Big Review

Scientific ComputingApproximations

Computer Arithmetic

Scientific Computing: An Introductory SurveyChapter 1 – Scientific Computing

Prof. Michael T. Heath

Department of Computer ScienceUniversity of Illinois at Urbana-Champaign

Copyright c© 2002. Reproduction permittedfor noncommercial, educational use only.

Michael T. Heath Scientific Computing 1 / 46


Computer Arithmetic

Outline

1 Scientific Computing

2 Approximations

3 Computer Arithmetic



Computer Arithmetic

IntroductionComputational ProblemsGeneral Strategy

Scientific Computing

What is scientific computing?

Design and analysis of algorithms for numerically solvingmathematical problems in science and engineeringTraditionally called numerical analysis

Distinguishing features of scientific computing

Deals with continuous quantitiesConsiders effects of approximations

Why scientific computing?

Simulation of natural phenomenaVirtual prototyping of engineering designs



Computer Arithmetic


Well-Posed Problems

Problem is well-posed if solutionexistsis uniquedepends continuously on problem data

Otherwise, problem is ill-posed

Even if problem is well posed, solution may still besensitive to input data

Computational algorithm should not make sensitivity worse



Computer Arithmetic


General Strategy

Replace difficult problem by easier one having same orclosely related solution

infinite → finitedifferential → algebraicnonlinear → linearcomplicated → simple

Solution obtained may only approximate that of originalproblem



Computer Arithmetic

Sources of ApproximationError AnalysisSensitivity and Conditioning

Sources of Approximation

Before computationmodelingempirical measurementsprevious computations

During computationtruncation or discretizationrounding

Accuracy of final result reflects all these

Uncertainty in input may be amplified by problem

Perturbations during computation may be amplified byalgorithm



Computer Arithmetic


Example: Approximations

Computing surface area of Earth using formula A = 4πr2

involves several approximations

Earth is modeled as sphere, idealizing its true shape

Value for radius is based on empirical measurements andprevious computations

Value for π requires truncating infinite process

Values for input data and results of arithmetic operationsare rounded in computer



Computer Arithmetic


Absolute Error and Relative Error

Absolute error : approximate value − true value

Relative error :absolute error

true value

Equivalently, approx value = (true value) × (1 + rel error)

True value usually unknown, so we estimate or bounderror rather than compute it exactly

Relative error often taken relative to approximate value,rather than (unknown) true value



Computer Arithmetic


Data Error and Computational Error

Typical problem: compute value of function f : R → R forgiven argument

x = true value of inputf(x) = desired resultx = approximate (inexact) inputf = approximate function actually computed

Total error: f(x)− f(x) =

f(x)− f(x) + f(x)− f(x)computational error + propagated data error

Algorithm has no effect on propagated data error



Computer Arithmetic


Truncation Error and Rounding Error

Truncation error : difference between true result (for actualinput) and result produced by given algorithm using exactarithmetic

Due to approximations such as truncating infinite series orterminating iterative sequence before convergence

Rounding error : difference between result produced bygiven algorithm using exact arithmetic and result producedby same algorithm using limited precision arithmetic

Due to inexact representation of real numbers andarithmetic operations upon them

Computational error is sum of truncation error androunding error, but one of these usually dominates

< interactive example >


http://www.cse.uiuc.edu/iem/floating_point/rounding_truncation/


Computer Arithmetic


Example: Finite Difference Approximation

Error in finite difference approximation

f ′(x) ≈ f(x + h)− f(x)h

exhibits tradeoff between rounding error and truncationerror

Truncation error bounded by Mh/2, where M bounds|f ′′(t)| for t near x

Rounding error bounded by 2ε/h, where error in functionvalues bounded by ε

Total error minimized when h ≈ 2√

ε/M

Error increases for smaller h because of rounding errorand increases for larger h because of truncation error



Computer Arithmetic


Example: Finite Difference Approximation

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Computer Arithmetic


Forward and Backward Error

Suppose we want to compute y = f(x), where f : R → R,but obtain approximate value y

Forward error : ∆y = y − y

Backward error : ∆x = x− x, where f(x) = y



Computer Arithmetic


Example: Forward and Backward Error

As approximation to y =√

2, y = 1.4 has absolute forwarderror

|∆y| = |y − y| = |1.4− 1.41421 . . . | ≈ 0.0142

or relative forward error of about 1 percent

Since√

1.96 = 1.4, absolute backward error is

|∆x| = |x− x| = |1.96− 2| = 0.04

or relative backward error of 2 percent



Computer Arithmetic


Backward Error Analysis

Idea: approximate solution is exact solution to modifiedproblem

How much must original problem change to give resultactually obtained?

How much data error in input would explain all error incomputed result?

Approximate solution is good if it is exact solution to nearbyproblem

Backward error is often easier to estimate than forwarderror



Computer Arithmetic


Example: Backward Error Analysis

Approximating cosine function f(x) = cos(x) by truncatingTaylor series after two terms gives

y = f(x) = 1− x2/2

Forward error is given by

∆y = y − y = f(x)− f(x) = 1− x2/2− cos(x)

To determine backward error, need value x such thatf(x) = f(x)

For cosine function, x = arccos(f(x)) = arccos(y)



Computer Arithmetic


Example, continued

For x = 1,

y = f(1) = cos(1) ≈ 0.5403y = f(1) = 1− 12/2 = 0.5x = arccos(y) = arccos(0.5) ≈ 1.0472

Forward error: ∆y = y − y ≈ 0.5− 0.5403 = −0.0403

Backward error: ∆x = x− x ≈ 1.0472− 1 = 0.0472



Computer Arithmetic


Sensitivity and Conditioning

Problem is insensitive, or well-conditioned, if relativechange in input causes similar relative change in solution

Problem is sensitive, or ill-conditioned, if relative change insolution can be much larger than that in input data

Condition number :

cond =|relative change in solution||relative change in input data|

=|[f(x)− f(x)]/f(x)|

|(x− x)/x|=|∆y/y||∆x/x|

Problem is sensitive, or ill-conditioned, if cond � 1



Computer Arithmetic


Condition Number

Condition number is amplification factor relating relativeforward error to relative backward error∣∣∣∣ relative

forward error

∣∣∣∣ = cond ×∣∣∣∣ relativebackward error

∣∣∣∣Condition number usually is not known exactly and mayvary with input, so rough estimate or upper bound is usedfor cond, yielding∣∣∣∣ relative

forward error

∣∣∣∣ / cond ×∣∣∣∣ relativebackward error

∣∣∣∣Michael T. Heath Scientific Computing 19 / 46


Computer Arithmetic


Example: Evaluating Function

Evaluating function f for approximate input x = x + ∆xinstead of true input x gives

Absolute forward error: f(x + ∆x)− f(x) ≈ f ′(x)∆x

Relative forward error:f(x + ∆x)− f(x)

f(x)≈ f ′(x)∆x

f(x)

Condition number: cond ≈∣∣∣∣f ′(x)∆x/f(x)

∆x/x

∣∣∣∣ = ∣∣∣∣xf ′(x)f(x)

∣∣∣∣Relative error in function value can be much larger orsmaller than that in input, depending on particular f and x



Computer Arithmetic


Example: Sensitivity

Tangent function is sensitive for arguments near π/2

tan(1.57079) ≈ 1.58058× 105

tan(1.57078) ≈ 6.12490× 104

Relative change in output is quarter million times greaterthan relative change in input

For x = 1.57079, cond ≈ 2.48275× 105



Computer Arithmetic


Stability

Algorithm is stable if result produced is relativelyinsensitive to perturbations during computation

Stability of algorithms is analogous to conditioning ofproblems

From point of view of backward error analysis, algorithm isstable if result produced is exact solution to nearbyproblem

For stable algorithm, effect of computational error is noworse than effect of small data error in input



Computer Arithmetic


Accuracy

Accuracy : closeness of computed solution to true solutionof problem

Stability alone does not guarantee accurate results

Accuracy depends on conditioning of problem as well asstability of algorithm

Inaccuracy can result from applying stable algorithm toill-conditioned problem or unstable algorithm towell-conditioned problem

Applying stable algorithm to well-conditioned problemyields accurate solution



Computer Arithmetic

Floating-Point NumbersFloating-Point Arithmetic

Floating-Point Numbers

Floating-point number system is characterized by fourintegers

β base or radixp precision[L,U ] exponent range

Number x is represented as

x = ±(

d0 +d1

β+

d2

β2+ · · ·+ dp−1

βp−1

)βE

where 0 ≤ di ≤ β − 1, i = 0, . . . , p− 1, and L ≤ E ≤ U



Computer Arithmetic


Floating-Point Numbers, continued

Portions of floating-poing number designated as follows

exponent : E

mantissa : d0d1 · · · dp−1

fraction : d1d2 · · · dp−1

Sign, exponent, and mantissa are stored in separatefixed-width fields of each floating-point word



Computer Arithmetic


Typical Floating-Point Systems

Parameters for typical floating-point systemssystem β p L U

IEEE SP 2 24 −126 127IEEE DP 2 53 −1022 1023Cray 2 48 −16383 16384HP calculator 10 12 −499 499IBM mainframe 16 6 −64 63

Most modern computers use binary (β = 2) arithmetic

IEEE floating-point systems are now almost universal indigital computers



Computer Arithmetic


Normalization

Floating-point system is normalized if leading digit d0 isalways nonzero unless number represented is zero

In normalized systems, mantissa m of nonzerofloating-point number always satisfies 1 ≤ m < β

Reasons for normalizationrepresentation of each number uniqueno digits wasted on leading zerosleading bit need not be stored (in binary system)



Computer Arithmetic


Properties of Floating-Point Systems

Floating-point number system is finite and discrete

Total number of normalized floating-point numbers is

2(β − 1)βp−1(U − L + 1) + 1

Smallest positive normalized number: UFL = βL

Largest floating-point number: OFL = βU+1(1− β−p)

Floating-point numbers equally spaced only betweensuccessive powers of β

Not all real numbers exactly representable; those that areare called machine numbers



Computer Arithmetic


Example: Floating-Point System

Tick marks indicate all 25 numbers in floating-point systemhaving β = 2, p = 3, L = −1, and U = 1

OFL = (1.11)2 × 21 = (3.5)10UFL = (1.00)2 × 2−1 = (0.5)10

At sufficiently high magnification, all normalizedfloating-point systems look grainy and unequally spaced



http://www.cse.uiuc.edu/iem/floating_point/fp_system/


Computer Arithmetic


Rounding Rules

If real number x is not exactly representable, then it isapproximated by “nearby” floating-point number fl(x)

This process is called rounding, and error introduced iscalled rounding error

Two commonly used rounding ruleschop : truncate base-β expansion of x after (p− 1)st digit;also called round toward zeroround to nearest : fl(x) is nearest floating-point number tox, using floating-point number whose last stored digit iseven in case of tie; also called round to even

Round to nearest is most accurate, and is default roundingrule in IEEE systems



http://www.cse.uiuc.edu/iem/floating_point/rounding_rules/


Computer Arithmetic


Machine Precision

Accuracy of floating-point system characterized by unitroundoff (or machine precision or machine epsilon)denoted by εmach

With rounding by chopping, εmach = β1−p

With rounding to nearest, εmach = 12β1−p

Alternative definition is smallest number ε such thatfl(1 + ε) > 1

Maximum relative error in representing real number xwithin range of floating-point system is given by∣∣∣∣fl(x)− x

x

∣∣∣∣ ≤ εmach



Computer Arithmetic


Machine Precision, continued

For toy system illustrated earlier

εmach = (0.01)2 = (0.25)10 with rounding by choppingεmach = (0.001)2 = (0.125)10 with rounding to nearest

For IEEE floating-point systems

εmach = 2−24 ≈ 10−7 in single precisionεmach = 2−53 ≈ 10−16 in double precision

So IEEE single and double precision systems have about 7and 16 decimal digits of precision, respectively



Computer Arithmetic


Machine Precision, continued

Though both are “small,” unit roundoff εmach should not beconfused with underflow level UFL

Unit roundoff εmach is determined by number of digits inmantissa of floating-point system, whereas underflow levelUFL is determined by number of digits in exponent field

In all practical floating-point systems,

0 < UFL < εmach < OFL



Computer Arithmetic


Subnormals and Gradual Underflow

Normalization causes gap around zero in floating-pointsystem

If leading digits are allowed to be zero, but only whenexponent is at its minimum value, then gap is “filled in” byadditional subnormal or denormalized floating-pointnumbers

Subnormals extend range of magnitudes representable,but have less precision than normalized numbers, and unitroundoff is no smaller

Augmented system exhibits gradual underflow



Computer Arithmetic


Exceptional Values

IEEE floating-point standard provides special values toindicate two exceptional situations

Inf, which stands for “infinity,” results from dividing a finitenumber by zero, such as 1/0NaN, which stands for “not a number,” results fromundefined or indeterminate operations such as 0/0, 0 ∗ Inf,or Inf/Inf

Inf and NaN are implemented in IEEE arithmetic throughspecial reserved values of exponent field



Computer Arithmetic


Floating-Point Arithmetic

Addition or subtraction : Shifting of mantissa to makeexponents match may cause loss of some digits of smallernumber, possibly all of them

Multiplication : Product of two p-digit mantissas contains upto 2p digits, so result may not be representable

Division : Quotient of two p-digit mantissas may containmore than p digits, such as nonterminating binaryexpansion of 1/10

Result of floating-point arithmetic operation may differ fromresult of corresponding real arithmetic operation on sameoperands



Computer Arithmetic


Example: Floating-Point Arithmetic

Assume β = 10, p = 6

Let x = 1.92403× 102, y = 6.35782× 10−1

Floating-point addition gives x + y = 1.93039× 102,assuming rounding to nearest

Last two digits of y do not affect result, and with evensmaller exponent, y could have had no effect on result

Floating-point multiplication gives x ∗ y = 1.22326× 102,which discards half of digits of true product



Computer Arithmetic


Floating-Point Arithmetic, continued

Real result may also fail to be representable because itsexponent is beyond available range

Overflow is usually more serious than underflow becausethere is no good approximation to arbitrarily largemagnitudes in floating-point system, whereas zero is oftenreasonable approximation for arbitrarily small magnitudes

On many computer systems overflow is fatal, but anunderflow may be silently set to zero



Computer Arithmetic


Example: Summing Series

Infinite series∞∑

n=1

1n

has finite sum in floating-point arithmetic even though realseries is divergentPossible explanations

Partial sum eventually overflows1/n eventually underflowsPartial sum ceases to change once 1/n becomes negligiblerelative to partial sum

1n

< εmach

n−1∑k=1

1k



http://www.cse.uiuc.edu/iem/floating_point/infinite_series/


Computer Arithmetic


Floating-Point Arithmetic, continued

Ideally, x flop y = fl(x op y), i.e., floating-point arithmeticoperations produce correctly rounded results

Computers satisfying IEEE floating-point standard achievethis ideal as long as x op y is within range of floating-pointsystem

But some familiar laws of real arithmetic are notnecessarily valid in floating-point system

Floating-point addition and multiplication are commutativebut not associative

Example: if ε is positive floating-point number slightlysmaller than εmach, then (1 + ε) + ε = 1, but 1 + (ε + ε) > 1



Computer Arithmetic


Cancellation

Subtraction between two p-digit numbers having same signand similar magnitudes yields result with fewer than pdigits, so it is usually exactly representable

Reason is that leading digits of two numbers cancel (i.e.,their difference is zero)

For example,

1.92403× 102 − 1.92275× 102 = 1.28000× 10−1

which is correct, and exactly representable, but has onlythree significant digits



Computer Arithmetic


Cancellation, continued

Despite exactness of result, cancellation often impliesserious loss of information

Operands are often uncertain due to rounding or otherprevious errors, so relative uncertainty in difference may belarge

Example: if ε is positive floating-point number slightlysmaller than εmach, then (1 + ε)− (1− ε) = 1− 1 = 0 infloating-point arithmetic, which is correct for actualoperands of final subtraction, but true result of overallcomputation, 2ε, has been completely lost

Subtraction itself is not at fault: it merely signals loss ofinformation that had already occurred



Computer Arithmetic


Cancellation, continued

Digits lost to cancellation are most significant, leadingdigits, whereas digits lost in rounding are least significant,trailing digits

Because of this effect, it is generally bad idea to computeany small quantity as difference of large quantities, sincerounding error is likely to dominate result

For example, summing alternating series, such as

ex = 1 + x +x2

2!+

x3

3!+ · · ·

for x < 0, may give disastrous results due to catastrophiccancellation



Computer Arithmetic


Example: Cancellation

Total energy of helium atom is sum of kinetic and potentialenergies, which are computed separately and have oppositesigns, so suffer cancellation

Year Kinetic Potential Total1971 13.0 −14.0 −1.01977 12.76 −14.02 −1.261980 12.22 −14.35 −2.131985 12.28 −14.65 −2.371988 12.40 −14.84 −2.44

Although computed values for kinetic and potential energieschanged by only 6% or less, resulting estimate for total energychanged by 144%



Computer Arithmetic


Example: Quadratic Formula

Two solutions of quadratic equation ax2 + bx + c = 0 aregiven by

x =−b±

√b2 − 4ac

2aNaive use of formula can suffer overflow, or underflow, orsevere cancellationRescaling coefficients avoids overflow or harmful underflowCancellation between −b and square root can be avoidedby computing one root using alternative formula

x =2c

−b∓√

b2 − 4ac

Cancellation inside square root cannot be easily avoidedwithout using higher precision



http://www.cse.uiuc.edu/iem/floating_point/quadratic_formula/


Computer Arithmetic


Example: Standard Deviation

Mean and standard deviation of sequence xi, i = 1, . . . , n,are given by

x =1n

n∑i=1

xi and σ =

[1

n− 1

n∑i=1

(xi − x)2] 1

2

Mathematically equivalent formula

σ =

[1

n− 1

(n∑

i=1

x2i − nx2

)] 12

avoids making two passes through dataSingle cancellation at end of one-pass formula is moredamaging numerically than all cancellations in two-passformula combined


Nonlinear EquationsNumerical Methods in One Dimension

Methods for Systems of Nonlinear Equations

Scientific Computing: An Introductory SurveyChapter 5 – Nonlinear Equations







Outline

1 Nonlinear Equations

2 Numerical Methods in One Dimension

3 Methods for Systems of Nonlinear Equations




Nonlinear EquationsSolutions and SensitivityConvergence

Nonlinear Equations

Given function f , we seek value x for which

f(x) = 0

Solution x is root of equation, or zero of function f

So problem is known as root finding or zero finding





Nonlinear Equations

Two important cases

Single nonlinear equation in one unknown, where

f : R → R

Solution is scalar x for which f(x) = 0

System of n coupled nonlinear equations in n unknowns,where

f : Rn → Rn

Solution is vector x for which all components of f are zerosimultaneously, f(x) = 0





Examples: Nonlinear Equations

Example of nonlinear equation in one dimension

x2 − 4 sin(x) = 0

for which x = 1.9 is one approximate solution

Example of system of nonlinear equations in twodimensions

x21 − x2 + 0.25 = 0

−x1 + x22 + 0.25 = 0

for which x =[0.5 0.5

]T is solution vector





Existence and Uniqueness

Existence and uniqueness of solutions are morecomplicated for nonlinear equations than for linearequations

For function f : R → R, bracket is interval [a, b] for whichsign of f differs at endpoints

If f is continuous and sign(f(a)) 6= sign(f(b)), thenIntermediate Value Theorem implies there is x∗ ∈ [a, b]such that f(x∗) = 0

There is no simple analog for n dimensions





Examples: One Dimension

Nonlinear equations can have any number of solutions

exp(x) + 1 = 0 has no solution

exp(−x)− x = 0 has one solution

x2 − 4 sin(x) = 0 has two solutions

x3 + 6x2 + 11x− 6 = 0 has three solutions

sin(x) = 0 has infinitely many solutions





Example: Systems in Two Dimensionsx2

1 − x2 + γ = 0−x1 + x2

2 + γ = 0





Multiplicity

If f(x∗) = f ′(x∗) = f ′′(x∗) = · · · = f (m−1)(x∗) = 0 butf (m)(x∗) 6= 0 (i.e., mth derivative is lowest derivative of fthat does not vanish at x∗), then root x∗ has multiplicity m

If m = 1 (f(x∗) = 0 and f ′(x∗) 6= 0), then x∗ is simple root






Conditioning of root finding problem is opposite to that forevaluating function

Absolute condition number of root finding problem for rootx∗ of f : R → R is 1/|f ′(x∗)|

Root is ill-conditioned if tangent line is nearly horizontal

In particular, multiple root (m > 1) is ill-conditioned

Absolute condition number of root finding problem for rootx∗ of f : Rn → Rn is ‖J−1

f (x∗)‖, where Jf is Jacobianmatrix of f ,

{Jf (x)}ij = ∂fi(x)/∂xj

Root is ill-conditioned if Jacobian matrix is nearly singular











What do we mean by approximate solution x to nonlinearsystem,

‖f(x)‖ ≈ 0 or ‖x− x∗‖ ≈ 0 ?

First corresponds to “small residual,” second measurescloseness to (usually unknown) true solution x∗

Solution criteria are not necessarily “small” simultaneously

Small residual implies accurate solution only if problem iswell-conditioned





Convergence Rate

For general iterative methods, define error at iteration k by

ek = xk − x∗

where xk is approximate solution and x∗ is true solution

For methods that maintain interval known to containsolution, rather than specific approximate value forsolution, take error to be length of interval containingsolution

Sequence converges with rate r if

limk→∞

‖ek+1‖‖ek‖r

= C

for some finite nonzero constant C





Convergence Rate, continued

Some particular cases of interest

r = 1: linear (C < 1)

r > 1: superlinear

r = 2: quadratic

Convergence Digits gainedrate per iterationlinear constantsuperlinear increasingquadratic double




Bisection MethodFixed-Point Iteration and Newton’s MethodAdditional Methods

Interval Bisection Method

Bisection method begins with initial bracket and repeatedlyhalves its length until solution has been isolated as accuratelyas desired

while ((b− a) > tol) dom = a + (b− a)/2if sign(f(a)) = sign(f(m)) then

a = melse

b = mend

end



http://www.cse.uiuc.edu/iem/nonlinear_eqns/Bisection/




Example: Bisection Method

f(x) = x2 − 4 sin(x) = 0

a f(a) b f(b)1.000000 −2.365884 3.000000 8.4355201.000000 −2.365884 2.000000 0.3628101.500000 −1.739980 2.000000 0.3628101.750000 −0.873444 2.000000 0.3628101.875000 −0.300718 2.000000 0.3628101.875000 −0.300718 1.937500 0.0198491.906250 −0.143255 1.937500 0.0198491.921875 −0.062406 1.937500 0.0198491.929688 −0.021454 1.937500 0.0198491.933594 −0.000846 1.937500 0.0198491.933594 −0.000846 1.935547 0.0094911.933594 −0.000846 1.934570 0.0043201.933594 −0.000846 1.934082 0.0017361.933594 −0.000846 1.933838 0.000445Michael T. Heath Scientific Computing 16 / 55




Bisection Method, continuedBisection method makes no use of magnitudes of functionvalues, only their signs

Bisection is certain to converge, but does so slowly

At each iteration, length of interval containing solutionreduced by half, convergence rate is linear, with r = 1 andC = 0.5

One bit of accuracy is gained in approximate solution foreach iteration of bisection

Given starting interval [a, b], length of interval after kiterations is (b− a)/2k, so achieving error tolerance of tolrequires ⌈

log2

(b− a

tol

)⌉iterations, regardless of function f involved





Fixed-Point Problems

Fixed point of given function g : R → R is value x such that

x = g(x)

Many iterative methods for solving nonlinear equations usefixed-point iteration scheme of form

xk+1 = g(xk)

where fixed points for g are solutions for f(x) = 0

Also called functional iteration, since function g is appliedrepeatedly to initial starting value x0

For given equation f(x) = 0, there may be many equivalentfixed-point problems x = g(x) with different choices for g





Example: Fixed-Point Problems

If f(x) = x2 − x− 2, then fixed points of each of functions

g(x) = x2 − 2

g(x) =√

x + 2

g(x) = 1 + 2/x

g(x) =x2 + 22x− 1

are solutions to equation f(x) = 0





Example: Fixed-Point Problems





Example: Fixed-Point Iteration





Example: Fixed-Point Iteration





Convergence of Fixed-Point Iteration

If x∗ = g(x∗) and |g′(x∗)| < 1, then there is intervalcontaining x∗ such that iteration

xk+1 = g(xk)

converges to x∗ if started within that interval

If |g′(x∗)| > 1, then iterative scheme diverges

Asymptotic convergence rate of fixed-point iteration isusually linear, with constant C = |g′(x∗)|

But if g′(x∗) = 0, then convergence rate is at leastquadratic



http://www.cse.uiuc.edu/iem/nonlinear_eqns/FixedPoint/




Newton’s Method

Truncated Taylor series

f(x + h) ≈ f(x) + f ′(x)h

is linear function of h approximating f near x

Replace nonlinear function f by this linear function, whosezero is h = −f(x)/f ′(x)

Zeros of original function and linear approximation are notidentical, so repeat process, giving Newton’s method

xk+1 = xk −f(xk)f ′(xk)





Newton’s Method, continued

Newton’s method approximates nonlinear function f near xk bytangent line at f(xk)





Example: Newton’s MethodUse Newton’s method to find root of

f(x) = x2 − 4 sin(x) = 0

Derivative isf ′(x) = 2x− 4 cos(x)

so iteration scheme is

xk+1 = xk −x2

k − 4 sin(xk)2xk − 4 cos(xk)

Taking x0 = 3 as starting value, we obtainx f(x) f ′(x) h

3.000000 8.435520 9.959970 −0.8469422.153058 1.294772 6.505771 −0.1990191.954039 0.108438 5.403795 −0.0200671.933972 0.001152 5.288919 −0.0002181.933754 0.000000 5.287670 0.000000





Convergence of Newton’s Method

Newton’s method transforms nonlinear equation f(x) = 0into fixed-point problem x = g(x), where

g(x) = x− f(x)/f ′(x)

and henceg′(x) = f(x)f ′′(x)/(f ′(x))2

If x∗ is simple root (i.e., f(x∗) = 0 and f ′(x∗) 6= 0), theng′(x∗) = 0

Convergence rate of Newton’s method for simple root istherefore quadratic (r = 2)

But iterations must start close enough to root to converge



http://www.cse.uiuc.edu/iem/nonlinear_eqns/Newton/





For multiple root, convergence rate of Newton’s method is onlylinear, with constant C = 1− (1/m), where m is multiplicity

k f(x) = x2 − 1 f(x) = x2 − 2x + 10 2.0 2.01 1.25 1.52 1.025 1.253 1.0003 1.1254 1.00000005 1.06255 1.0 1.03125





Secant Method

For each iteration, Newton’s method requires evaluation ofboth function and its derivative, which may be inconvenientor expensive

In secant method, derivative is approximated by finitedifference using two successive iterates, so iterationbecomes

xk+1 = xk − f(xk)xk − xk−1

f(xk)− f(xk−1)

Convergence rate of secant method is normallysuperlinear, with r ≈ 1.618





Secant Method, continued

Secant method approximates nonlinear function f by secantline through previous two iterates



http://www.cse.uiuc.edu/iem/nonlinear_eqns/Secant/




Example: Secant Method

Use secant method to find root of

f(x) = x2 − 4 sin(x) = 0

Taking x0 = 1 and x1 = 3 as starting guesses, we obtain

x f(x) h

1.000000 −2.3658843.000000 8.435520 −1.5619301.438070 −1.896774 0.2867351.724805 −0.977706 0.3050292.029833 0.534305 −0.1077891.922044 −0.061523 0.0111301.933174 −0.003064 0.0005831.933757 0.000019 −0.0000041.933754 0.000000 0.000000





Higher-Degree Interpolation

Secant method uses linear interpolation to approximatefunction whose zero is sought

Higher convergence rate can be obtained by usinghigher-degree polynomial interpolation

For example, quadratic interpolation (Muller’s method) hassuperlinear convergence rate with r ≈ 1.839

Unfortunately, using higher degree polynomial also hasdisadvantages

interpolating polynomial may not have real rootsroots may not be easy to computechoice of root to use as next iterate may not be obvious





Inverse Interpolation

Good alternative is inverse interpolation, where xk areinterpolated as function of yk = f(xk) by polynomial p(y),so next approximate solution is p(0)

Most commonly used for root finding is inverse quadraticinterpolation





Inverse Quadratic Interpolation

Given approximate solution values a, b, c, with functionvalues fa, fb, fc, next approximate solution found by fittingquadratic polynomial to a, b, c as function of fa, fb, fc, thenevaluating polynomial at 0

Based on nontrivial derivation using Lagrangeinterpolation, we compute

u = fb/fc, v = fb/fa, w = fa/fc

p = v(w(u− w)(c− b)− (1− u)(b− a))

q = (w − 1)(u− 1)(v − 1)

then new approximate solution is b + p/q

Convergence rate is normally r ≈ 1.839



http://www.cse.uiuc.edu/iem/nonlinear_eqns/InverseInterpolation/




Example: Inverse Quadratic Interpolation

Use inverse quadratic interpolation to find root of

f(x) = x2 − 4 sin(x) = 0

Taking x = 1, 2, and 3 as starting values, we obtainx f(x) h

1.000000 −2.3658842.000000 0.3628103.000000 8.4355201.886318 −0.244343 −0.1136821.939558 0.030786 0.0532401.933742 −0.000060 −0.0058151.933754 0.000000 0.0000111.933754 0.000000 0.000000





Linear Fractional Interpolation

Interpolation using rational fraction of form

φ(x) =x− u

vx− w

is especially useful for finding zeros of functions havinghorizontal or vertical asymptotesφ has zero at x = u, vertical asymptote at x = w/v, andhorizontal asymptote at y = 1/v

Given approximate solution values a, b, c, with functionvalues fa, fb, fc, next approximate solution is c + h, where

h =(a− c)(b− c)(fa − fb)fc

(a− c)(fc − fb)fa − (b− c)(fc − fa)fb

Convergence rate is normally r ≈ 1.839, same as forquadratic interpolation (inverse or regular)





Example: Linear Fractional Interpolation

Use linear fractional interpolation to find root of

f(x) = x2 − 4 sin(x) = 0

Taking x = 1, 2, and 3 as starting values, we obtainx f(x) h

1.000000 −2.3658842.000000 0.3628103.000000 8.4355201.906953 −0.139647 −1.0930471.933351 −0.002131 0.0263981.933756 0.000013 −0.0004061.933754 0.000000 −0.000003



http://www.cse.uiuc.edu/iem/nonlinear_eqns/LinearFractional/




Safeguarded Methods

Rapidly convergent methods for solving nonlinearequations may not converge unless started close tosolution, but safe methods are slow

Hybrid methods combine features of both types ofmethods to achieve both speed and reliability

Use rapidly convergent method, but maintain bracketaround solution

If next approximate solution given by fast method fallsoutside bracketing interval, perform one iteration of safemethod, such as bisection





Safeguarded Methods, continued

Fast method can then be tried again on smaller intervalwith greater chance of success

Ultimately, convergence rate of fast method should prevail

Hybrid approach seldom does worse than safe method,and usually does much better

Popular combination is bisection and inverse quadraticinterpolation, for which no derivatives required





Zeros of Polynomials

For polynomial p(x) of degree n, one may want to find all nof its zeros, which may be complex even if coefficients arereal

Several approaches are available

Use root-finding method such as Newton’s or Muller’smethod to find one root, deflate it out, and repeatForm companion matrix of polynomial and use eigenvalueroutine to compute all its eigenvaluesUse method designed specifically for finding all roots ofpolynomial, such as Jenkins-Traub




Fixed-Point IterationNewton’s MethodSecant Updating Methods

Systems of Nonlinear Equations

Solving systems of nonlinear equations is much more difficultthan scalar case because

Wider variety of behavior is possible, so determiningexistence and number of solutions or good starting guessis much more complex

There is no simple way, in general, to guaranteeconvergence to desired solution or to bracket solution toproduce absolutely safe method

Computational overhead increases rapidly with dimensionof problem





Fixed-Point Iteration

Fixed-point problem for g : Rn → Rn is to find vector x suchthat

x = g(x)

Corresponding fixed-point iteration is

xk+1 = g(xk)

If ρ(G(x∗)) < 1, where ρ is spectral radius and G(x) isJacobian matrix of g evaluated at x, then fixed-pointiteration converges if started close enough to solution

Convergence rate is normally linear, with constant C givenby spectral radius ρ(G(x∗))

If G(x∗) = O, then convergence rate is at least quadratic





Newton’s Method

In n dimensions, Newton’s method has form

xk+1 = xk − J(xk)−1f(xk)

where J(x) is Jacobian matrix of f ,

{J(x)}ij =∂fi(x)∂xj

In practice, we do not explicitly invert J(xk), but insteadsolve linear system

J(xk)sk = −f(xk)

for Newton step sk, then take as next iterate

xk+1 = xk + sk





Example: Newton’s Method

Use Newton’s method to solve nonlinear system

f(x) =[x1 + 2x2 − 2x2

1 + 4x22 − 4

]= 0

Jacobian matrix is Jf (x) =[

1 22x1 8x2

]If we take x0 =

[1 2

]T , then

f(x0) =[

313

], Jf (x0) =

[1 22 16

]

Solving system[1 22 16

]s0 =

[−3−13

]gives s0 =

[−1.83−0.58

],

so x1 = x0 + s0 =[−0.83 1.42

]T





Example, continuedEvaluating at new point,

f(x1) =[

04.72

], Jf (x1) =

[1 2

−1.67 11.3

]

Solving system[

1 2−1.67 11.3

]s1 =

[0

−4.72

]gives

s1 =[0.64 −0.32

]T , so x2 = x1 + s1 =[−0.19 1.10

]T

Evaluating at new point,

f(x2) =[

00.83

], Jf (x2) =

[1 2

−0.38 8.76

]Iterations eventually convergence to solution x∗ =

[0 1

]T



http://www.cse.uiuc.edu/iem/nonlinear_eqns/Newton2D/




Convergence of Newton’s Method

Differentiating corresponding fixed-point operator

g(x) = x− J(x)−1f(x)

and evaluating at solution x∗ gives

G(x∗) = I − (J(x∗)−1J(x∗) +n∑

i=1

fi(x∗)Hi(x∗)) = O

where Hi(x) is component matrix of derivative of J(x)−1

Convergence rate of Newton’s method for nonlinearsystems is normally quadratic, provided Jacobian matrixJ(x∗) is nonsingular

But it must be started close enough to solution to converge





Cost of Newton’s Method

Cost per iteration of Newton’s method for dense problem in ndimensions is substantial

Computing Jacobian matrix costs n2 scalar functionevaluations

Solving linear system costs O(n3) operations





Secant Updating Methods

Secant updating methods reduce cost by

Using function values at successive iterates to buildapproximate Jacobian and avoiding explicit evaluation ofderivativesUpdating factorization of approximate Jacobian rather thanrefactoring it each iteration

Most secant updating methods have superlinear but notquadratic convergence rate

Secant updating methods often cost less overall thanNewton’s method because of lower cost per iteration





Broyden’s Method

Broyden’s method is typical secant updating method

Beginning with initial guess x0 for solution and initialapproximate Jacobian B0, following steps are repeateduntil convergence

x0 = initial guessB0 = initial Jacobian approximationfor k = 0, 1, 2, . . .

Solve Bk sk = −f(xk) for sk

xk+1 = xk + sk

yk = f(xk+1)− f(xk)Bk+1 = Bk + ((yk −Bksk)sT

k )/(sTk sk)

end





Broyden’s Method, continued

Motivation for formula for Bk+1 is to make least change toBk subject to satisfying secant equation

Bk+1(xk+1 − xk) = f(xk+1)− f(xk)

In practice, factorization of Bk is updated instead ofupdating Bk directly, so total cost per iteration is only O(n2)





Example: Broyden’s Method

Use Broyden’s method to solve nonlinear system

f(x) =[x1 + 2x2 − 2x2

1 + 4x22 − 4

]= 0

If x0 =[1 2

]T , then f(x0) =[3 13

]T , and we choose

B0 = Jf (x0) =[1 22 16

]Solving system [

1 22 16

]s0 =

[−3−13

]gives s0 =

[−1.83−0.58

], so x1 = x0 + s0 =

[−0.83

1.42

]Michael T. Heath Scientific Computing 51 / 55




Example, continued

Evaluating at new point x1 gives f(x1) =[

04.72

], so

y0 = f(x1)− f(x0) =[−3−8.28

]From updating formula, we obtain

B1 =[1 22 16

]+

[0 0

−2.34 −0.74

]=

[1 2

−0.34 15.3

]Solving system[

1 2−0.34 15.3

]s1 =

[0

−4.72

]gives s1 =

[0.59

−0.30

], so x2 = x1 + s1 =

[−0.241.120





Example, continued

Evaluating at new point x2 gives f(x2) =[

01.08

], so

y1 = f(x2)− f(x1) =[

0−3.64

]From updating formula, we obtain

B2 =[

1 2−0.34 15.3

]+

[0 0

1.46 −0.73

]=

[1 2

1.12 14.5

]

Iterations continue until convergence to solution x∗ =[01

]< interactive example >


http://www.cse.uiuc.edu/iem/nonlinear_eqns/Broyden/




Robust Newton-Like Methods

Newton’s method and its variants may fail to convergewhen started far from solution

Safeguards can enlarge region of convergence ofNewton-like methods

Simplest precaution is damped Newton method, in whichnew iterate is

xk+1 = xk + αksk

where sk is Newton (or Newton-like) step and αk is scalarparameter chosen to ensure progress toward solution

Parameter αk reduces Newton step when it is too large,but αk = 1 suffices near solution and still yields fastasymptotic convergence rate





Trust-Region Methods

Another approach is to maintain estimate of trust regionwhere Taylor series approximation, upon which Newton’smethod is based, is sufficiently accurate for resultingcomputed step to be reliable

Adjusting size of trust region to constrain step size whennecessary usually enables progress toward solution evenstarting far away, yet still permits rapid converge once nearsolution

Unlike damped Newton method, trust region method maymodify direction as well as length of Newton step

More details on this approach will be given in Chapter 6


Optimization ProblemsOne-Dimensional OptimizationMulti-Dimensional Optimization

DefinitionsExistence and UniquenessOptimality Conditions

Optimization

Given function f : Rn → R, and set S ⊆ Rn, find x∗ ∈ Ssuch that f(x∗) ≤ f(x) for all x ∈ S

x∗ is called minimizer or minimum of f

It suffices to consider only minimization, since maximum off is minimum of −f

Objective function f is usually differentiable, and may belinear or nonlinear

Constraint set S is defined by system of equations andinequalities, which may be linear or nonlinear

Points x ∈ S are called feasible points

If S = Rn, problem is unconstrained




Optimization Problems

General continuous optimization problem:

min f(x) subject to g(x) = 0 and h(x) ≤ 0

where f : Rn → R, g : Rn → Rm, h : Rn → Rp

Linear programming : f , g, and h are all linear

Nonlinear programming : at least one of f , g, and h isnonlinear




Examples: Optimization Problems

Minimize weight of structure subject to constraint on itsstrength, or maximize its strength subject to constraint onits weight

Minimize cost of diet subject to nutritional constraints

Minimize surface area of cylinder subject to constraint onits volume:

minx1,x2

f(x1, x2) = 2πx1(x1 + x2)

subject to g(x1, x2) = πx21x2 − V = 0

where x1 and x2 are radius and height of cylinder, and V isrequired volume




Local vs Global Optimization

x∗ ∈ S is global minimum if f(x∗) ≤ f(x) for all x ∈ S

x∗ ∈ S is local minimum if f(x∗) ≤ f(x) for all feasible x insome neighborhood of x∗




Global Optimization

Finding, or even verifying, global minimum is difficult, ingeneral

Most optimization methods are designed to find localminimum, which may or may not be global minimum

If global minimum is desired, one can try several widelyseparated starting points and see if all produce sameresult

For some problems, such as linear programming, globaloptimization is more tractable




Existence of Minimum

If f is continuous on closed and bounded set S ⊆ Rn, thenf has global minimum on S

If S is not closed or is unbounded, then f may have nolocal or global minimum on S

Continuous function f on unbounded set S ⊆ Rn iscoercive if

lim‖x‖→∞

f(x) = +∞

i.e., f(x) must be large whenever ‖x‖ is large

If f is coercive on closed, unbounded set S ⊆ Rn, then fhas global minimum on S




Level Sets

Level set for function f : S ⊆ Rn → R is set of all points inS for which f has some given constant value

For given γ ∈ R, sublevel set is

Lγ = {x ∈ S : f(x) ≤ γ}

If continuous function f on S ⊆ Rn has nonempty sublevelset that is closed and bounded, then f has global minimumon S

If S is unbounded, then f is coercive on S if, and only if, allof its sublevel sets are bounded




Uniqueness of Minimum

Set S ⊆ Rn is convex if it contains line segment betweenany two of its points

Function f : S ⊆ Rn → R is convex on convex set S if itsgraph along any line segment in S lies on or below chordconnecting function values at endpoints of segment

Any local minimum of convex function f on convex setS ⊆ Rn is global minimum of f on S

Any local minimum of strictly convex function f on convexset S ⊆ Rn is unique global minimum of f on S




First-Order Optimality Condition

For function of one variable, one can find extremum bydifferentiating function and setting derivative to zero

Generalization to function of n variables is to find criticalpoint, i.e., solution of nonlinear system

∇f(x) = 0

where ∇f(x) is gradient vector of f , whose ith componentis ∂f(x)/∂xi

For continuously differentiable f : S ⊆ Rn → R, any interiorpoint x∗ of S at which f has local minimum must be criticalpoint of f

But not all critical points are minima: they can also bemaxima or saddle points




Second-Order Optimality Condition

For twice continuously differentiable f : S ⊆ Rn → R, wecan distinguish among critical points by consideringHessian matrix Hf (x) defined by

{Hf (x)}ij =∂2f(x)∂xi∂xj

which is symmetric

At critical point x∗, if Hf (x∗) is

positive definite, then x∗ is minimum of f

negative definite, then x∗ is maximum of f

indefinite, then x∗ is saddle point of f

singular, then various pathological situations are possible




Constrained OptimalityIf problem is constrained, only feasible directions arerelevant

For equality-constrained problem

min f(x) subject to g(x) = 0

where f : Rn → R and g : Rn → Rm, with m ≤ n, necessarycondition for feasible point x∗ to be solution is that negativegradient of f lie in space spanned by constraint normals,

−∇f(x∗) = JTg (x∗)λ

where Jg is Jacobian matrix of g, and λ is vector ofLagrange multipliers

This condition says we cannot reduce objective functionwithout violating constraints




Constrained Optimality, continued

Lagrangian function L : Rn+m → R, is defined by

L(x,λ) = f(x) + λT g(x)

Its gradient is given by

∇L(x,λ) =[∇f(x) + JT

g (x)λg(x)

]Its Hessian is given by

HL(x,λ) =[B(x,λ) JT

g (x)Jg(x) O

]where

B(x,λ) = Hf (x) +m∑

i=1

λiHgi(x)





Together, necessary condition and feasibility imply criticalpoint of Lagrangian function,

∇L(x,λ) =[∇f(x) + JT

g (x)λg(x)

]= 0

Hessian of Lagrangian is symmetric, but not positivedefinite, so critical point of L is saddle point rather thanminimum or maximum

Critical point (x∗,λ∗) of L is constrained minimum of f ifB(x∗,λ∗) is positive definite on null space of Jg(x∗)

If columns of Z form basis for null space, then testprojected Hessian ZT BZ for positive definiteness





If inequalities are present, then KKT optimality conditionsalso require nonnegativity of Lagrange multiplierscorresponding to inequalities, and complementaritycondition





Function minimization and equation solving are closelyrelated problems, but their sensitivities differ

In one dimension, absolute condition number of root x∗ ofequation f(x) = 0 is 1/|f ′(x∗)|, so if |f(x)| ≤ ε, then|x− x∗| may be as large as ε/|f ′(x∗)|

For minimizing f , Taylor series expansion

f(x) = f(x∗ + h)= f(x∗) + f ′(x∗)h + 1

2 f ′′(x∗)h2 +O(h3)

shows that, since f ′(x∗) = 0, if |f(x)− f(x∗)| ≤ ε, then|x− x∗| may be as large as

√2ε/|f ′′(x∗)|

Thus, based on function values alone, minima can becomputed to only about half precision



Golden Section SearchSuccessive Parabolic InterpolationNewton’s Method

Unimodality

For minimizing function of one variable, we need “bracket”for solution analogous to sign change for nonlinearequation

Real-valued function f is unimodal on interval [a, b] if thereis unique x∗ ∈ [a, b] such that f(x∗) is minimum of f on[a, b], and f is strictly decreasing for x ≤ x∗, strictlyincreasing for x∗ ≤ x

Unimodality enables discarding portions of interval basedon sample function values, analogous to interval bisection




Golden Section Search

Suppose f is unimodal on [a, b], and let x1 and x2 be twopoints within [a, b], with x1 < x2

Evaluating and comparing f(x1) and f(x2), we can discardeither (x2, b] or [a, x1), with minimum known to lie inremaining subinterval

To repeat process, we need compute only one newfunction evaluation

To reduce length of interval by fixed fraction at eachiteration, each new pair of points must have samerelationship with respect to new interval that previous pairhad with respect to previous interval




Golden Section Search, continued

To accomplish this, we choose relative positions of twopoints as τ and 1− τ , where τ2 = 1− τ , soτ = (

√5− 1)/2 ≈ 0.618 and 1− τ ≈ 0.382

Whichever subinterval is retained, its length will be τrelative to previous interval, and interior point retained willbe at position either τ or 1− τ relative to new interval

To continue iteration, we need to compute only one newfunction value, at complementary point

This choice of sample points is called golden sectionsearch

Golden section search is safe but convergence rate is onlylinear, with constant C ≈ 0.618




Golden Section Search, continuedτ = (

√5− 1)/2

x1 = a + (1− τ)(b− a); f1 = f(x1)x2 = a + τ(b− a); f2 = f(x2)while ((b− a) > tol) do

if (f1 > f2) thena = x1

x1 = x2

f1 = f2

x2 = a + τ(b− a)f2 = f(x2)

elseb = x2

x2 = x1

f2 = f1

x1 = a + (1− τ)(b− a)f1 = f(x1)

endend




Example: Golden Section Search

Use golden section search to minimize

f(x) = 0.5− x exp(−x2)




Example, continued

x1 f1 x2 f2

0.764 0.074 1.236 0.2320.472 0.122 0.764 0.0740.764 0.074 0.944 0.1130.652 0.074 0.764 0.0740.584 0.085 0.652 0.0740.652 0.074 0.695 0.0710.695 0.071 0.721 0.0710.679 0.072 0.695 0.0710.695 0.071 0.705 0.0710.705 0.071 0.711 0.071



http://www.cse.uiuc.edu/iem/optimization/GoldenSection/



Successive Parabolic InterpolationFit quadratic polynomial to three function valuesTake minimum of quadratic to be new approximation tominimum of function

New point replaces oldest of three previous points andprocess is repeated until convergenceConvergence rate of successive parabolic interpolation issuperlinear, with r ≈ 1.324




Example: Successive Parabolic Interpolation

Use successive parabolic interpolation to minimize

f(x) = 0.5− x exp(−x2)




Example, continued

xk f(xk)0.000 0.5000.600 0.0811.200 0.2160.754 0.0730.721 0.0710.692 0.0710.707 0.071



http://www.cse.uiuc.edu/iem/optimization/SuccessiveParabolic/



Newton’s MethodAnother local quadratic approximation is truncated Taylorseries

f(x + h) ≈ f(x) + f ′(x)h +f ′′(x)

2h2

By differentiation, minimum of this quadratic function of h isgiven by h = −f ′(x)/f ′′(x)

Suggests iteration scheme

xk+1 = xk − f ′(xk)/f ′′(xk)

which is Newton’s method for solving nonlinear equationf ′(x) = 0

Newton’s method for finding minimum normally hasquadratic convergence rate, but must be started closeenough to solution to converge < interactive example >


http://www.cse.uiuc.edu/iem/optimization/Newton_Opt/



Example: Newton’s MethodUse Newton’s method to minimize f(x) = 0.5− x exp(−x2)

First and second derivatives of f are given by

f ′(x) = (2x2 − 1) exp(−x2)

andf ′′(x) = 2x(3− 2x2) exp(−x2)

Newton iteration for zero of f ′ is given by

xk+1 = xk − (2x2k − 1)/(2xk(3− 2x2

k))

Using starting guess x0 = 1, we obtain

xk f(xk)1.000 0.1320.500 0.1110.700 0.0710.707 0.071




Safeguarded Methods

As with nonlinear equations in one dimension,slow-but-sure and fast-but-risky optimization methods canbe combined to provide both safety and efficiency

Most library routines for one-dimensional optimization arebased on this hybrid approach

Popular combination is golden section search andsuccessive parabolic interpolation, for which no derivativesare required



Unconstrained OptimizationNonlinear Least SquaresConstrained Optimization

Direct Search Methods

Direct search methods for multidimensional optimizationmake no use of function values other than comparing them

For minimizing function f of n variables, Nelder-Meadmethod begins with n + 1 starting points, forming simplexin Rn

Then move to new point along straight line from currentpoint having highest function value through centroid ofother points

New point replaces worst point, and process is repeated

Direct search methods are useful for nonsmooth functionsor for small n, but expensive for larger n



http://www.cse.uiuc.edu/iem/optimization/NelderMead/



Steepest Descent Method

Let f : Rn → R be real-valued function of n real variables

At any point x where gradient vector is nonzero, negativegradient, −∇f(x), points downhill toward lower values of f

In fact, −∇f(x) is locally direction of steepest descent: fdecreases more rapidly along direction of negativegradient than along any other

Steepest descent method: starting from initial guess x0,successive approximate solutions given by

xk+1 = xk − αk∇f(xk)

where αk is line search parameter that determines how farto go in given direction




Steepest Descent, continued

Given descent direction, such as negative gradient,determining appropriate value for αk at each iteration isone-dimensional minimization problem

minαk

f(xk − αk∇f(xk))

that can be solved by methods already discussed

Steepest descent method is very reliable: it can alwaysmake progress provided gradient is nonzero

But method is myopic in its view of function’s behavior, andresulting iterates can zigzag back and forth, making veryslow progress toward solution

In general, convergence rate of steepest descent is onlylinear, with constant factor that can be arbitrarily close to 1




Example: Steepest Descent

Use steepest descent method to minimize

f(x) = 0.5x21 + 2.5x2

2

Gradient is given by ∇f(x) =[

x1

5x2

]Taking x0 =

[51

], we have ∇f(x0) =

[55

]Performing line search along negative gradient direction,

minα0

f(x0 − α0∇f(x0))

exact minimum along line is given by α0 = 1/3, so next

approximation is x1 =[

3.333−0.667




Example, continued

xk f(xk) ∇f(xk)5.000 1.000 15.000 5.000 5.0003.333 −0.667 6.667 3.333 −3.3332.222 0.444 2.963 2.222 2.2221.481 −0.296 1.317 1.481 −1.4810.988 0.198 0.585 0.988 0.9880.658 −0.132 0.260 0.658 −0.6580.439 0.088 0.116 0.439 0.4390.293 −0.059 0.051 0.293 −0.2930.195 0.039 0.023 0.195 0.1950.130 −0.026 0.010 0.130 −0.130




Example, continued



http://www.cse.uiuc.edu/iem/optimization/SteepestDescent/



Newton’s Method

Broader view can be obtained by local quadraticapproximation, which is equivalent to Newton’s method

In multidimensional optimization, we seek zero of gradient,so Newton iteration has form

xk+1 = xk −H−1f (xk)∇f(xk)

where Hf (x) is Hessian matrix of second partialderivatives of f ,

{Hf (x)}ij =∂2f(x)∂xi∂xj





Do not explicitly invert Hessian matrix, but instead solvelinear system

Hf (xk)sk = −∇f(xk)

for Newton step sk, then take as next iterate

xk+1 = xk + sk

Convergence rate of Newton’s method for minimization isnormally quadratic

As usual, Newton’s method is unreliable unless startedclose enough to solution to converge



http://www.cse.uiuc.edu/iem/optimization/Newton_Opt2D/



Example: Newton’s Method

Use Newton’s method to minimize

f(x) = 0.5x21 + 2.5x2

2

Gradient and Hessian are given by

∇f(x) =[

x1

5x2

]and Hf (x) =

[1 00 5

]

Taking x0 =[51

], we have ∇f(x0) =

[55

]Linear system for Newton step is

[1 00 5

]s0 =

[−5−5

], so

x1 = x0 + s0 =[51

]+[−5−1

]=[00

], which is exact solution

for this problem, as expected for quadratic function





In principle, line search parameter is unnecessary withNewton’s method, since quadratic model determineslength, as well as direction, of step to next approximatesolution

When started far from solution, however, it may still beadvisable to perform line search along direction of Newtonstep sk to make method more robust (damped Newton)

Once iterates are near solution, then αk = 1 should sufficefor subsequent iterations





If objective function f has continuous second partialderivatives, then Hessian matrix Hf is symmetric, andnear minimum it is positive definite

Thus, linear system for step to next iterate can be solved inonly about half of work required for LU factorization

Far from minimum, Hf (xk) may not be positive definite, soNewton step sk may not be descent direction for function,i.e., we may not have

∇f(xk)T sk < 0

In this case, alternative descent direction can becomputed, such as negative gradient or direction ofnegative curvature, and then perform line search




Trust Region Methods

Alternative to line search is trust region method, in whichapproximate solution is constrained to lie within regionwhere quadratic model is sufficiently accurate

If current trust radius is binding, minimizing quadraticmodel function subject to this constraint may modifydirection as well as length of Newton step

Accuracy of quadratic model is assessed by comparingactual decrease in objective function with that predicted byquadratic model, and trust radius is increased ordecreased accordingly




Trust Region Methods, continued




Quasi-Newton Methods

Newton’s method costs O(n3) arithmetic and O(n2) scalarfunction evaluations per iteration for dense problem

Many variants of Newton’s method improve reliability andreduce overhead

Quasi-Newton methods have form

xk+1 = xk − αkB−1k ∇f(xk)

where αk is line search parameter and Bk is approximationto Hessian matrix

Many quasi-Newton methods are more robust thanNewton’s method, are superlinearly convergent, and havelower overhead per iteration, which often more than offsetstheir slower convergence rate




Secant Updating Methods

Could use Broyden’s method to seek zero of gradient, butthis would not preserve symmetry of Hessian matrix

Several secant updating formulas have been developed forminimization that not only preserve symmetry inapproximate Hessian matrix, but also preserve positivedefiniteness

Symmetry reduces amount of work required by about half,while positive definiteness guarantees that quasi-Newtonstep will be descent direction




BFGS Method

One of most effective secant updating methods for minimizationis BFGS

x0 = initial guessB0 = initial Hessian approximationfor k = 0, 1, 2, . . .

Solve Bk sk = −∇f(xk) for sk

xk+1 = xk + sk

yk = ∇f(xk+1)−∇f(xk)Bk+1 = Bk + (yky

Tk )/(yT

k sk) − (BksksTk Bk)/(sT

k Bksk)end




BFGS Method, continued

In practice, factorization of Bk is updated rather than Bk

itself, so linear system for sk can be solved at cost of O(n2)rather than O(n3) work

Unlike Newton’s method for minimization, no secondderivatives are required

Can start with B0 = I, so initial step is along negativegradient, and then second derivative information isgradually built up in approximate Hessian matrix oversuccessive iterations

BFGS normally has superlinear convergence rate, eventhough approximate Hessian does not necessarilyconverge to true Hessian

Line search can be used to enhance effectivenessMichael T. Heath Scientific Computing 46 / 74



Example: BFGS Method

Use BFGS to minimize f(x) = 0.5x21 + 2.5x2

2


x1

5x2

]Taking x0 =

[5 1

]T and B0 = I, initial step is negativegradient, so

x1 = x0 + s0 =[51

]+[−5−5

]=[

0−4

]Updating approximate Hessian using BFGS formula, weobtain

B1 =[0.667 0.3330.333 0.667

]Then new step is computed and process is repeated




Example: BFGS Method

xk f(xk) ∇f(xk)5.000 1.000 15.000 5.000 5.0000.000 −4.000 40.000 0.000 −20.000

−2.222 0.444 2.963 −2.222 2.2220.816 0.082 0.350 0.816 0.408

−0.009 −0.015 0.001 −0.009 −0.077−0.001 0.001 0.000 −0.001 0.005

Increase in function value can be avoided by using linesearch, which generally enhances convergence

For quadratic objective function, BFGS with exact linesearch finds exact solution in at most n iterations, where nis dimension of problem < interactive example >


http://www.cse.uiuc.edu/iem/optimization/BFGS/



Conjugate Gradient Method

Another method that does not require explicit secondderivatives, and does not even store approximation toHessian matrix, is conjugate gradient (CG) method

CG generates sequence of conjugate search directions,implicitly accumulating information about Hessian matrix

For quadratic objective function, CG is theoretically exactafter at most n iterations, where n is dimension of problem

CG is effective for general unconstrained minimization aswell




Conjugate Gradient Method, continued

x0 = initial guessg0 = ∇f(x0)s0 = −g0

for k = 0, 1, 2, . . .Choose αk to minimize f(xk + αksk)xk+1 = xk + αksk

gk+1 = ∇f(xk+1)βk+1 = (gT

k+1gk+1)/(gTk gk)

sk+1 = −gk+1 + βk+1sk

end

Alternative formula for βk+1 is

βk+1 = ((gk+1 − gk)T gk+1)/(gTk gk)




Example: Conjugate Gradient Method

Use CG method to minimize f(x) = 0.5x21 + 2.5x2

2


x1

5x2

]Taking x0 =

[5 1

]T , initial search direction is negativegradient,

s0 = −g0 = −∇f(x0) =[−5−5

]Exact minimum along line is given by α0 = 1/3, so nextapproximation is x1 =

[3.333 −0.667

]T , and we computenew gradient,

g1 = ∇f(x1) =[

3.333−3.333




Example, continued

So far there is no difference from steepest descent method

At this point, however, rather than search along newnegative gradient, we compute instead

β1 = (gT1 g1)/(gT

0 g0) = 0.444

which gives as next search direction

s1 = −g1 + β1s0 =[−3.333

3.333

]+ 0.444

[−5−5

]=[−5.556

1.111

]Minimum along this direction is given by α1 = 0.6, whichgives exact solution at origin, as expected for quadraticfunction



http://www.cse.uiuc.edu/iem/optimization/ConjugateGradient/



Truncated Newton Methods

Another way to reduce work in Newton-like methods is tosolve linear system for Newton step by iterative method

Small number of iterations may suffice to produce step asuseful as true Newton step, especially far from overallsolution, where true Newton step may be unreliableanyway

Good choice for linear iterative solver is CG method, whichgives step intermediate between steepest descent andNewton-like step

Since only matrix-vector products are required, explicitformation of Hessian matrix can be avoided by using finitedifference of gradient along given vector




Nonlinear Least Squares

Given data (ti, yi), find vector x of parameters that gives“best fit” in least squares sense to model function f(t, x),where f is nonlinear function of x

Define components of residual function

ri(x) = yi − f(ti,x), i = 1, . . . ,m

so we want to minimize φ(x) = 12r

T (x)r(x)

Gradient vector is ∇φ(x) = JT (x)r(x) and Hessian matrixis

Hφ(x) = JT (x)J(x) +m∑

i=1

ri(x)Hi(x)

where J(x) is Jacobian of r(x), and Hi(x) is Hessian ofri(x)




Nonlinear Least Squares, continued

Linear system for Newton step is(JT (xk)J(xk) +

m∑i=1

ri(xk)Hi(xk)

)sk = −JT (xk)r(xk)

m Hessian matrices Hi are usually inconvenient andexpensive to compute

Moreover, in Hφ each Hi is multiplied by residualcomponent ri, which is small at solution if fit of modelfunction to data is good




Gauss-Newton Method

This motivates Gauss-Newton method for nonlinear leastsquares, in which second-order term is dropped and linearsystem

JT (xk)J(xk)sk = −JT (xk)r(xk)

is solved for approximate Newton step sk at each iteration

This is system of normal equations for linear least squaresproblem

J(xk)sk∼= −r(xk)

which can be solved better by QR factorization

Next approximate solution is then given by

xk+1 = xk + sk

and process is repeated until convergence




Example: Gauss-Newton Method

Use Gauss-Newton method to fit nonlinear model function

f(t, x) = x1 exp(x2t)

to datat 0.0 1.0 2.0 3.0y 2.0 0.7 0.3 0.1

For this model function, entries of Jacobian matrix ofresidual function r are given by

{J(x)}i,1 =∂ri(x)∂x1

= − exp(x2ti)

{J(x)}i,2 =∂ri(x)∂x2

= −x1ti exp(x2ti)




Example, continued

If we take x0 =[1 0

]T , then Gauss-Newton step s0 isgiven by linear least squares problem

−1 0−1 −1−1 −2−1 −3

s0∼=

−10.30.70.9

whose solution is s0 =

[0.69

−0.61

]Then next approximate solution is given by x1 = x0 + s0,and process is repeated until convergence




Example, continued

xk ‖r(xk)‖221.000 0.000 2.3901.690 −0.610 0.2121.975 −0.930 0.0071.994 −1.004 0.0021.995 −1.009 0.0021.995 −1.010 0.002



http://www.cse.uiuc.edu/iem/optimization/GaussNewton/



Gauss-Newton Method, continued

Gauss-Newton method replaces nonlinear least squaresproblem by sequence of linear least squares problemswhose solutions converge to solution of original nonlinearproblem

If residual at solution is large, then second-order termomitted from Hessian is not negligible, and Gauss-Newtonmethod may converge slowly or fail to converge

In such “large-residual” cases, it may be best to usegeneral nonlinear minimization method that takes intoaccount true full Hessian matrix




Levenberg-Marquardt Method

Levenberg-Marquardt method is another useful alternativewhen Gauss-Newton approximation is inadequate or yieldsrank deficient linear least squares subproblem

In this method, linear system at each iteration is of form

(JT (xk)J(xk) + µkI)sk = −JT (xk)r(xk)

where µk is scalar parameter chosen by some strategy

Corresponding linear least squares problem is[J(xk)√

µkI

]sk∼=[−r(xk)

0

]With suitable strategy for choosing µk, this method can bevery robust in practice, and it forms basis for severaleffective software packages < interactive example >


http://www.cse.uiuc.edu/iem/optimization/LevenbergMarquardt/



Equality-Constrained Optimization

For equality-constrained minimization problem

min f(x) subject to g(x) = 0

where f : Rn → R and g : Rn → Rm, with m ≤ n, we seekcritical point of Lagrangian L(x,λ) = f(x) + λT g(x)

Applying Newton’s method to nonlinear system

∇L(x,λ) =[∇f(x) + JT

g (x)λg(x)

]= 0

we obtain linear system[B(x,λ) JT

g (x)Jg(x) O

] [sδ

]= −

[∇f(x) + JT

g (x)λg(x)

]for Newton step (s, δ) in (x,λ) at each iteration




Sequential Quadratic Programming

Foregoing block 2× 2 linear system is equivalent toquadratic programming problem, so this approach isknown as sequential quadratic programming

Types of solution methods include

Direct solution methods, in which entire block 2× 2 systemis solved directlyRange space methods, based on block elimination in block2× 2 linear systemNull space methods, based on orthogonal factorization ofmatrix of constraint normals, JT

g (x)



http://www.cse.uiuc.edu/iem/optimization/SQP_ConstrainedOptimization/



Merit Function

Once Newton step (s, δ) determined, we need meritfunction to measure progress toward overall solution foruse in line search or trust region

Popular choices include penalty function

φρ(x) = f(x) + 12 ρ g(x)T g(x)

and augmented Lagrangian function

Lρ(x,λ) = f(x) + λT g(x) + 12 ρ g(x)T g(x)

where parameter ρ > 0 determines relative weighting ofoptimality vs feasibility

Given starting guess x0, good starting guess for λ0 can beobtained from least squares problem

JTg (x0) λ0

∼= −∇f(x0)




Inequality-Constrained Optimization

Methods just outlined for equality constraints can beextended to handle inequality constraints by using activeset strategy

Inequality constraints are provisionally divided into thosethat are satisfied already (and can therefore be temporarilydisregarded) and those that are violated (and are thereforetemporarily treated as equality constraints)

This division of constraints is revised as iterations proceeduntil eventually correct constraints are identified that arebinding at solution




Penalty Methods

Merit function can also be used to convertequality-constrained problem into sequence ofunconstrained problems

If x∗ρ is solution to

minx

φρ(x) = f(x) + 12 ρ g(x)T g(x)

then under appropriate conditions

limρ→∞

x∗ρ = x∗

This enables use of unconstrained optimization methods,but problem becomes ill-conditioned for large ρ, so wesolve sequence of problems with gradually increasingvalues of ρ, with minimum for each problem used asstarting point for next problem < interactive example >


http://www.cse.uiuc.edu/iem/optimization/Penalty_ConstrainedOptimization/



Barrier MethodsFor inequality-constrained problems, another alternative isbarrier function, such as

φµ(x) = f(x)− µ

p∑i=1

1hi(x)

or

φµ(x) = f(x)− µ

p∑i=1

log(−hi(x))

which increasingly penalize feasible points as theyapproach boundary of feasible regionAgain, solutions of unconstrained problem approach x∗ asµ → 0, but problems are increasingly ill-conditioned, sosolve sequence of problems with decreasing values of µBarrier functions are basis for interior point methods forlinear programming




Example: Constrained Optimization

Consider quadratic programming problem

minx

f(x) = 0.5x21 + 2.5x2

2

subject tog(x) = x1 − x2 − 1 = 0

Lagrangian function is given by

L(x, λ) = f(x) + λ g(x) = 0.5x21 + 2.5x2

2 + λ(x1 − x2 − 1)

Since

∇f(x) =[

x1

5x2

]and Jg(x) =

[1 −1

]we have

∇xL(x, λ) = ∇f(x) + JTg (x)λ =

[x1

5x2

]+ λ

[1

−1




Example, continued

So system to be solved for critical point of Lagrangian is

x1 + λ = 05x2 − λ = 0x1 − x2 = 1

which in this case is linear system1 0 10 5 −11 −1 0

x1

x2

λ

=

001

Solving this system, we obtain solution

x1 = 0.833, x2 = −0.167, λ = −0.833




Example, continued




Linear Programming

One of most important and common constrainedoptimization problems is linear programming

One standard form for such problems is

min f(x) = cT x subject to Ax = b and x ≥ 0

where m < n, A ∈ Rm×n, b ∈ Rm, and c,x ∈ Rn

Feasible region is convex polyhedron in Rn, and minimummust occur at one of its vertices

Simplex method moves systematically from vertex tovertex until minimum point is found




Linear Programming, continued

Simplex method is reliable and normally efficient, able tosolve problems with thousands of variables, but canrequire time exponential in size of problem in worst case

Interior point methods for linear programming developed inrecent years have polynomial worst case solution time

These methods move through interior of feasible region,not restricting themselves to investigating only its vertices

Although interior point methods have significant practicalimpact, simplex method is still predominant method instandard packages for linear programming, and itseffectiveness in practice is excellent




Example: Linear Programming

To illustrate linear programming, consider

minx

= cT x = −8x1 − 11x2

subject to linear inequality constraints

5x1 + 4x2 ≤ 40, −x1 + 3x2 ≤ 12, x1 ≥ 0, x2 ≥ 0

Minimum value must occur at vertex of feasible region, inthis case at x1 = 3.79, x2 = 5.26, where objective functionhas value −88.2




Example, continued


AMS527: Numerical Analysis IISupplementary Material onNumerical Optimization

Xiangmin Jiao

SUNY Stony Brook

Xiangmin Jiao (SUNY Stony Brook) AMS527: Numerical Analysis II 1 / 21

Outline

1 BFGS Method

2 Conjugate Gradient Methods

3 Constrained Optimization


BFGS Method

BFGS Method is one of most effective secant updating methods forminimizationNamed after Broyden, Fletcher, Goldfarb, and ShannoUnlike Broyden’s method, BFGS preserves the symmetry ofapproximate Hessian matrixIn addition, BFGS preserves the positive definiteness of theapproximate Hessian matrixReference: J. Nocedal, S. J. Wright, Numerical Optimization, 2ndedition, Springer, 2006. Section 6.1.


Algorithm

x0 =initial guessB0 =initial Hessian approximationfor k =0, 1, 2, . . .

Solve Bksk = −∇f (xk) for skxk+1 = xk + skyk = ∇f (xk+1)−∇f (xk)Bk+1 = Bk + (ykyT

k )/(yTk sk)− (BksksT

k Bk)/(sTk Bksk)


Motivation of BFGSLet sk = xk+1 − xk and yk = ∇f (xk+1)−∇f (xk)

Matrix Bk+1 should satisfy secant equation

Bk+1sk = yk

In addition, Bk+1 is positive definition, which requires sTk yk > 0

There are infinite number of Bk+1 that satisfies secant equationDavidon (1950s) proposed to choose Bk+1 to be closest to Bk , i.e.,

minB‖B − Bk‖

Subject to B = BT , Bsk = yk .

BFGS proposed to choose Bk+1 so that B−1k+1 is closest to B−1

k , i.e.,

minB‖B−1 − B−1

k ‖

Subject to B = BT , Bsk = yk .


Properties of BFGS

BFGS normally has superlinear convergence rate, even thoughapproximate Hessian does not necessarily converge to true HessianApproximate Hessian preserves positive definiteness

I Key idea of proof: Let Hk denote B−1k . For any vector z 6= 0, and let

w = z − ρkyk(sTk z), where ρk > 0. Then it can be shown that

zTHk+1z = wTHkw + ρk(sTk z)2 ≥ 0.

If sTk z = 0, then w = z 6= 0. So zTHk+1z > 0.

Line search can be used to enhance effectiveness of BFGS. If exact linesearch is performed at each iteration, BFGS terminates at exactsolution in at most n iterations for a quadratic objective function


Outline

1 BFGS Method




Motivation of Conjugate Gradients

Conjugate gradient can be used to solve a linear system Ax = b,where A is symmetric positive definite (SPD)If A is m ×m SPD, then quadratic function

ϕ(x) =12xTAx − xTb

has unique minimumNegative gradient of this function is residual vector

−∇ϕ(x) = b − Ax = r

so minimum is obtained precisely when Ax = b


Search Direction in Conjugate Gradients

Optimization methods have form

xn+1 = xn + αnpn

where pn is search direction and α is step length chosen to minimizeϕ(xn + αnpn)

Line search parameter can be determined analytically asαn = rT

n pn/pTn Apn

In CG, pn is chosen to be A-conjugate (or A-orthogonal) to previoussearch directions, i.e., pT

n Apj = 0 for j < n


Optimality of Step LengthSelect step length αn over vector pn−1 to minimizeϕ(x) = 1

2xTAx − xTb

Let xn = xn−1 + αnpn−1,

ϕ(xn) =12(xn−1 + αnpn−1)

TA(xn−1 + αnpn−1)− (xn−1 + αnpn−1)Tb

=12α2

npTn−1Apn−1 + αnpT

n−1Axn−1 − αnpTn−1b + constant

=12α2

npTn−1Apn−1 − αnpT

n−1rn−1 + constant

Therefore,

dϕdαn

= 0⇒ αnpTn−1Apn−1 − pT

n−1rn−1 = 0⇒ αn =pT

n−1rn−1

pTn−1Apn−1

.

In addition, pTn−1rn−1 = rT

n−1rn−1 because pn−1 = rn−1 + βnpn−2and rT

n−1pn−2 = 0 due to the following theorem.


Conjugate Gradient Method

Algorithm: Conjugate Gradient Methodx0 = 0, r0 = b, p0 = r0for n = 1 to 1, 2, 3, . . .

αn = (rTn−1rn−1)/(pT

n−1Apn−1) step lengthxn = xn−1 + αnpn−1 approximate solutionrn = rn−1 − αnApn−1 residualβn = (rT

n rn)/(rTn−1rn−1) improvement this step

pn = rn + βnpn−1 search direction

Only one matrix-vector product Apn−1 per iterationApart from matrix-vector product, #operations per iteration is O(m)

CG can be viewed as minimization of quadratic functionϕ(x) = 1

2xTAx − xTb by modifying steepest descent

First proposed by Hestens and Stiefel in 1950s


An Alternative Interpretation of CG

Algorithm: CGx0 = 0, r0 = b, p0 = r0for n =1, 2, 3, . . .αn = rT

n−1rn−1/(pTn−1Apn−1)

xn = xn−1 + αnpn−1rn = rn−1 − αnApn−1βn = rT

n rn/(rTn−1rn−1)

pn = rn + βnpn−1

Algorithm: A non-standard CGx0 = 0, r0 = b, p0 = r0for n =1, 2, 3, . . .αn = rT

n−1pn−1/(pTn−1Apn−1)

xn = xn−1 + αnpn−1rn = b − Axnβn = −rT

n Apn−1/(pTn−1Apn−1)

pn = rn + βnpn−1

The non-standard one is less efficient but easier to understandIt is easy to see rn = rn−1 − αnApn−1 = b − Axn


Comparison of Linear and Nonlinear CG

Algorithm: Linear CGx0 = 0, r0 = b,p0 = r0for n =1, 2, 3, . . .αn = rT

n−1rn−1/(pTn−1Apn−1)

xn = xn−1 + αnpn−1rn = rn−1 − αnApn−1βn = rT

n rn/(rTn−1rn−1)

pn = rn + βnpn−1

Algorithm: Non-linear CGx0 = initial guess, g0 = ∇f (x0),s0 = −g0for k = 0, 1, 2, . . .

Choose αk to min f (xk + αksk)xk+1 = xk + αkskgk+1 = ∇f (xk+1)βk+1 = (gT

k+1gk+1)/(gTk gk)

sk+1 = −gk+1 + βk+1sk

βk+1 = (gTk+1gk+1)/(gT

k gk) was due to Fletcher and Reeves (1964)

An alternative formula βk+1 = (gk+1 − gk)Tgk+1/(gT

k gk) was dueto Polak and Riebiere (1969)


Properties of Conjugate GradientsKrylov subspaces for Ax = b is Kn = {b,Ab, . . . ,An−1b}.

TheoremIf rn−1 6= 0, spaces spanned by approximate solutions xn, search directionspn, and residuals rn are all equal to Krylov subspaces

Kn = 〈x1, x2, . . . , xn〉 = 〈p0,p1, . . . ,pn−1〉= 〈r0, r1, . . . , rn−1〉 = 〈b,Ab, . . . ,An−1b〉

The residual are orthogonal (i.e., rTn r j = 0 for j < n) and search directions

are A-conjugate (i.e, pTn Apj = 0 for j < n).

TheoremIf rn−1 6= 0, then error en = x∗ − xn are minimized in A-norm in Kn.

Because Kn grows monotonically, error decreases monotonically.


Rate of Convergence

Some important convergence resultsI If A has n distinct eigenvalues, CG converges in at most n stepsI If A has 2-norm condition number κ, the errors are

‖en‖A‖e0‖A

≤ 2(√

κ− 1√κ+ 1

)n

which is ≈ 2(1− 2√

κ

)nas κ→∞. So convergence is expected in

O(√κ) iterations.

In general, CG performs well with clustered eigenvalues


Outline

1 BFGS Method




Equality-Constrained MinimizationEquality-constrained problem has form

minx∈Rn

f (x) subject to g(x) = 0

where objective function f : Rn → R and constraints g : Rn → Rm,where m ≤ nNecessary condition for feasible point x to be solution is that negativegradient of f lie in space spanned by constraint normals, i.e.,

−∇f (x∗) = JTg (x

∗)λ,

where Jg is Jacobian matrix of g , and λ is vector of LagrangemultipliersTherefore, constrained local minimum must be critical point ofLagrangian function

L(x ,λ) = f (x) + λTg(x)


First-Order and Second-Order Optimality ConditionsEquality-constrained minimization can be reduced to solving

∇L(x ,λ) =[∇f (x) + JT

g (x)λg(x)

]= 0,

which is known as Karush-Kuhn-Tucker (or KKT) condition forconstrained local minimum.Hessian of Lagrangian function

HL(x ,λ) =[

B(x ,λ) JTg (x)

Jg (x) 0

]where B(x ,λ) = H f (x) +

∑mi=1 λiHgi (x). HL is sometimes called

KKT (Karush-Kuhn-Tucker) matrix. HL is symmetric, but not ingeneral positive definiteCritical point (x∗,λ∗) of L is constrained minimum if B(x∗,λ∗) ispositive definite on null space of Jg (x∗).Let Z form basis of null (Jg (x∗)), then projected Hessian ZTBZshould be positive definite


Sequential Quadratic Programming∇L(x ,λ) = 0 can be solved using Newton’s method. kth iteration ofNewton’s step is[

B(xk ,λk) JTg (xk)

Jg (xk) 0

] [skδk

]= −

[∇f (xk) + JT

g (xk)λkg(xk)

],

and then xk+1 = xk + sk and λk+1 = λk + δk

Above system of equations is first-order optimality condition forconstrained optimization problem

mins12sTB(xk ,λk)s + sT

(∇f (xk) + JT

g (xk)λk

)subject to

Jg (xk)s + g(xk) = 0.

This problem is quadratic programming problem, so approach usingNewton’s method is known as sequential quadratic programing


Solving KKT System

KKT system[

B JT

J 0

] [sδ

]= −

[wg

]can be solved in several

waysDirect solution

I Solve system using method for symmetric indefinite factorization, suchas LDLT with pivoting, or

I Use iterative method such as GMRES, MINRES

Range-space methodI Use block elimination and obtain symmetric system(

JB−1JT)δ = g − JB−1w

and thenBs = −w − JTδ

I First equation finds δ in range space of JI It is attractive when number of constraints m is relatively small,

because JB−1JT is m ×mI However, it requires B to be nonsingular and J has full rank. Also,

condition number of JB−1JT may be largeXiangmin Jiao (SUNY Stony Brook) AMS527: Numerical Analysis II 20 / 21

Solving KKT System Cont’dNull space method

Let Z be composed of null space of J , and it can be obtained by QRfactorization of JT . Then JZ = 0Let JY = RT , and write s = Yu + Zv . Second block row yields

Js = J(Yu + Zv) = RTu = −g

and premultiplying first block row by ZT yields(ZTBZ

)v = −ZT (w − BYu)

Finally,Y TJTδ = Rδ = −Y T (w + Bs)

This method method is advantageous when n −m is smallIt is more stable than range-space method. Also, B does not need tobe nonsingular


AMS527: Numerical Analysis IILinear Programming

Xiangmin Jiao

SUNY Stony Brook


Linear Programming

Linear programming has linear objective function and linear equalityand inequality constraintsExample: Maximize profit of combination of wheat and barley, butwith limited budget of land, fertilizer, and insecticide. Let x1 and x2be areas planted for wheat and barley, we have linear programmingproblem

maximize c1x1 + c2x2 {maximize revenue}0 ≤ x1 + x2 ≤ L {limit on area}

F1x1 + F2x2 ≤ F {limit on fertilizer}P1x1 + P2x2 ≤ P {limit on insecticide}

x1 ≥ 0, x2 ≥ 0 {nonnegative land}

Linear programming is typically solved by simplex methods or interiorpoint methods


Standard Form of Linear Programming

Linear programming has many forms. A standard form (called slackform) is

min cx subject to Ax = b and x ≥ 0

Simplex method and interior-point method requires slack formPrevious example can be converted into standard form

minimize (−c1) x1 + (−c2)x2 {maximize revenue}x1 + x2 + x3 = L {limit on area}

F1x1 + F2x2 + x4 = F {limit on fertilizer}P1x1 + P2x2 + x5 = P {limit on insecticide}

x1, x2, x3, x4, x5 ≥ 0 {nonnegativity}

Here, x3, x4, and x5 are called slack variables


Duality

m equations Ax = b have m corresponding Lagrange multipliers in yPrimal problem

Minimize cTx subject to Ax = b and x ≥ 0

Dual problem

Maximize bTy subject to ATy ≤ c

Weak duality: bTy ≤ cTx for any feasible x and y

I because bTy = (Ax)T y = xT(ATy

)≤ xTc = cTx

Strong duality: If both feasible sets of primal and dual problems arenonempty, then cTx∗ = bTy∗ at optimal x∗ and y∗


Simplex Methods

Developed by George Dantzig in 1947Key observation: Feasible region is convex polytope in Rn, andminimum must occur at one of its verticesBasic idea: Construct a feasible solution at a vertex of the polytope,walk along a path on the edges of the polytope to vertices withnon-decreasing values of the objective function, until an optimum isreachedSimplex method in the worst case can be slow, because number ofcorners is exponential with m and nHowever, its average-case complexity is polynomial time, and inpractice, best corner is often found in 2m steps


Interior Point Methods

First proposed by Narendra Karmarkar in 1984In contrast to simplex methods, interior point methods move throughthe interior of the feasible regionBarrier problem

minimize cTx − θ(log x1 + · · ·+ log xn) with Ax = b

When any xi touches zero, extra cost −θ log xi blows upBarrier problem gives approximate problem for each θ. Its Lagrangianis

L(x , y , θ) = cTx − θ(∑

log xi

)− yT (Ax − b)

The derivatives ∂L/∂xj = cj − θxj− (ATy)j = 0, or xjsj = θ, where

s = c − ATy


Newton Step

n optimality equations xjsj = θ are nonlinear, and are solved iterativelyusing Newton’s methodTo determine increment ∆x , ∆y , and ∆s, we need to solve(xi + ∆xi )(si + ∆si ) = θ. It is typical to ignore second order term∆xi∆si . Then linear equations become

A∆x = 0

AT ∆y + ∆s = 0sj∆xj + xj∆sj = θ − xjsj .

The iteration has quadratic convergence for each θ, and θ approacheszeroGilbert Strang, Computational Science and Engineering, WellesleyCambridge, 2007. Section 8.6, Linear Programming and Duality.


ExampleMinimize cTx = 5x1 + 3x2 + 8x3 with xi ≥ 0 and Ax = x1 + x2 + 2x3 = 4.

Barrier Lagrangian isL = (5x1 +3x2 +8x3)+θ(log x1 + log x2 + log x3)−y(x1 +x2 +2x3−4)Optimality equation gives us:

s = c − ATy s1 = 5− y , s2 = 3− y , s3 = 8− 2y∂L/∂xi = 0 x1s1 = x2s2 = x3s3 = θ∂L/∂y = 0 x1 + x2 + 2x3 = 4

.

Start from an interior point x1 = x2 = x3 = 1, y = 1, and s = (3, 1, 4).From A∆x = 0 and xjsj .+ sj∆xj + xj∆sj = θ, we obtain equations

3∆x1 − 1∆y = θ − 31∆x2 − 1∆y = θ − 14∆x3 − 2∆y = θ − 4

∆x1 + ∆x2 + 2∆x3 = 0.

Given θ = 4/3, we then obtain xnew = (2/3, 2, 2/3) and ynew = 8/3,whereas x∗ = (0, 4, 0) and y∗ = 3


InterpolationPolynomial Interpolation

Piecewise Polynomial Interpolation

Scientific Computing: An Introductory SurveyChapter 7 – Interpolation







Outline

1 Interpolation

2 Polynomial Interpolation

3 Piecewise Polynomial Interpolation




MotivationChoosing InterpolantExistence and Uniqueness

Interpolation

Basic interpolation problem: for given data

(t1, y1), (t2, y2), . . . (tm, ym) with t1 < t2 < · · · < tm

determine function f : R → R such that

f(ti) = yi, i = 1, . . . ,m

f is interpolating function, or interpolant, for given data

Additional data might be prescribed, such as slope ofinterpolant at given points

Additional constraints might be imposed, such assmoothness, monotonicity, or convexity of interpolant

f could be function of more than one variable, but we willconsider only one-dimensional case





Purposes for Interpolation

Plotting smooth curve through discrete data points

Reading between lines of table

Differentiating or integrating tabular data

Quick and easy evaluation of mathematical function

Replacing complicated function by simple one





Interpolation vs Approximation

By definition, interpolating function fits given data pointsexactly

Interpolation is inappropriate if data points subject tosignificant errors

It is usually preferable to smooth noisy data, for exampleby least squares approximation

Approximation is also more appropriate for special functionlibraries





Issues in Interpolation

Arbitrarily many functions interpolate given set of data points

What form should interpolating function have?

How should interpolant behave between data points?

Should interpolant inherit properties of data, such asmonotonicity, convexity, or periodicity?

Are parameters that define interpolating functionmeaningful?

If function and data are plotted, should results be visuallypleasing?





Choosing Interpolant

Choice of function for interpolation based on

How easy interpolating function is to work with

determining its parametersevaluating interpolantdifferentiating or integrating interpolant

How well properties of interpolant match properties of datato be fit (smoothness, monotonicity, convexity, periodicity,etc.)





Functions for Interpolation

Families of functions commonly used for interpolationinclude

PolynomialsPiecewise polynomialsTrigonometric functionsExponential functionsRational functions

For now we will focus on interpolation by polynomials andpiecewise polynomials

We will consider trigonometric interpolation (DFT) later





Basis Functions

Family of functions for interpolating given data points isspanned by set of basis functions φ1(t), . . . , φn(t)

Interpolating function f is chosen as linear combination ofbasis functions,

f(t) =n∑

j=1

xjφj(t)

Requiring f to interpolate data (ti, yi) means

f(ti) =n∑

j=1

xjφj(ti) = yi, i = 1, . . . ,m

which is system of linear equations Ax = y for n-vector xof parameters xj , where entries of m× n matrix A aregiven by aij = φj(ti)





Existence, Uniqueness, and Conditioning

Existence and uniqueness of interpolant depend onnumber of data points m and number of basis functions n

If m > n, interpolant usually doesn’t exist

If m < n, interpolant is not unique

If m = n, then basis matrix A is nonsingular provided datapoints ti are distinct, so data can be fit exactly

Sensitivity of parameters x to perturbations in datadepends on cond(A), which depends in turn on choice ofbasis functions




Monomial, Lagrange, and Newton InterpolationOrthogonal PolynomialsAccuracy and Convergence

Polynomial Interpolation

Simplest and most common type of interpolation usespolynomials

Unique polynomial of degree at most n− 1 passes throughn data points (ti, yi), i = 1, . . . , n, where ti are distinct

There are many ways to represent or compute interpolatingpolynomial, but in theory all must give same result



http://www.cse.uiuc.edu/iem/interpolation/sumbasis/




Monomial Basis

Monomial basis functions

φj(t) = tj−1, j = 1, . . . , n

give interpolating polynomial of form

pn−1(t) = x1 + x2t + · · ·+ xntn−1

with coefficients x given by n× n linear system

Ax =

1 t1 · · · tn−1

1

1 t2 · · · tn−12

......

. . ....

1 tn · · · tn−1n

x1

x2...

xn

=

y1

y2...

yn

= y

Matrix of this form is called Vandermonde matrix





Example: Monomial Basis

Determine polynomial of degree two interpolating threedata points (−2,−27), (0,−1), (1, 0)

Using monomial basis, linear system is

Ax =

1 t1 t211 t2 t221 t3 t23

x1

x2

x3

=

y1

y2

y3

= y

For these particular data, system is1 −2 41 0 01 1 1

x1

x2

x3

=

−27−1

0

whose solution is x =

[−1 5 −4

]T , so interpolatingpolynomial is

p2(t) = −1 + 5t− 4t2





Monomial Basis, continued


Solving system Ax = y using standard linear equationsolver to determine coefficients x of interpolatingpolynomial requires O(n3) work


http://www.cse.uiuc.edu/iem/interpolation/polybase/





For monomial basis, matrix A is increasingly ill-conditionedas degree increases

Ill-conditioning does not prevent fitting data points well,since residual for linear system solution will be small

But it does mean that values of coefficients are poorlydetermined

Both conditioning of linear system and amount ofcomputational work required to solve it can be improved byusing different basis

Change of basis still gives same interpolating polynomialfor given data, but representation of polynomial will bedifferent






Conditioning with monomial basis can be improved byshifting and scaling independent variable t

φj(t) =(

t− c

d

)j−1

where, c = (t1 + tn)/2 is midpoint and d = (tn − t1)/2 ishalf of range of data

New independent variable lies in interval [−1, 1], which alsohelps avoid overflow or harmful underflow

Even with optimal shifting and scaling, monomial basisusually is still poorly conditioned, and we must seek betteralternatives







Evaluating Polynomials

When represented in monomial basis, polynomial

pn−1(t) = x1 + x2t + · · ·+ xntn−1

can be evaluated efficiently using Horner’s nestedevaluation scheme

pn−1(t) = x1 + t(x2 + t(x3 + t(· · · (xn−1 + txn) · · · )))

which requires only n additions and n multiplications

For example,

1− 4t + 5t2 − 2t3 + 3t4 = 1 + t(−4 + t(5 + t(−2 + 3t)))

Other manipulations of interpolating polynomial, such asdifferentiation or integration, are also relatively easy withmonomial basis representation





Lagrange Interpolation

For given set of data points (ti, yi), i = 1, . . . , n, Lagrangebasis functions are defined by

`j(t) =n∏

k=1,k 6=j

(t− tk) /

n∏k=1,k 6=j

(tj − tk), j = 1, . . . , n

For Lagrange basis,

`j(ti) ={

1 if i = j0 if i 6= j

, i, j = 1, . . . , n

so matrix of linear system Ax = y is identity matrix

Thus, Lagrange polynomial interpolating data points (ti, yi)is given by

pn−1(t) = y1`1(t) + y2`2(t) + · · ·+ yn`n(t)





Lagrange Basis Functions


Lagrange interpolant is easy to determine but moreexpensive to evaluate for given argument, compared withmonomial basis representation

Lagrangian form is also more difficult to differentiate,integrate, etc.






Example: Lagrange Interpolation

Use Lagrange interpolation to determine interpolatingpolynomial for three data points (−2,−27), (0,−1), (1, 0)

Lagrange polynomial of degree two interpolating threepoints (t1, y1), (t2, y2), (t3, y3) is given by p2(t) =

y1(t− t2)(t− t3)

(t1 − t2)(t1 − t3)+ y2

(t− t1)(t− t3)(t2 − t1)(t2 − t3)

+ y3(t− t1)(t− t2)

(t3 − t1)(t3 − t2)

For these particular data, this becomes

p2(t) = −27t(t− 1)

(−2)(−2− 1)+ (−1)

(t + 2)(t− 1)(2)(−1)





Newton Interpolation

For given set of data points (ti, yi), i = 1, . . . , n, Newtonbasis functions are defined by

πj(t) =j−1∏k=1

(t− tk), j = 1, . . . , n

where value of product is taken to be 1 when limits make itvacuous

Newton interpolating polynomial has form

pn−1(t) = x1 + x2(t− t1) + x3(t− t1)(t− t2) +· · ·+ xn(t− t1)(t− t2) · · · (t− tn−1)

For i < j, πj(ti) = 0, so basis matrix A is lower triangular,where aij = πj(ti)





Newton Basis Functions







Newton Interpolation, continued

Solution x to system Ax = y can be computed byforward-substitution in O(n2) arithmetic operations

Moreover, resulting interpolant can be evaluated efficientlyfor any argument by nested evaluation scheme similar toHorner’s method

Newton interpolation has better balance between cost ofcomputing interpolant and cost of evaluating it





Example: Newton Interpolation

Use Newton interpolation to determine interpolatingpolynomial for three data points (−2,−27), (0,−1), (1, 0)

Using Newton basis, linear system is1 0 01 t2 − t1 01 t3 − t1 (t3 − t1)(t3 − t2)

x1

x2

x3

=

y1

y2

y3

For these particular data, system is1 0 0

1 2 01 3 3

x1

x2

x3

=

−27−1

0

whose solution by forward substitution isx =

[−27 13 −4

]T , so interpolating polynomial is

p(t) = −27 + 13(t + 2)− 4(t + 2)tMichael T. Heath Scientific Computing 24 / 56




Newton Interpolation, continued

If pj(t) is polynomial of degree j − 1 interpolating j givenpoints, then for any constant xj+1,

pj+1(t) = pj(t) + xj+1πj+1(t)

is polynomial of degree j that also interpolates same jpoints

Free parameter xj+1 can then be chosen so that pj+1(t)interpolates yj+1,

xj+1 =yj+1 − pj(tj+1)

πj+1(tj+1)

Newton interpolation begins with constant polynomialp1(t) = y1 interpolating first data point and thensuccessively incorporates each remaining data point intointerpolant < interactive example >


http://www.cse.uiuc.edu/iem/interpolation/nwtnntrp/




Divided Differences

Given data points (ti, yi), i = 1, . . . , n, divided differences,denoted by f [ ], are defined recursively by

f [t1, t2, . . . , tk] =f [t2, t3, . . . , tk]− f [t1, t2, . . . , tk−1]

tk − t1

where recursion begins with f [tk] = yk, k = 1, . . . , n

Coefficient of jth basis function in Newton interpolant isgiven by

xj = f [t1, t2, . . . , tj ]

Recursion requires O(n2) arithmetic operations to computecoefficients of Newton interpolant, but is less prone tooverflow or underflow than direct formation of triangularNewton basis matrix





Orthogonal Polynomials

Inner product can be defined on space of polynomials oninterval [a, b] by taking

〈p, q〉 =∫ b

ap(t)q(t)w(t)dt

where w(t) is nonnegative weight function

Two polynomials p and q are orthogonal if 〈p, q〉 = 0

Set of polynomials {pi} is orthonormal if

〈pi, pj〉 ={

1 if i = j0 otherwise

Given set of polynomials, Gram-Schmidt orthogonalizationcan be used to generate orthonormal set spanning samespace





Orthogonal Polynomials, continued

For example, with inner product given by weight functionw(t) ≡ 1 on interval [−1, 1], applying Gram-Schmidtprocess to set of monomials 1, t, t2, t3, . . . yields Legendrepolynomials

1, t, (3t2 − 1)/2, (5t3 − 3t)/2, (35t4 − 30t2 + 3)/8,

(63t5 − 70t3 + 15t)/8, . . .

first n of which form an orthogonal basis for space ofpolynomials of degree at most n− 1

Other choices of weight functions and intervals yield otherorthogonal polynomials, such as Chebyshev, Jacobi,Laguerre, and Hermite





Orthogonal Polynomials, continued

Orthogonal polynomials have many useful properties

They satisfy three-term recurrence relation of form

pk+1(t) = (αkt + βk)pk(t)− γkpk−1(t)

which makes them very efficient to generate and evaluate

Orthogonality makes them very natural for least squaresapproximation, and they are also useful for generatingGaussian quadrature rules, which we will see later





Chebyshev Polynomials

kth Chebyshev polynomial of first kind, defined on interval[−1, 1] by

Tk(t) = cos(k arccos(t))

are orthogonal with respect to weight function (1− t2)−1/2

First few Chebyshev polynomials are given by

1, t, 2t2− 1, 4t3− 3t, 8t4− 8t2 + 1, 16t5− 20t3 + 5t, . . .

Equi-oscillation property : successive extrema of Tk areequal in magnitude and alternate in sign, which distributeserror uniformly when approximating arbitrary continuousfunction





Chebyshev Basis Functions



http://www.cse.uiuc.edu/iem/interpolation/chbshvp/




Chebyshev PointsChebyshev points are zeros of Tk, given by

ti = cos(

(2i− 1)π2k

), i = 1, . . . , k

or extrema of Tk, given by

ti = cos(

iπ

k

), i = 0, 1, . . . , k

Chebyshev points are abscissas of points equally spacedaround unit circle in R2

Chebyshev points have attractive properties forinterpolation and other problems





Interpolating Continuous Functions

If data points are discrete sample of continuous function,how well does interpolant approximate that functionbetween sample points?

If f is smooth function, and pn−1 is polynomial of degree atmost n− 1 interpolating f at n points t1, . . . , tn, then

f(t)− pn−1(t) =f (n)(θ)

n!(t− t1)(t− t2) · · · (t− tn)

where θ is some (unknown) point in interval [t1, tn]

Since point θ is unknown, this result is not particularlyuseful unless bound on appropriate derivative of f isknown





Interpolating Continuous Functions, continued

If |f (n)(t)| ≤ M for all t ∈ [t1, tn], andh = max{ti+1 − ti : i = 1, . . . , n− 1}, then

maxt∈[t1,tn]

|f(t)− pn−1(t)| ≤Mhn

4n

Error diminishes with increasing n and decreasing h, butonly if |f (n)(t)| does not grow too rapidly with n



http://www.cse.uiuc.edu/iem/interpolation/errorbnd/




High-Degree Polynomial Interpolation

Interpolating polynomials of high degree are expensive todetermine and evaluate

In some bases, coefficients of polynomial may be poorlydetermined due to ill-conditioning of linear system to besolved

High-degree polynomial necessarily has lots of “wiggles,”which may bear no relation to data to be fit

Polynomial passes through required data points, but it mayoscillate wildly between data points





Convergence

Polynomial interpolating continuous function may notconverge to function as number of data points andpolynomial degree increases

Equally spaced interpolation points often yieldunsatisfactory results near ends of interval

If points are bunched near ends of interval, moresatisfactory results are likely to be obtained withpolynomial interpolation

Use of Chebyshev points distributes error evenly andyields convergence throughout interval for any sufficientlysmooth function





Example: Runge’s Function

Polynomial interpolants of Runge’s function at equallyspaced points do not converge



http://www.cse.uiuc.edu/iem/interpolation/cnvrgnc/




Example: Runge’s Function

Polynomial interpolants of Runge’s function at Chebyshevpoints do converge



http://www.cse.uiuc.edu/iem/interpolation/cnvrgnc/




Taylor Polynomial

Another useful form of polynomial interpolation for smoothfunction f is polynomial given by truncated Taylor series

pn(t) = f(a)+f ′(a)(t−a)+f ′′(a)

2(t−a)2+· · ·+f (n)(a)

n!(t−a)n

Polynomial interpolates f in that values of pn and its first nderivatives match those of f and its first n derivativesevaluated at t = a, so pn(t) is good approximation to f(t)for t near a

We have already seen examples in Newton’s method fornonlinear equations and optimization



http://www.cse.uiuc.edu/iem/interpolation/taylor/



Piecewise Polynomial InterpolationHermite Cubic InterpolationCubic Spline Interpolation


Fitting single polynomial to large number of data points islikely to yield unsatisfactory oscillating behavior ininterpolant

Piecewise polynomials provide alternative to practical andtheoretical difficulties with high-degree polynomialinterpolation

Main advantage of piecewise polynomial interpolation isthat large number of data points can be fit with low-degreepolynomials

In piecewise interpolation of given data points (ti, yi),different function is used in each subinterval [ti, ti+1]

Abscissas ti are called knots or breakpoints, at whichinterpolant changes from one function to another





Piecewise Interpolation, continued

Simplest example is piecewise linear interpolation, inwhich successive pairs of data points are connected bystraight lines

Although piecewise interpolation eliminates excessiveoscillation and nonconvergence, it appears to sacrificesmoothness of interpolating function

We have many degrees of freedom in choosing piecewisepolynomial interpolant, however, which can be exploited toobtain smooth interpolating function despite its piecewisenature



http://www.cse.uiuc.edu/iem/interpolation/plyspcmp/




Hermite Interpolation

In Hermite interpolation, derivatives as well as values ofinterpolating function are taken into account

Including derivative values adds more equations to linearsystem that determines parameters of interpolatingfunction

To have unique solution, number of equations must equalnumber of parameters to be determined

Piecewise cubic polynomials are typical choice for Hermiteinterpolation, providing flexibility, simplicity, and efficiency





Hermite Cubic Interpolation

Hermite cubic interpolant is piecewise cubic polynomialinterpolant with continuous first derivative

Piecewise cubic polynomial with n knots has 4(n− 1)parameters to be determined

Requiring that it interpolate given data gives 2(n− 1)equations

Requiring that it have one continuous derivative gives n− 2additional equations, or total of 3n− 4, which still leaves nfree parameters

Thus, Hermite cubic interpolant is not unique, andremaining free parameters can be chosen so that resultsatisfies additional constraints





Cubic Spline Interpolation

Spline is piecewise polynomial of degree k that is k − 1times continuously differentiable

For example, linear spline is of degree 1 and has 0continuous derivatives, i.e., it is continuous, but notsmooth, and could be described as “broken line”

Cubic spline is piecewise cubic polynomial that is twicecontinuously differentiable

As with Hermite cubic, interpolating given data andrequiring one continuous derivative imposes 3n− 4constraints on cubic spline

Requiring continuous second derivative imposes n− 2additional constraints, leaving 2 remaining free parameters





Cubic Splines, continued

Final two parameters can be fixed in various ways

Specify first derivative at endpoints t1 and tn

Force second derivative to be zero at endpoints, whichgives natural spline

Enforce “not-a-knot” condition, which forces twoconsecutive cubic pieces to be same

Force first derivatives, as well as second derivatives, tomatch at endpoints t1 and tn (if spline is to be periodic)





Example: Cubic Spline Interpolation

Determine natural cubic spline interpolating three datapoints (ti, yi), i = 1, 2, 3

Required interpolant is piecewise cubic function defined byseparate cubic polynomials in each of two intervals [t1, t2]and [t2, t3]

Denote these two polynomials by

p1(t) = α1 + α2t + α3t2 + α4t

3

p2(t) = β1 + β2t + β3t2 + β4t

3

Eight parameters are to be determined, so we need eightequations





Example, continued

Requiring first cubic to interpolate data at end points of firstinterval [t1, t2] gives two equations

α1 + α2t1 + α3t21 + α4t

31 = y1

α1 + α2t2 + α3t22 + α4t

32 = y2

Requiring second cubic to interpolate data at end points ofsecond interval [t2, t3] gives two equations

β1 + β2t2 + β3t22 + β4t

32 = y2

β1 + β2t3 + β3t23 + β4t

33 = y3

Requiring first derivative of interpolant to be continuous att2 gives equation

α2 + 2α3t2 + 3α4t22 = β2 + 2β3t2 + 3β4t

22





Example, continued

Requiring second derivative of interpolant function to becontinuous at t2 gives equation

2α3 + 6α4t2 = 2β3 + 6β4t2

Finally, by definition natural spline has second derivativeequal to zero at endpoints, which gives two equations

2α3 + 6α4t1 = 0

2β3 + 6β4t3 = 0

When particular data values are substituted for ti and yi,system of eight linear equations can be solved for eightunknown parameters αi and βi





Hermite Cubic vs Spline Interpolation

Choice between Hermite cubic and spline interpolationdepends on data to be fit and on purpose for doinginterpolation

If smoothness is of paramount importance, then splineinterpolation may be most appropriate

But Hermite cubic interpolant may have more pleasingvisual appearance and allows flexibility to preservemonotonicity if original data are monotonic

In any case, it is advisable to plot interpolant and data tohelp assess how well interpolating function capturesbehavior of original data



http://www.cse.uiuc.edu/iem/interpolation/pcwcubic/




Hermite Cubic vs Spline Interpolation





B-splines

B-splines form basis for family of spline functions of givendegree

B-splines can be defined in various ways, includingrecursion (which we will use), convolution, and divideddifferences

Although in practice we use only finite set of knotst1, . . . , tn, for notational convenience we will assumeinfinite set of knots

· · · < t−2 < t−1 < t0 < t1 < t2 < · · ·Additional knots can be taken as arbitrarily defined pointsoutside interval [t1, tn]

We will also use linear functions

vki (t) = (t− ti)/(ti+k − ti)





B-splines, continued

To start recursion, define B-splines of degree 0 by

B0i (t) =

{1 if ti ≤ t < ti+1

0 otherwise

and then for k > 0 define B-splines of degree k by

Bki (t) = vk

i (t)Bk−1i (t) + (1− vk

i+1(t))Bk−1i+1 (t)

Since B0i is piecewise constant and vk

i is linear, B1i is

piecewise linear

Similarly, B2i is in turn piecewise quadratic, and in general,

Bki is piecewise polynomial of degree k








http://www.cse.uiuc.edu/iem/interpolation/bspline/





Important properties of B-spline functions Bki

1 For t < ti or t > ti+k+1, Bki (t) = 0

2 For ti < t < ti+k+1, Bki (t) > 0

3 For all t,∑∞

i=−∞Bki (t) = 1

4 For k ≥ 1, Bki has k − 1 continuous derivatives

5 Set of functions {Bk1−k, . . . , B

kn−1} is linearly independent

on interval [t1, tn] and spans space of all splines of degreek having knots ti






Properties 1 and 2 together say that B-spline functionshave local support

Property 3 gives normalization

Property 4 says that they are indeed splines

Property 5 says that for given k, these functions form basisfor set of all splines of degree k






If we use B-spline basis, linear system to be solved forspline coefficients will be nonsingular and banded

Use of B-spline basis yields efficient and stable methodsfor determining and evaluating spline interpolants, andmany library routines for spline interpolation are based onthis approach

B-splines are also useful in many other contexts, such asnumerical solution of differential equations, as we will seelater


AMS527: Numerical Analysis IIReview for Test 1

Xiangmin Jiao

SUNY Stony Brook

Xiangmin Jiao SUNY Stony Brook AMS527: Numerical Analysis II 1 / 5

Approximations in Scientific Computations

Concepts

Absolute error, relative errorComputational error, propagated data errorTruncation error, rounding errorForward error, backward errorCondition number, stabilityCancellation


Solutions of Nonlinear Equations

Concepts

MultiplicitySensitivityConvergence rate

Basic methods

Interval bisection methodFixed-point iterationNewton’s methodSecant method, Broyden’s methodOther Newton-like method


Numerical Optimization

Concepts

Unconstrained optimization, constrained optimization (linearvs. nonlinear programming)Global vs. local minimumFirst- and second-order optimality conditionCoercive, convex, unimodality

Methods for unconstrained optimization

Golden section searchNewton’s method, Quasi-Newton methods (basic ideas)Steepest descent, conjugate gradient (basic ideas)

Methods for constrained optimization (especiallyequality-constrained optimization)

Lagrange multiplier for constrained optimizationLagrange function and its solutionLinear programming


Polynomial interpolation

Concepts

Existence and uniquenessInterpolation vs. approximationAccuracy; Runge’s phenomena

Methods

Monomial basisLagrange interpolantNewton interpolation and divided differenceOrthogonal polynomials


Documents

Big Review