MPSolve benchmarks

We report the CPU time, expressed in seconds, needed to compute the roots of univariate polynomials with the following functions:

The experiments have been performed on a Pentium 200 MMX with 64MB RAM under the Linux operating system.
The set of test polynomials has been divided into the following classes:

Polynomials with roots in geometric progressions

Spiral p(x) = (x+1) (x+1+a) (x+1+a+a²) ... (x+1+a+a²+... +aⁿ), a = i/1000, i² = -1, n = 10, 15, 20, 25 The polynomial has roots along a spiral centered in 1/(1-a). Their distance from the center decreases exponentially.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
10	1.51	1.53	26.3	20.4	24.5	17.5	13	16	1.63	73	250	59	45.6	156	36.8	10
15	13.8	13.7	231	102	129	16.8	7.4	9.4	13.8	-	957	238	-	69	17	15
20	91	84.4	1860	400	F	22	4.7	-	92.4	-	-	-	-	-	-	20
25	279	227	-	1557	-	-	6.8	-	276	-	-	-	-	-	-	25
Timings on a Pentium 200 MMX

Geometric - 1 p(x) = (x+1) (x+a) (x+a²) ... (x+a^n-1), a = 100 i, i² = -1, n = 10, 15, 20, 40 The polynomial has roots in geometric progression of ratio a, \|a\| > 1.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
10	.	0.04	1	7	3.1	25	175	75	.	.	.	.				10
15	.	0.05	2.9	32.4	14.8	58	648	296	.	.	.	.				15
20	.	0.08	F	280	61.8	-	3496	772	.	.	.	.				20
25	.	0.45	F	F	>36000	-	-	>45000	.	.	.	.				25
Timings on a Pentium 200 MMX

Geometric - 2 p(x) = (x+1)(x+a)(x+a²)...(x+a^n-1), a = i/100, i² = -1, n = 10, 15, 20, 40 The polynomial has roots in geometric progression of ratio a, \|a\| < 1.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
10	.	0.03	5.9	6	14.4	197	200	480	.	.	.	.				10
15	.	0.07	45	54.4	38	643	777	543	.	.	.	.				15
20	.	0.11	255	424	205	2322	3853	1859	.	.	.	.				20
25	.	0.6	F	>36000	F	-	>60000	-	.	.	.	.				25
Timings on a Pentium 200 MMX

Polynomials with roots along a curve

Truncated exponential p(x) = 1 + x/1! + x²/2! + ... + xⁿ/n!, n = 50, 100, 200 The polynomial has roots along a curve.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
50	.	0.7	17.6	8.2	27.6	25	12	39	.	.	.	.				50
100	.	7.4	364	726	243	49	98	33	.	.	.	.				100
200	.	66	F	-	2434	-	-	37	.	.	.	.				200
.Timings on a Pentium 200 MMX

Easy polynomial p(x) = 1 + 2x + 3x² + ... + (n+1)xⁿ, n = 50, 100, 200, 400, 800, 1600 The polynomial has roots along a curve, the roots are well conditioned.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
50	.	0.4	15.7	5.7	21.3	39	14	53	.	.	.	.				50
100	.	1.7	164	25	156	97	15	92	.	.	.	.				100
200	.	8.5	772	169	>9600	91	20	>1136	.	.	.	.				200
400	.	35								.	.	.				400
800	.	150							.	.	.	.				800
1600	.	725							.	.	.	.				1600
Timings on a Pentium 200 MMX

Roots of unity p(x) = xⁿ -1, n = 50, 100, 200, 400, 800 The polynomial has roots on the unit circle, all the roots are well conditioned.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
50	.	0.2	0.2	0.7	20	1	3.5	100	.	.	.	.				50
100	.	0.5	0.4	1.4	155	0.8	2.8	310	.	.	.	.				100
200	.	1.7	1	3.6	1823	0.6	2	1072	.	.	.	.				200
400	.	7.4	3.5	10	-	0.5	1.4	-	.	.	.	.				400
800	.	25.7	12.8	43	-	0.5	1.7	-	.	.	.	.				800
Timings on a Pentium 200 MMX

Wilkinson's polynomial p(x) = (x-1) (x-2) ... (x-n), n = 20, 40, 80, 160 All the roots are ill conditioned.
	Isolate	Approximate 50 digits							Approximate 200 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
20	.	0.2	1.5	15.1	3.5	8	76	18	.	.	.	.				20
40	.	1.6	40.6	181	30.1	25	113	19	.	.	.	.				40
80	.	19	277	1266	357	15	67	19	.	.	.	.				80
160	.	243	-	-	-	-	-	-	.	.	.	.				160
Timings on a Pentium 200 MMX

Characteristic polynomial - 1

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-i, -3, 0, 10, 1, i, 2 i, -3, 1),
n = 128, 256
Many roots cluster into accumulation points.

Isolate

Approximate 50 digits

Approximate 200 digits

CPU

Speedup

CPU

Speedup

MPSolve

Math

Maple

Pari

Math

Maple

Pari

MPSolve

Math

Maple

Pari

Math

Maple

Pari

128

8.5

424

128

256

3017

256

Timings on a Pentium 200 MMX

Characteristic polynomial - 2

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-7 i, -32, 0, 10, 0, 0, 0, -32, 1),
n = 128, 256
Many roots cluster into accumulation points.

Isolate

Approximate 50 digits

Approximate 200 digits

CPU

Speedup

CPU

Speedup

MPSolve

Math

Maple

Pari

Math

Maple

Pari

MPSolve

Math

Maple

Pari

Math

Maple

Pari

128

439

128

256

301

>3600

>12

256

Timings on a Pentium 200 MMX

Polynomials with few clustered roots and/or large and small moduli

Kameny polynomial p(x) = (c² x² - 3)² + c² x⁹, c = 10ⁿ, n = 6, 10, 20 Two extremely close real roots (with 20, 70, 247 common decimal digits), a complex pair with small imaginary parts (10^-27, 10^-90, 10^-315). The remaining roots are well separated, one of them is real.
	Isolate	Approximate 50 digits							Approximate 400 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
6	0.02	0.02	3.7	4	1.8	185	200	90	0.2	83	36	24	415	180	120	6
10	0.12	0.05	3.2	67	5.2	64	1340	104	0.3	F	55	29	-	165	87	10
20	0.3	0.05	3.2	>2000	36.5	64	>40000	730	0.7	F	>1300	77	-	1857	110	20
Timings on a Pentium 200 MMX

Mignotte-like polynomial p(x) = xⁿ + (a x+ 1)^m, a = 100 i, m = 3, n =20, 100, 200, 500 The polynomial is a modification of Mignotte's polynomials having clustered roots whose distance is close to the Mahler bound. It has m roots close to -1/a and the remaining are roughly displaced along a circle.
	Isolate	Approximate 50 digits							Approximate 400 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
20	0.04	0.09	12	13	9.3	120	130	93	0.2	83	36	24	415	180	120	20
100	0.42	0.77	1650	>2000	587	2142	>2597	726	0.3	F	55	29	-	165	87	100
200	1.5	2.13	-	-	-	-	-	-	0.7	F	>1300	77	-	1857	110	200
500	16.7	19.9	-	-	-	-	-	-	.							500
Timings on a Pentium 200 MMX

Roots with large and small moduli - 1

p(x) = x⁵⁰⁰ + 10²⁰⁰ x⁴⁹⁹ + 10⁵⁰⁰ x⁴⁹⁵ + 10⁴⁹⁹ x⁴⁸⁰ + 10³ x³ + 3 10² x² + 1.

Isolate

Approximate 50 digits

Approximate 400 digits

CPU

Speedup

CPU

Speedup

MPSolve

Math

Maple

Pari

Math

Maple

Pari

MPSolve

Math

Maple

Pari

Math

Maple

Pari

23.1

19.7

Timings on a Pentium 200 MMX

Roots with large and small moduli - 2

p(x) = (x¹² - (10²⁰ x - 1)⁴) (1 + (10²⁰ + x)⁴ x⁸)
4 roots of modulus 10^-20, 4 roots of modulus 10²⁰, 60 common digits.

Isolate

Approximate 50 digits

Approximate 1000 digits

CPU

Speedup

CPU

Speedup

MPSolve

Math

Maple

Pari

Math

Maple

Pari

MPSolve

Math

Maple

Pari

Math

Maple

Pari

0.64

0.33

1699

>3000

5048

>9090

251

5.4

1485

275

Timings on a Pentium 200 MMX

Polynomials with the roots in a fractal (Mandelbrot set)

Mandelbrot polynomial - 1 p(x) defined according to the recurrence equation p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1, i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ... The polynomials have roots in the Mandelbrot set.
	Isolate	Approximate 50 digits							Approximate 1000 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
31	.	0.25	2.6	1.8	4.5	10	7.2	18	.	.	.	.				31
63	.	1.83	26	175	26	14	95	14	.	.	.	.				63
127	.	18.5	363	1382	356	19	74	19	.	.	.	.				127
255	.	213	2781	-	2544	13	-	12	.	.	.	.				255
511	.	3272							.	.	.	.				511
Timings on a Pentium 200 MMX

Mandelbrot polynomial - 2 p(x) defined according to the recurrence equation p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1, i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ... Polynomial defined by means of a straight line program Uses a special feature of MPSolve: polynomial defined by a C program.
	Isolate	Approximate 50 digits							Approximate 1000 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
31	.	0.12							.	.	.	.				31
63	.	0.6							.	.	.	.				63
127	.	3.33							.	.	.	.				127
255	.	23.7							.	.	.	.				255
511	.	163							.	.	.	.				511
1023	.	1122							.	.	.	.				1023
2047	.	8537							.	.	.	.				2047
Timings on a Pentium 200 MMX

Polynomials with many multiple roots

Kirrinnis polynomial p(x) = (x⁴ - 1 / 16)^m (x⁴ - (1 / 2 + e)⁴) 16^m/e⁴, e = 1/4096, m = 10, 20, 40 4 multiple roots 1/2, -1/2, i/2, -i/2, of multiplicity m, 4 roots 1/2+e, -(1/2+e), i(1/2+e), -i (1/2+e) pretty close to the former. Between parentheses is reported the time needed after applying a symbolic preprocessing.
	Isolate	Approximate 50 digits							Approximate 400 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
10	.	6.9 (0.01)	54.3	0.02	0.23	8 (5430)	0.003 (2)	0.03 (23)	.	.	.	.				10
20	.	7.2 (0.02)	>3600	0.4	0.23	>500 (>180000)	0.05 (20)	0.03 (11)	.	.	.	.				20
40	.	39.9 (0.3)	-	1.1	7.7	-	0.03 (3.6)	0.2 (25)	.	.	.	.				40
Timings on a Pentium 200 MMX

Polynomials with all the roots grouped into few clusters

Modified Kirrinnis polynomial p(x) = (x⁴ - 1 / 16)^m (x⁴ - (1 / 2 + e)⁴)16^m/e⁴+x^4m+4, e = 1/4096, m = 10, 20, 40 The polynomial is obtained by a tiny relative perturbation of the leading coefficient of the Kirrinnis polynomials. They have simple roots clustered into 4 groups, and degree 4 m + 4.
	Isolate	Approximate 50 digits							Approximate 400 digits
	CPU	CPU				Speedup			CPU				Speedup
n	MPSolve	MPSolve	Math	Maple	Pari	Math	Maple	Pari	MPSolve	Math	Maple	Pari	Math	Maple	Pari	n
10	.	2.4	4	110	29	1.6	45	12	.	.	.	.				10
20	.	11	27	-	174	2.4	-	16	.	.	.	.				20
40	.	81	227	-	1512	2.8	-	19	.	.	.	.				40
Timings on a Pentium 200 MMX

MPSolve benchmarks

Polynomials with roots in geometric progressions

Spiral

p(x) = (x+1) (x+1+a) (x+1+a+a²) ... (x+1+a+a²+... +aⁿ),
a = i/1000, i² = -1, n = 10, 15, 20, 25
The polynomial has roots along a spiral centered in 1/(1-a).
Their distance from the center decreases exponentially.

Geometric - 1

p(x) = (x+1) (x+a) (x+a²) ... (x+a^n-1),
a = 100 i, i² = -1, n = 10, 15, 20, 40
The polynomial has roots in geometric progression of ratio a, |a| > 1.

Geometric - 2

p(x) = (x+1)(x+a)(x+a²)...(x+a^n-1),
a = i/100, i² = -1, n = 10, 15, 20, 40
The polynomial has roots in geometric progression of ratio a, |a| < 1.

Polynomials with roots along a curve

Truncated exponential

p(x) = 1 + x/1! + x²/2! + ... + xⁿ/n!,
n = 50, 100, 200
The polynomial has roots along a curve.

Easy polynomial

p(x) = 1 + 2x + 3x² + ... + (n+1)xⁿ,
n = 50, 100, 200, 400, 800, 1600
The polynomial has roots along a curve, the roots are well conditioned.

Roots of unity

p(x) = xⁿ -1,
n = 50, 100, 200, 400, 800
The polynomial has roots on the unit circle, all the roots are well conditioned.

Wilkinson's polynomial

p(x) = (x-1) (x-2) ... (x-n),
n = 20, 40, 80, 160
All the roots are ill conditioned.

Characteristic polynomial - 1

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-i, -3, 0, 10, 1, i, 2 i, -3, 1),
n = 128, 256
Many roots cluster into accumulation points.

Characteristic polynomial - 2

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-7 i, -32, 0, 10, 0, 0, 0, -32, 1),
n = 128, 256
Many roots cluster into accumulation points.

Polynomials with few clustered roots and/or large and small moduli

Kameny polynomial

p(x) = (c² x² - 3)² + c² x⁹,
c = 10ⁿ, n = 6, 10, 20
Two extremely close real roots (with 20, 70, 247 common decimal digits),
a complex pair with small imaginary parts (10^-27, 10^-90, 10^-315).
The remaining roots are well separated, one of them is real.

Mignotte-like polynomial

p(x) = xⁿ + (a x+ 1)^m,
a = 100 i, m = 3, n =20, 100, 200, 500
The polynomial is a modification of Mignotte's polynomials having
clustered roots whose distance is close to the Mahler bound.
It has m roots close to -1/a and the remaining are roughly displaced along a circle.

Roots with large and small moduli - 1

p(x) = x⁵⁰⁰ + 10²⁰⁰ x⁴⁹⁹ + 10⁵⁰⁰ x⁴⁹⁵ + 10⁴⁹⁹ x⁴⁸⁰ + 10³ x³ + 3 10² x² + 1.

Roots with large and small moduli - 2

p(x) = (x¹² - (10²⁰ x - 1)⁴) (1 + (10²⁰ + x)⁴ x⁸)
4 roots of modulus 10^-20, 4 roots of modulus 10²⁰, 60 common digits.

Polynomials with the roots in a fractal (Mandelbrot set)

Mandelbrot polynomial - 1

p(x) defined according to the recurrence equation
p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1,
i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ...
The polynomials have roots in the Mandelbrot set.

Mandelbrot polynomial - 2

p(x) defined according to the recurrence equation
p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1,
i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ...
Polynomial defined by means of a straight line program
Uses a special feature of MPSolve: polynomial defined by a C program.

Polynomials with many multiple roots

Kirrinnis polynomial

Polynomials with all the roots grouped into few clusters

Modified Kirrinnis polynomial

p(x) = (x⁴ - 1 / 16)^m (x⁴ - (1 / 2 + e)⁴)16^m/e⁴+x^4m+4,
e = 1/4096, m = 10, 20, 40
The polynomial is obtained by a tiny relative perturbation
of the leading coefficient of the Kirrinnis polynomials.
They have simple roots clustered into 4 groups, and degree 4 m + 4.

MPSolve benchmarks

Polynomials with roots in geometric progressions

Spiral

p(x) = (x+1) (x+1+a) (x+1+a+a2) ... (x+1+a+a2+... +an), a = i/1000, i2 = -1, n = 10, 15, 20, 25 The polynomial has roots along a spiral centered in 1/(1-a). Their distance from the center decreases exponentially.

Geometric - 1

p(x) = (x+1) (x+a) (x+a2) ... (x+an-1), a = 100 i, i2 = -1, n = 10, 15, 20, 40 The polynomial has roots in geometric progression of ratio a, |a| > 1.

Geometric - 2

p(x) = (x+1)(x+a)(x+a2)...(x+an-1), a = i/100, i2 = -1, n = 10, 15, 20, 40 The polynomial has roots in geometric progression of ratio a, |a| < 1.

Polynomials with roots along a curve

Truncated exponential

p(x) = 1 + x/1! + x2/2! + ... + xn/n!, n = 50, 100, 200 The polynomial has roots along a curve.

Easy polynomial

p(x) = 1 + 2x + 3x2 + ... + (n+1)xn, n = 50, 100, 200, 400, 800, 1600 The polynomial has roots along a curve, the roots are well conditioned.

Roots of unity

p(x) = xn -1, n = 50, 100, 200, 400, 800 The polynomial has roots on the unit circle, all the roots are well conditioned.

Wilkinson's polynomial

p(x) = (x-1) (x-2) ... (x-n), n = 20, 40, 80, 160 All the roots are ill conditioned.

Characteristic polynomial - 1

Characteristic polynomial of a n×n band Toeplitz matrix whose entries along the diagonals are given by (a-2, a-1, a0, a1, ..., a6) = (-i, -3, 0, 10, 1, i, 2 i, -3, 1), n = 128, 256 Many roots cluster into accumulation points.

Characteristic polynomial - 2

Characteristic polynomial of a n×n band Toeplitz matrix whose entries along the diagonals are given by (a-2, a-1, a0, a1, ..., a6) = (-7 i, -32, 0, 10, 0, 0, 0, -32, 1), n = 128, 256 Many roots cluster into accumulation points.

Polynomials with few clustered roots and/or large and small moduli

Kameny polynomial

p(x) = (c2 x2 - 3)2 + c2 x9, c = 10n, n = 6, 10, 20 Two extremely close real roots (with 20, 70, 247 common decimal digits), a complex pair with small imaginary parts (10-27, 10-90, 10-315). The remaining roots are well separated, one of them is real.

Mignotte-like polynomial

p(x) = xn + (a x + 1)m, a = 100 i, m = 3, n =20, 100, 200, 500 The polynomial is a modification of Mignotte's polynomials having clustered roots whose distance is close to the Mahler bound. It has m roots close to -1/a and the remaining are roughly displaced along a circle.

Roots with large and small moduli - 1

p(x) = x500 + 10200 x499 + 10500 x495 + 10499 x480 + 103 x3 + 3 102 x2 + 1.

Roots with large and small moduli - 2

p(x) = (x12 - (1020 x - 1)4) (1 + (1020 + x)4 x8) 4 roots of modulus 10-20, 4 roots of modulus 1020, 60 common digits.

Polynomials with the roots in a fractal (Mandelbrot set)

Mandelbrot polynomial - 1

p(x) defined according to the recurrence equation p0(x) = 1, pi+1(x) = x pi2(x) + 1, i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ... The polynomials have roots in the Mandelbrot set.

Mandelbrot polynomial - 2

p(x) defined according to the recurrence equation p0(x) = 1, pi+1(x) = x pi2(x) + 1, i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ... Polynomial defined by means of a straight line program Uses a special feature of MPSolve: polynomial defined by a C program.

Polynomials with many multiple roots

Kirrinnis polynomial

p(x) = (x4 - 1 / 16)m (x4 - (1 / 2 + e)4) 16m/e4, e = 1/4096, m = 10, 20, 40 4 multiple roots 1/2, -1/2, i/2, -i/2, of multiplicity m, 4 roots 1/2+e, -(1/2+e), i(1/2+e), -i (1/2+e) pretty close to the former. Between parentheses is reported the time needed after applying a symbolic preprocessing.

Polynomials with all the roots grouped into few clusters

Modified Kirrinnis polynomial

p(x) = (x4 - 1 / 16)m (x4 - (1 / 2 + e)4)16m/e4+x4m+4, e = 1/4096, m = 10, 20, 40 The polynomial is obtained by a tiny relative perturbation of the leading coefficient of the Kirrinnis polynomials. They have simple roots clustered into 4 groups, and degree 4 m + 4.

p(x) = (x+1) (x+1+a) (x+1+a+a²) ... (x+1+a+a²+... +aⁿ),
a = i/1000, i² = -1, n = 10, 15, 20, 25
The polynomial has roots along a spiral centered in 1/(1-a).
Their distance from the center decreases exponentially.

p(x) = (x+1) (x+a) (x+a²) ... (x+a^n-1),
a = 100 i, i² = -1, n = 10, 15, 20, 40
The polynomial has roots in geometric progression of ratio a, |a| > 1.

p(x) = (x+1)(x+a)(x+a²)...(x+a^n-1),
a = i/100, i² = -1, n = 10, 15, 20, 40
The polynomial has roots in geometric progression of ratio a, |a| < 1.

p(x) = 1 + x/1! + x²/2! + ... + xⁿ/n!,
n = 50, 100, 200
The polynomial has roots along a curve.

p(x) = 1 + 2x + 3x² + ... + (n+1)xⁿ,
n = 50, 100, 200, 400, 800, 1600
The polynomial has roots along a curve, the roots are well conditioned.

p(x) = xⁿ -1,
n = 50, 100, 200, 400, 800
The polynomial has roots on the unit circle, all the roots are well conditioned.

p(x) = (x-1) (x-2) ... (x-n),
n = 20, 40, 80, 160
All the roots are ill conditioned.

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-i, -3, 0, 10, 1, i, 2 i, -3, 1),
n = 128, 256
Many roots cluster into accumulation points.

Characteristic polynomial of a n×n band Toeplitz matrix
whose entries along the diagonals are given by
(a_-2, a_-1, a₀, a₁, ..., a₆) = (-7 i, -32, 0, 10, 0, 0, 0, -32, 1),
n = 128, 256
Many roots cluster into accumulation points.

p(x) = (c² x² - 3)² + c² x⁹,
c = 10ⁿ, n = 6, 10, 20
Two extremely close real roots (with 20, 70, 247 common decimal digits),
a complex pair with small imaginary parts (10^-27, 10^-90, 10^-315).
The remaining roots are well separated, one of them is real.

p(x) = xⁿ + (a x+ 1)^m,
a = 100 i, m = 3, n =20, 100, 200, 500
The polynomial is a modification of Mignotte's polynomials having
clustered roots whose distance is close to the Mahler bound.
It has m roots close to -1/a and the remaining are roughly displaced along a circle.

p(x) = x⁵⁰⁰ + 10²⁰⁰ x⁴⁹⁹ + 10⁵⁰⁰ x⁴⁹⁵ + 10⁴⁹⁹ x⁴⁸⁰ + 10³ x³ + 3 10² x² + 1.

p(x) = (x¹² - (10²⁰ x - 1)⁴) (1 + (10²⁰ + x)⁴ x⁸)
4 roots of modulus 10^-20, 4 roots of modulus 10²⁰, 60 common digits.

p(x) defined according to the recurrence equation
p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1,
i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ...
The polynomials have roots in the Mandelbrot set.

p(x) defined according to the recurrence equation
p₀(x) = 1, p_i+1(x) = x p_i²(x) + 1,
i = 0, 1, ... the degrees are n = 1, 3, 7, 15, ...
Polynomial defined by means of a straight line program
Uses a special feature of MPSolve: polynomial defined by a C program.

p(x) = (x⁴ - 1 / 16)^m (x⁴ - (1 / 2 + e)⁴)16^m/e⁴+x^4m+4,
e = 1/4096, m = 10, 20, 40
The polynomial is obtained by a tiny relative perturbation
of the leading coefficient of the Kirrinnis polynomials.
They have simple roots clustered into 4 groups, and degree 4 m + 4.