0 (f x1)) (< (f x1) 0 (f x0)))

then the root returned will be between

`x0`and`x1`, and`f`will not be passed an argument outside of that interval.`secant:find-bracketed-root`

will return`#f`

unless`x0`and`x1`bracket a root.The secant method is used until a bracketing interval is found, at which point a modified

*regula falsi*method is used.If

`prec`is instead a negative integer,`secant:find-root`

returns the result of -`prec`iterations.If

`prec`is a procedure it should accept 5 arguments:`x0``f0``x1``f1`and`count`, where`f0`will be`(f x0)`

,`f1``(f x1)`

, and`count`the number of iterations performed so far.`prec`should return non-false if the iteration should be stopped.

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