Cases

Conditions

Bounds

First Sparse’s case

(same as Cauchy)

| a 0 | ≥ 1

| λ | ≤ max { | a 0 | ; 1 + | a 1 | ; ⋯ ; 1 + | a 4 | }

Second Sparse’s case

| a 0 | < 1

| a 0 | + | a 1 | ≥ 1

| λ | ≤ max { | a 0 | + | a 1 | ; 1 + | a 2 | ; 1 + | a 3 | ; 1 + | a 4 | }

Third Sparse’s case

| a 0 | + | a 1 | < 1

| a 0 | + | a 1 | + | a 2 | ≥ 1

| λ | ≤ max { | a 0 | + | a 1 | + | a 2 | ; 1 + | a 3 | ; 1 + | a 4 | }

Fourth Sparse’s case

| a 0 | + | a 1 | + | a 2 | < 1

| a 0 | + ⋯ + | a 3 | ≥ 1

| λ | ≤ max { | a 0 | + ⋯ + | a 3 | ; 1 + | a 4 | }

fifth Sparse’s case

(Same as Montel)

| a 0 | + ⋯ + | a 3 | < 1

| λ | ≤ max { 1 ; | a 0 | + ⋯ + | a 4 | }

Cauchy

‖   .   ‖ ∞

| λ | ≤ max { | a 0 | ; 1 + | a 1 | ; ⋯ ; 1 + | a 4 | }

Montel

‖   .   ‖ 1

| λ | ≤ max { 1 ; | a 0 | + ⋯ + | a 4 | }

Carmichel-Mason

‖   .   ‖ 2

| λ | ≤ 1 + | a 0 | 2 + ⋯ + | a 4 | 2