Newton's method fails when the function is flat because the local slope (derivative) is zero or near-zero. Since the method relies on drawing a tangent line and finding its x-intercept, a horizontal tangent does not intersect the x-axis at a useful new point, preventing progress toward the root.
Conditions: The function is locally flat at the current point.; The slope is too small or zero to provide a useful Newton step.
Newton's method fails when the function is flat because the local slope (derivative) is zero or near-zero. Since the method relies on drawing a tangent line and finding its x-intercept, a horizontal tangent does not intersect the x-axis at a useful new point, preventing progress toward the root.
Conditions: The function is locally flat at the current point.; The slope is too small or zero to provide a useful Newton step.