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Probabilistic Versus Worst-Case Rounding Error Analysis Nick Higham School of Mathematics The University of Manchester www.maths.manchester.ac.uk/~higham Advances in Numerical Linear Algebra University of Manchester, May 29-30, 2019 Joint work with Theo Mary

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Page 1: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Research Matters

February 25, 2009

Nick HighamDirector of Research

School of Mathematics

1 / 6

Probabilistic Versus Worst-CaseRounding Error Analysis

Nick HighamSchool of Mathematics

The University of Manchesterwww.maths.manchester.ac.uk/~higham

Advances in Numerical Linear AlgebraUniversity of Manchester, May 29-30, 2019

Joint work with Theo Mary

Page 2: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

James Hardy Wilkinson (1919–1986)

Nick Higham Probabilistic Rounding Error Analysis 2 / 28

Page 3: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Today’s Floating-Point Arithmetics

Type Bits Range u = 2−t

bfloat16 half 16 10±38 2−8 ≈ 3.9× 10−3

fp16 half 16 10±5 2−11 ≈ 4.9× 10−4

fp32 single 32 10±38 2−24 ≈ 6.0× 10−8

fp64 double 64 10±308 2−53 ≈ 1.1× 10−16

fp128 quadruple 128 10±4932 2−113 ≈ 9.6× 10−35

fp* forms all IEEE standard.bfloat16 used by Google TPU.bfloat16 will be in Intel Xeon Cooper Lake (2020).

Nick Higham Probabilistic Rounding Error Analysis 3 / 28

Page 4: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Today’s Floating-Point Arithmetics

Type Bits Range u = 2−t

bfloat16 half 16 10±38 2−8 ≈ 3.9× 10−3

fp16 half 16 10±5 2−11 ≈ 4.9× 10−4

fp32 single 32 10±38 2−24 ≈ 6.0× 10−8

fp64 double 64 10±308 2−53 ≈ 1.1× 10−16

fp128 quadruple 128 10±4932 2−113 ≈ 9.6× 10−35

fp* forms all IEEE standard.bfloat16 used by Google TPU.bfloat16 will be in Intel Xeon Cooper Lake (2020).

Nick Higham Probabilistic Rounding Error Analysis 3 / 28

Page 5: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Rounding

fl(x) denotes rounding x to the nearest fl pt number.

TheoremFor x ∈ R, excluding overflow and underflow,

fl(x) = x(1 + δ), |δ| ≤ u.

u := 12β

1−t is the unit roundoff, or machine precision.

The normalized nonneg numbers for β = 2, t = 3:

0 0.5 1.0 2.0 3.0 4.0 5.0 6.0 7.0

Nick Higham Probabilistic Rounding Error Analysis 4 / 28

Page 6: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Model for Rounding Error Analysis

For x , y ∈ F

fl(x op y) = (x op y)(1 + δ), |δ| ≤ u, op = +,−, ∗, /.

Also for op =√

.

Wilkinson (1960):

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Page 7: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Model for Rounding Error Analysis

For x , y ∈ F

fl(x op y) = (x op y)(1 + δ), |δ| ≤ u, op = +,−, ∗, /.

Also for op =√

.

Wilkinson (1960):

Nick Higham Probabilistic Rounding Error Analysis 5 / 28

Page 8: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Weaknesses of the Model

Worst-case bound on |δ|.

Does not account for some modern hardwarefeatures.

Weaker than fl(x op y) being correctly rounded.

Does it matter?Yes.Yes.No (usually).

Nick Higham Probabilistic Rounding Error Analysis 6 / 28

Page 9: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Weaknesses of the Model

Worst-case bound on |δ|.

Does not account for some modern hardwarefeatures.

Weaker than fl(x op y) being correctly rounded.

Does it matter?Yes.Yes.No (usually).

Nick Higham Probabilistic Rounding Error Analysis 6 / 28

Page 10: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Model vs Correctly Rounded Resulty = x(1 + δ), with |δ| ≤ u does not imply y = fl(x).

β = 10,t = 2

x y |δ| u = 12101−t

9.185 8.7 5.28e-2 5.00e-29.185 8.8 4.19e-2 5.00e-29.185 8.9 3.10e-2 5.00e-29.185 9.0 2.01e-2 5.00e-29.185 9.1 9.25e-3 5.00e-29.185 9.2 1.63e-3 5.00e-29.185 9.3 1.25e-2 5.00e-29.185 9.4 2.34e-2 5.00e-29.185 9.5 3.43e-2 5.00e-29.185 9.6 4.52e-2 5.00e-29.185 9.7 5.61e-2 5.00e-2

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Page 11: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Hardware Features That Reduce Error

Fused multiply-add (FMA) may be available (NVIDIAV100), with just one rounding error:

fl(x + y ∗ z) = (x + y ∗ z)(1 + δ), |δ| ≤ u.

Reduces error bound by factor 2.

80-bit registers on Intel chips may be exploited bythe compiler in evaluating inner products, etc.

“Tensor units” with block FMA on Google TPU andNVIDIA Volta and Turing GPUs.

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Page 12: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Traditional Bounds Are Pessimistic (1)Traditional bounds are worst-case bounds and arepessimistic on average. For Uniform [−1,1] data:

Matrix–vector product (fp32)

10 1 10 2 10 3 10 410 -8

10 -7

10 -6

10 -5

10 -4

10 -3

Solution of Ax = b (fp32)

10 1 10 2 10 3 10 410 -8

10 -6

10 -4

10 -2

Nick Higham Probabilistic Rounding Error Analysis 9 / 28

Page 13: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Traditional Bounds Are Pessimistic (2)Matrix–vector product (fp16)

10 0 10 1 10 2 10 310 -4

10 -3

10 -2

10 -1

10 0

Matrix–vector product (fp8)

10 0 10 1 10 2 10 3

10 -1

10 0

Traditional bounds do not provide a realistic picture of thetypical behavior of numerical computations

Nick Higham Probabilistic Rounding Error Analysis 10 / 28

Page 14: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

The Rule of Thumb

“In general, the statistical distribution ofthe rounding errors will reduceconsiderably the function of n occurring inthe relative errors. We might expect ineach case that this function should bereplaced by something which is no biggerthan its square root.”

— Wilkinson, 1961

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Page 15: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Statistical Argument

Consider sum of rounding errors s =∑n

i=1 δi , |δi | ≤ u.

Worst-case bound |s| ≤ nu is attainable—unlikely!

Assume δi are independent random variables ofmean zeroCentral limit theorem: for sufficiently large n,

s/√

n ∼ N (0, σ2), σ ≤ u,

hence |s| ≤ λ√

nu , with λ a small constant, holds withhigh probability (e.g., 99.7% with λ = 3).

s is only linear part of error.What n is “sufficiently large”?

Nick Higham Probabilistic Rounding Error Analysis 12 / 28

Page 16: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Objective

Fundamental lemma in backward error analysis (H, 2002)If |δi | ≤ u for i = 1 : n and nu < 1 then

n∏i=1

(1 + δi) = 1 + θn,

where|θn| ≤ γn :=

nu1− nu

= nu + O(u2).

The basis of most rounding error analyses.We seek an analogous result with a smaller, butprobabilistic, bound on θn.We will focus on backward error results.

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Page 17: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Probabilistic Model of Rounding Errors

In the computation of interest, the quantities δ infl(a op b) = (a op b)(1 + δ), |δ| ≤ u, op ∈ {+,−,×, /}

are independent random variables of mean zero.

“There is no claim that ordinary rounding andchopping are random processes, or thatsuccessive errors are independent. Thequestion to be decided is whether or notthese particular probabilistic models of theprocesses will adequately describe whatactually happens.”

— Hull & Swenson, 1966

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Page 18: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Why Might the Model Not be Realistic?

In some cases δ ≡ 0, e.g., integer operands in x + y ,xy , or an operand a power of 2 in xy , x/y .

Pairs of operands might be repeated, so different δare the same.

Non-pathological examples can be found whererounding errors are strongly correlated (Kahan).

If an operand comes from an earlier computation it willdepend on an earlier δ and so the new δ will dependon a previous one.

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Page 19: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

The Key IdeasTransform the product into a sum by taking the logarithm:

S = logn∏

i=1

(1 + δi) =n∑

i=1

log(1 + δi).

Hoeffding’s concentration inequalityLet X1, . . . , Xn be random independent variables satisfying|Xi | ≤ ci . Then the sum S =

∑ni=1 Xi satisfies

Pr(|S − E(S)| ≥ ξ) ≤ 2exp(− ξ2

2∑n

i=1 c2i

).

Apply to Xi = log(1 + δi)⇒ requires boundinglog(1 + δi) and E (log(1 + δi)) using Taylor expansions.Retrieve the result by taking the exponential of S.

Nick Higham Probabilistic Rounding Error Analysis 16 / 28

Page 20: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

The Key IdeasTransform the product into a sum by taking the logarithm:

S = logn∏

i=1

(1 + δi) =n∑

i=1

log(1 + δi).

Hoeffding’s concentration inequalityLet X1, . . . , Xn be random independent variables satisfying|Xi | ≤ ci . Then the sum S =

∑ni=1 Xi satisfies

Pr(|S − E(S)| ≥ ξ) ≤ 2exp(− ξ2

2∑n

i=1 c2i

).

Apply to Xi = log(1 + δi)⇒ requires boundinglog(1 + δi) and E (log(1 + δi)) using Taylor expansions.Retrieve the result by taking the exponential of S.

Nick Higham Probabilistic Rounding Error Analysis 16 / 28

Page 21: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

The Key IdeasTransform the product into a sum by taking the logarithm:

S = logn∏

i=1

(1 + δi) =n∑

i=1

log(1 + δi).

Hoeffding’s concentration inequalityLet X1, . . . , Xn be random independent variables satisfying|Xi | ≤ ci . Then the sum S =

∑ni=1 Xi satisfies

Pr(|S − E(S)| ≥ ξ) ≤ 2exp(− ξ2

2∑n

i=1 c2i

).

Apply to Xi = log(1 + δi)⇒ requires boundinglog(1 + δi) and E (log(1 + δi)) using Taylor expansions.Retrieve the result by taking the exponential of S.

Nick Higham Probabilistic Rounding Error Analysis 16 / 28

Page 22: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Probabilistic Error Bound

Theorem (H & Mary, 2018)Let δi , i = 1 : n, be independent random variables of meanzero such that |δi | ≤ u. For any constant λ > 0, the relation∏n

i=1(1 + δi) = 1 + θn holds with

|θn| ≤ γn(λ) := exp(λ√

nu +nu2

1− u

)− 1

≤ λ√

nu + O(u2)

with prob of failure P(λ) = 2exp(−λ2(1− u)2/2

)

Key features:Exact bound, not first order.nu < 1 not required.No “n is sufficiently large” assumption.Small values of λ suffice: P(1) ≈ 0.27, P(5) ≤ 10−5.

Nick Higham Probabilistic Rounding Error Analysis 17 / 28

Page 23: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Probabilistic Error Bound

Theorem (H & Mary, 2018)Let δi , i = 1 : n, be independent random variables of meanzero such that |δi | ≤ u. For any constant λ > 0, the relation∏n

i=1(1 + δi) = 1 + θn holds with

|θn| ≤ γn(λ) := exp(λ√

nu +nu2

1− u

)− 1

≤ λ√

nu + O(u2)

with prob of failure P(λ) = 2exp(−λ2(1− u)2/2

)Key features:

Exact bound, not first order.nu < 1 not required.No “n is sufficiently large” assumption.Small values of λ suffice: P(1) ≈ 0.27, P(5) ≤ 10−5.

Nick Higham Probabilistic Rounding Error Analysis 17 / 28

Page 24: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Inner Products

TheoremLet y = aT b, where a,b ∈ Rn. No matter what the order ofevaluation the computed y satisfies

y = (a + ∆a)T b,

|∆a| ≤ γn(λ)|a| ≤ λ√

nu|a|+ O(u2),

with probability of failure nP(λ).

Now a factor n in front of P(λ).Can choose any λ > 0.Analogous result for matrix–vector products.

Nick Higham Probabilistic Rounding Error Analysis 18 / 28

Page 25: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

LU Factorization

TheoremThe computed LU factors from Gaussian elimination onA ∈ Rn×n satisfy LU = A + ∆A, where

|∆A| ≤ γn(λ)|L||U|, |γn(λ)| ≤ λ√

nu + O(u2)

holds with probability of failure (n3/3 + n2/2 + 7n/6)P(λ).

Want probabilities independent of n! Fortunately:

O(n3)P(λ) = O(1) ⇒ λ = O(√

logn)

⇒ error grows no faster than√

n logn u. . . and the constant hidden in the big O is small:

n3

3P(13) ≤ 10−5 for n ≤ 1010.

Nick Higham Probabilistic Rounding Error Analysis 19 / 28

Page 26: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

LU Factorization

TheoremThe computed LU factors from Gaussian elimination onA ∈ Rn×n satisfy LU = A + ∆A, where

|∆A| ≤ γn(λ)|L||U|, |γn(λ)| ≤ λ√

nu + O(u2)

holds with probability of failure (n3/3 + n2/2 + 7n/6)P(λ).

Want probabilities independent of n! Fortunately:

O(n3)P(λ) = O(1) ⇒ λ = O(√

logn)

⇒ error grows no faster than√

n logn u. . . and the constant hidden in the big O is small:

n3

3P(13) ≤ 10−5 for n ≤ 1010.

Nick Higham Probabilistic Rounding Error Analysis 19 / 28

Page 27: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Probabilities of Success for A = LU Bounds

λ n = 102 n = 103 n = 104

7.0 9.9998e-01 9.8474e-01 −1.4265e+017.5 1.0000e+00 9.9959e-01 5.9320e-018.0 1.0000e+00 9.9999e-01 9.9156e-018.5 1.0000e+00 1.0000e+00 9.9986e-019.0 1.0000e+00 1.0000e+00 1.0000e+00

Nick Higham Probabilistic Rounding Error Analysis 20 / 28

Page 28: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

MATLAB Experiments

Simulate fp16 and fp8 with MATLAB functionchop (H & Pranesh, 2019).

Compare the bounds γn and γn(λ) with thecomponentwise backward error εbwd (Oettli–Prager):

Matrix–vector product y = Ax : εbwd = maxi|y−y |i(|A||x |)i

Solution to Ax = b via LU factorization:εbwd = maxi |Ax − b|i/(|L||U||x |)i

A and x are chosen.

Random entries are Uniform [−1,1] or Uniform [0,1].

Codes available online: https://gitlab.com/theo.andreas.mary/proberranalysis

Nick Higham Probabilistic Rounding Error Analysis 21 / 28

Page 29: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Random [−1,1] Entries

Matrix–vector product (fp32)

10 1 10 2 10 3 10 410 -8

10 -7

10 -6

10 -5

10 -4

10 -3

Linear system Ax = b (fp32)

10 1 10 2 10 3 10 410 -8

10 -6

10 -4

10 -2

Probabilistic bound (λ = 1) much closer to the actualerror.But for [−1,1] entries it’s still pessimistic

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Page 30: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Random [−1,1] Entries

Matrix–vector product (fp32)

10 1 10 2 10 3 10 410 -8

10 -7

10 -6

10 -5

10 -4

10 -3

Linear system Ax = b (fp32)

10 1 10 2 10 3 10 410 -8

10 -6

10 -4

10 -2

Probabilistic bound (λ = 1) much closer to the actualerror.But for [−1,1] entries it’s still pessimistic

Nick Higham Probabilistic Rounding Error Analysis 22 / 28

Page 31: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Random [0,1] EntriesMatrix–vector product (fp32)

10 1 10 2 10 3 10 410 -8

10 -7

10 -6

10 -5

10 -4

10 -3

Linear system Ax = b (fp32)

10 1 10 2 10 3 10 410 -8

10 -6

10 -4

10 -2

Prob bound has λ = 1⇒ P(λ) pessimistic . . .. . . but γn bound itself can be sharp and successfullycaptures the

√n error growth

⇒ the bounds cannot be improved without furtherassumptions

Nick Higham Probabilistic Rounding Error Analysis 23 / 28

Page 32: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Low Precisions, Random [−1,1] EntriesMatrix–vector product (fp16)

10 0 10 1 10 2 10 310 -4

10 -3

10 -2

10 -1

10 0

Matrix–vector product (fp8)

10 0 10 1 10 2 10 3

10 -1

10 0

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Page 33: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Low Precisions, Random [0,1] EntriesMatrix–vector product (fp16)

10 0 10 1 10 2 10 310 -4

10 -3

10 -2

10 -1

10 0

Matrix–vector product (fp8)

10 0 10 1 10 2 10 3

10 -1

10 0

Importance of the probabilistic bound becomes evenclearer for lower precisions

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Page 34: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Real-Life MatricesSolution of Ax = b (fp64), b from Uniform [0,1],

for 943 matrices from SuiteSparse collection (λ = 1).

101

102

103

104

10-16

10-15

10-14

10-13

10-12

Nick Higham Probabilistic Rounding Error Analysis 26 / 28

Page 35: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Nonzero MeanInner product of two very large nonnegative vectors:

si+1 = si + aibi ⇒ si+1 = (si + aibi)(1 + δi)

100

102

104

106

108

10-10

10-5

100

Top: 1 ≤ n ≤ 106

Bottom: 106 ≤ n ≤ 108

Explanation: si ⇑ and at some point it is so large thatsi+1 = si ⇒ δi = −aibi/(si + aibi) < 0

Nick Higham Probabilistic Rounding Error Analysis 27 / 28

Page 36: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Nonzero MeanInner product of two very large nonnegative vectors:

si+1 = si + aibi ⇒ si+1 = (si + aibi)(1 + δi)

100

102

104

106

108

10-10

10-5

100

Top: 1 ≤ n ≤ 106

Bottom: 106 ≤ n ≤ 108

Explanation: si ⇑ and at some point it is so large thatsi+1 = si ⇒ δi = −aibi/(si + aibi) < 0

Nick Higham Probabilistic Rounding Error Analysis 27 / 28

Page 37: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Nonzero MeanInner product of two very large nonnegative vectors:

si+1 = si + aibi ⇒ si+1 = (si + aibi)(1 + δi)

100

102

104

106

108

10-10

10-5

100 Top: 1 ≤ n ≤ 106

Bottom: 106 ≤ n ≤ 108

Explanation: si ⇑ and at some point it is so large thatsi+1 = si ⇒ δi = −aibi/(si + aibi) < 0

Nick Higham Probabilistic Rounding Error Analysis 27 / 28

Page 38: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Conclusions

Given first rigorous justification of Wilkinson’s “takethe square root of the constant in error bound” rule ofthumb—and our results hold for any n!Experiments show prob bounds give betterpredictions than deterministic ones for both randomand real-life matrices and can be sharp. Consistentwith

The fact that rounding errors are neitherrandom nor uncorrelated will not in itselfpreclude the possibility of modelling themusefully by uncorrelated random variables.

— Kahan, 1996

H & Mary, A New Approach to Probabilistic RoundingError Analysis, MIMS EPrint 2018.33, Nov 2018.

Nick Higham Probabilistic Rounding Error Analysis 28 / 28

Page 39: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

References I

N. J. Higham.Accuracy and Stability of Numerical Algorithms.Society for Industrial and Applied Mathematics,Philadelphia, PA, USA, second edition, 2002.ISBN 0-89871-521-0.xxx+680 pp.

N. J. Higham and T. Mary.A new approach to probabilistic rounding error analysis.MIMS EPrint 2018.33, Manchester Institute forMathematical Sciences, The University of Manchester,UK, Nov. 2018.22 pp.Revised March 2019.

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Page 40: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

References II

N. J. Higham and S. Pranesh.Simulating low precision floating-point arithmetic.MIMS EPrint 2019.4, Manchester Institute forMathematical Sciences, The University of Manchester,UK, Mar. 2019.17 pp.

T. E. Hull and J. R. Swenson.Tests of probabilistic models for propagation of roundofferrors.Comm. ACM, 9(2):108–113, 1966.

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Page 41: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

References III

W. Kahan.The improbability of probabilistic error analyses fornumerical computations.Manuscript, Mar. 1996.

J. H. Wilkinson.Error analysis of direct methods of matrix inversion.J. Assoc. Comput. Mach., 8:281–330, 1961.

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Page 42: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Not IndependentInner product of twoconstant vectors:

si+1 = si + aibi = si + c⇒ si+1 = (si + c)(1 + δi)

⇒ δi = θ is constant withinintervals [2q−1; 2q]

10 2 10 3 10 4

10 -6

10 -5

10 -4

10 -3

10 -2

2q−1 2q

×

si si+1 si+2 si+3

+c

××θ

+c

××θ

+c

××θ

Nick Higham Probabilistic Rounding Error Analysis 4 / 4

Page 43: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Not IndependentInner product of twoconstant vectors:

si+1 = si + aibi = si + c⇒ si+1 = (si + c)(1 + δi)

⇒ δi = θ is constant withinintervals [2q−1; 2q]

10 2 10 3 10 4

10 -6

10 -5

10 -4

10 -3

10 -2

2q−1 2q

×

si si+1 si+2 si+3

+c

××θ

+c

××θ

+c

××θ

Nick Higham Probabilistic Rounding Error Analysis 4 / 4

Page 44: Research Matters Nick Higham February 25, 2009 School of … · 2019. 6. 4. · the square root of the constant in error bound” rule of thumb—and our results hold for any n!,

Example: Rounding Errors Not IndependentInner product of twoconstant vectors:

si+1 = si + aibi = si + c⇒ si+1 = (si + c)(1 + δi)

⇒ δi = θ is constant withinintervals [2q−1; 2q]

10 2 10 3 10 4

10 -6

10 -5

10 -4

10 -3

10 -2

2q−1 2q

×

si si+1 si+2 si+3

+c

××θ

+c

××θ

+c

××θ

Nick Higham Probabilistic Rounding Error Analysis 4 / 4