CS 5220 Slides and Notes

Von Neumann and Goldstine

“Numerical Inverting of Matrices of High Order” (1947)

... matrices of the orders 15, 50, 150 can usually be inverted with a (relative) precision of 8, 10, 12 decimal digits less, respectively, than the number of digits carried throughout.

Turing

“Rounding-Off Errors in Matrix Processes” (1948)

Carrying \(d\) digits is equivalent to changing input data in the \(d\)th place (backward error analysis).

Wilkinson

“Error Analysis of Direct Methods of Matrix Inversion” (1961)
Modern error analysis of Gaussian elimination

For his research in numerical analysis to facilitiate the use of the high-speed digital computer, having received special recognition for his work in computations in linear algebra and “backward” error analysis. — 1970 Turing Award citation

Kahan

IEEE-754/854 (1985, revised 2008, 2018)

For his fundamental contributions to numerical analysis. One of the foremost experts on floating-point computations. Kahan has dedicated himself to “making the world safe for numerical computations.” — 1989 Turing Award citation

IEEE floating point reminder

Normalized numbers: \[(-1)^s \times (1.b_1 b_2 \ldots b_p)_2 \times 2^e\] 32-bit single, 64-bit double numbers consisting of

Sign \(s\)
Precision \(p\) (\(p = 23\) or \(52\))
Exponent \(e\) (\(-126 \leq e \leq 126\) or \(-1022 \leq e \leq 1023\))

Newer 16-bit formats: fp16 (\(p = 10\)); bfloat16 (\(p = 7\))

Beyond normalized

What if we can’t represent an exact result?
What about \(2^{e_{\max}+1} \leq x < \infty\) or \(0 \leq x < 2^{e_{\min}}\)?
What if we compute \(1/0\)?
What if we compute \(\sqrt{-1}\)?

Rounding

Basic ops (\(+, -, \times, /, \sqrt{}\)), require correct rounding

As if computed to infinite precision, then rounded.
- Don’t actually need infinite precision for this!
Different rounding rules possible:
- Round to nearest even (default)
- Round up, down, to 0 – error bds + intervals
754 recommends (does not require) correct rounding for a few transcendentals as well (sine, cosine, etc).

Inexact

If rounded result \(\neq\) exact result, have inexact exception
- Which most people seem not to know about...
- ... and which most of us who do usually ignore

Denormalization and underflow

Denormalized numbers: \[(-1)^s \times (0.b_1 b_2 \ldots b_p)_2 \times 2^{e_{\min}}\]

Evenly fill in space between \(\pm 2^{e_{\min}}\)
Gradually lose bits of precision as we approach zero
Denormalization results in an underflow exception
- Except when an exact zero is generated

Infinity and NaN

Other things can happen:

\(2^{e_{\max}} + 2^{e_{\max}}\) generates \(\infty\) (overflow exception)
\(1/0\) generates \(\infty\) (divide by zero exception)
- ... should really be called “exact infinity”
\(\sqrt{-1}\) generates Not-a-Number (invalid exception)

But every basic op produces something well defined.

Basic rounding model

Model of roundoff in a basic op: \[\mathrm{fl}(a \odot b) = (a \odot b)(1 + \delta), \quad |\delta| \leq \epsilon.\]

This model is not complete
- Misses overflow, underflow, divide by zero
- Also, some things are done exactly!
- Example: \(2x\) exact, as is \(x+y\) if \(x/2 \leq y \leq 2x\)
But useful as a basis for backward error analysis

Example: Horner’s rule

Evaluate \(p(x) = \sum_{k=0}^n c_k x^k\):

p = c(n)
for k = n-1 downto 0
  p = x*p + c(k)

Example: Horner’s rule

Can show backward error result: \[\mathrm{fl}(p) = \sum_{k=0}^n \hat{c}_k x^k\] where \(|\hat{c}_k-c_k| \leq (n+1) \epsilon |c_k|\).

Backward error + sensitivity gives forward error. Can even compute running error estimates!

Hooray for the modern era!

Everyone almost implements IEEE 754
- Old Cray arithmetic is essentially extinct
We teach backward error analysis in basic classes
Good libraries for LA, elementary functions

Back to the future?

But GPUs have funky (low-precision) formats!
Hard to write portable exception handlers
Exception flags may be inaccessible
Some features might be slow
Compiler might not do what you expected

Back to the future?

We teach backward error analysis in basic classes
- ... which are often no longer required!
- And anyhow, bwd error isn’t everything.
Good libraries for LA, elementary functions
- But people will still roll their own.

Arithmetic speed

Single faster than double precision

Actual arithmetic cost may be comparable (on CPU)
But GPUs generally prefer single (or lower)
And AVX instructions do more per cycle with single
And memory bandwidth is lower

NB: FP16 originally intended for storage only!

Mixed-precision arithmetic

Idea: use double precision only where needed

Example: iterative refinement and relatives
Or use double-precision arithmetic between single-precision representations (may be a good idea regardless)

Example: Mixed-precision iterative refinement

Factor \(A = LU\): \(O(n^3)\) single-precision work
Solve \(x = U^{-1} (L^{-1} b)\): \(O(n^2)\) single-precision work
\(r = b-Ax\): \(O(n^2)\) double-precision work
While \(\|r\|\) too large
- \(d = U^{-1} (L^{-1} r)\): \(O(n^2)\) single-precision work
- \(x = x+d\): \(O(n)\) single-precision work
- \(r = b-Ax\): \(O(n^2)\) double-precision work

Example: Helpful extra precision

/*
 * Assuming all coordinates are in [1,2), check on which
 * side of the line through A and B is the point C.
 */
int check_side(float ax, float ay, float bx, float by, 
               float cx, float cy)
{
    double abx = bx-ax, aby = by-ay;
    double acx = cx-ax, acy = cy-ay;
    double det = acx*aby-abx*aby;
    if (det == 0) return  0;
    if (det <  0) return -1;
    if (det >  0) return  1;
}

This is not robust if the inputs are double precision!

Single or double?

What to use for:

Large data sets? (single for performance, if possible)
Local calculations? (double by default, except GPU?)
Physically measured inputs? (probably single)
Nodal coordinates? (probably single)
Stiffness matrices? (maybe single, maybe double)
Residual computations? (probably double)
Checking geometric predicates? (double or more)

Simulating extra precision

What if we want higher precision than is fast?

Double precision on a GPU?
Quad precision on a CPU?

Simulating extra precision

Can simulate extra precision. Example:

  // s1, s2 = two_sum(a, b) -- Dekker's version
  if abs(a) < abs(b) { swap(&a, &b); }
  double s1 = a+b;         /* May suffer roundoff */
  double s2 = (a-s1) + b;  /* No roundoff! */

Simulating extra precision

Idea applies more broadly (Bailey, Bohlender, Dekker, Demmel, Hida, Kahan, Li, Linnainmaa, Priest, Shewchuk, ...)

Used in fast extra-precision packages
And in robust geometric predicate code
And in XBLAS

Exceptional arithmetic speed

Time to sum 1000 doubles on my laptop:

Initialized to 1: 1.3 microseconds
Initialized to inf/nan: 1.3 microseconds
Initialized to \(10^{-312}\): 67 microseconds

\(50 \times\) performance penalty for gradual underflow!

Exceptional arithmetic

Why worry? One reason:

if (x != y)
    z = x/(x-y);

Also limits range of simulated extra precision.

Exceptional algorithms, take 2

A general idea (works outside numerics, too):

Try something fast but risky
If something breaks, retry more carefully

If risky usually works and doesn’t cost too much extra, this improves performance.

(See Demmel and Li; Hull, Farfrieve, and Tang.)

Three problems

What goes wrong with floating point in parallel (or just high performance) environments?

Problem 0: Mis-attributed Blame

To blame is human. To fix is to engineer. — Unknown

Three variants:

“Probably no worries about floating point error.”
“This is probably due to floating point error.”
“Floating point error makes this untrustworthy.”

Problem 1: Repeatability

Floating point addition is not associative: \[\mathrm{fl}(a + \mathrm{fl}(b + c)) \neq \mathrm{fl}(\mathrm{fl}(a + b) + c)\]

So answers depends on the inputs, but also

How blocking is done in multiply or other kernels
Maybe compiler optimizations
Order in which reductions are computed
Order in which critical sections are reached

Problem 1: Repeatability

Worst case: with nontrivial probability we get an answer too bad to be useful, not bad enough for the program to barf — and garbage comes out.

Problem 1: Repeatability

What can we do?

Apply error analysis agnostic to ordering
Write slower debug version with specific ordering
Soon(?): Call the reproducible BLAS

Problem 2: Heterogeneity

Local arithmetic faster than communication
So be redundant about some computation
What if redundant computations use different HW?
- Different nodes in the cloud?
- GPU and CPU?
Problems
- Different exception handling on different nodes
- Different branches due to different rounding

Problem 2: Heterogeneity

What can we do?

Avoid FP-dependent branches
Communicate FP results affecting branches
Use reproducible kernels

New World Order

Claim: DNNs robust to low precision!

Overflow an issue (hence bfloat16)
Same pressure has revived block FP?
More experiments than analysis

Recap

So why care about the vagaries of floating point?

Might actually care about error analysis
Or using single precision for speed
Or maybe just reproducibility
Or avoiding crashes from inconsistent decisions!

References

“What Every Computer Scientist Should Know About Floating Point Arithmetic” (David Goldberg + addendum by Doug Priest)
“Revisiting ‘What Every Computer Scientist Should Know About Floating Point Arithmetic’” (Lafage)
Numerical Computing with IEEE Floating Point Arithmetic (Overton)
Handbook of Floating Point Arithmetic (Muller et al)
Accuracy and Stability of Numerical Algorithms (Higham)