Cauchy Bound

analysis math

Table of Contents

The Cauchy bound limits possible roots of a polynomial to an interval.

Let cauchy_bound_6087d57885dad44f2ddccf638c361750d44789d2.svg be not the zero-polynomial and cauchy_bound_6568ddd3939fdc147c46632f40f278416a97d281.svg .

For every cauchy_bound_901e0914ba7f398cdc691b171171496482c36551.svg with cauchy_bound_4f79f08b52b2b5cf3d873c0594dbc8e87f158683.svg we have

cauchy_bound_b318437dd0e7502b3b6f36cc674a333273c94f2a.svg

All roots of cauchy_bound_a97390dc87a628420b23faa53d6d6d4d17e34c82.svg lie within the interval cauchy_bound_17b0c003c2d82f1e29de379fa1450ce12aac84a6.svg .

Proof

Divide all cauchy_bound_cf6f9e2ab576fe62626d4f2d764bfec3836982fb.svg by cauchy_bound_2720c61019cc406a7f3befc29502a90e855bc8c9.svg , then prove for cauchy_bound_cc5607a7d1d984a0adeb9bc00a8afd4268ad2b65.svg .

Let cauchy_bound_36ecc87976108b00711259d6cdf24e4c9d1c9c3d.svg and cauchy_bound_90d46c739fb4f0958d7a9e89813ca3c61434ae91.svg .

Properties of cauchy_bound_a0617e019657c050199ce168300299aac6a94ca3.svg

  • cauchy_bound_e427e0f28906d0d4a644fbdb03201199be9c0876.svg
  • cauchy_bound_0f7a0e4e9fc8c933c99f4021323d809bb5c6c415.svg
  • cauchy_bound_0e9dbe57f0bd483698081360c8d93301d1bc6e71.svg
cauchy_bound_bc5808fe92c28596a498a5a98e9ee5417434ca7d.svg

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