Mètodes Numèrics: A First Course on Finite Elements

© Numerical Factory

Numerical Methods in Engineering

Dept. Matemàtiques ETSEIB - UPC BarcelonaTech

Linear Systems

1


Linear Systems: Iterative Methods

11 1 1 1

1 1

n n

m mn n m

a x a x b

a x a x b

+ + = + + =

a Coefficients

x Unknowns

b Independent terms

Linear System

2


Introduction

11 1 1 1

1

n

m mn n n

a a x b

a a x b

=

Ax b=

Matrix Form

3


Matrix Properties

4


MATRIX PROPERTIES

• Matrix Algebra:

– Matrix-Vector product: b=Ax

– Linear Mapping

5


MATRIX PROPERTIES

– Columns

– Alternative view of matrix-vector product

– b is a linear combination of the columns of A

6


MATRIX PROPERTIES

– Matrix- matrix product B=AC

– Matrix-vector product for each column of C

– Transpose of a matrix : changing rows by columns

𝐴𝑇

(𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵𝑇

(𝐴𝐵)𝑇 = 𝐵𝑇𝐴𝑇

(𝐴𝑇)𝑇 = 𝐴

7


MATRIX PROPERTIES

• Determinant:

– 2x2 dim

– Principal minors:

𝑎11 … 𝑎1𝑛⋮ ⋱ ⋮

𝑎𝑚1 ⋯ 𝑎𝑚𝑛

𝑎11 𝑎12𝑎21 𝑎22

= 𝑎11 · 𝑎22−𝑎12 · 𝑎21

Ak

8


MATRIX PROPERTIES

• Det. Properties:

• A is symmetric iff

• Trace: Sum of the diagonal elements

1

n

ii

i

a=

=

det( 𝐴 · 𝐵) = det( 𝐴) · det( 𝐵)

det( 𝐴𝑇) = det( 𝐴)

𝐴𝑇 = 𝐴

9


MATRIX PROPERTIES

• Rank: number of linear independent columns

– Maximum order of the nonvanishing determinant extracted from A.

– A has full rank if rank(A)= min(m,n)

– column rank = row rank

10


MATRIX PROPERTIES

• Rank: number of linear independent columns

– Maximum order of the nonvanishing determinant extracted from A.

– A has full rank if rank(A)= min(m,n)

– column rank = row rank

• Kernel: all solutions of Ax=0

rank + dim(Kernel) = m

11


MATRIX PROPERTIES

• Eigenvalues and Eigenvectors:

– Directions preserved by A

– Deformation induced by A

• Linear homogeneous system:

𝐴𝑣 = λ𝑣

(𝐴 − λ𝐼𝑑)𝑣 = 0

12


MATRIX PROPERTIES

• Inverse of a matrix: (for square matrices)

– If rank(A)=n (or det(A) ≠ 0), A is invertible.

– If A is not invertible is called singular

• A is orthogonal iff

(𝐴 · 𝐵)−1 = 𝐵−1 · 𝐴−1

(𝐴𝑇)−1 = (𝐴−1)𝑇

𝐴𝑇 = 𝐴−1

𝐴 · 𝐴−1 =𝐼𝑑

13


Matrix Types

14


Special Matrices Types

• Diagonal and block diagonal:

– Nonzero entries only at the diagonal

11

mn

a

a

11

mn

A

A

15



• Triangular Matrices:

– Lower triangular

11

21 22

31 32 33

1 1m mn mn

l

l l

l l l

l l l−

16



• Triangular Matrices:

– Upper triangular

11 12 1

22 23 2

33 3

n

n

n

nn

u u u

u u u

u u

u

17



• Tridiagonal Matrices:

11 12

21

1

1

n n

nn nn

a a

a

a

a a

−

−

18



• Hessemberg Matrices:

11 12 1

21

1

1

n

n n

nn nn

a a a

a

a

a a

−

−

19



• Banded Matrices:

11 1

1

q

p pn

mq mn

a a

a a

a a

20



• Sparse Matrix

Mainly zero elements

• Special data structures

– Compression

– Efficiency

21


Matrix Norms

22


MATRIX NORMS

• Vector norms:

(1,0)(-1,0)

(0,-1)

(0,1)

23


MATRIX NORMS

• Vector norms:

(1,0)(-1,0)

(0,-1)

(0,1)

24


MATRIX NORMS

• Vector norms:

(1,0)(-1,0)

(0,-1)

(0,1)

25


MATRIX NORMS

• Vector norms:

26


MATRIX NORMS

• Induced matrix norms: ∥ 𝐴 ∥𝑘= sup∥ 𝐴𝑥 ∥𝑘∥ 𝑥 ∥𝑘

27


MATRIX NORMS

28


Solving Linear Systems

29


Solving linear systems

• DIRECT METHODS:

– Solution is obtained after a finite number of steps

• Gaussian Elimination, LU, Cholesky, QR, SVD

• ITERATIVE METHODS:

– Solution is obtained as successive approaches

• Jacobi, Gauss-Seidel, Conjugate Gradient

30


Direct Methods

31


DIRECT METHODS

• Naïf approach:

• Easy solutions

– If A is Diagonal:

– If A is Orthogonal:

• Already Solved

– If A is Triangular:

1Ax b x A b−= ⎯⎯→ =

Tx A b=

1x D b−=

11 12 1 1 1

22 23 2

33 3

n

n

n

nn n n

u u u x b

u u u

u u

u x b

=

32


DIRECT METHODS

• If A is Triangular:11 12 1 1 1

22 23 2

33 3

n

n

n

nn n n

u u u x b

u u u

u u

u x b

=

33


DIRECT METHODS

• Gaussian Elimination-LU

– Inverse matrix

– Determinant

• Cholesky

• QR

• SVDAx b=

11 1 1 1

1

n

m mn n n

a a x b

a a x b

=

34


Gaussian Elimination - LU

35


Gaussian Elimination-LU

• Transform the original matrix to an easy form:

11 12 1

22 23 2

33 3

n

n

n

nn

u u u

u u u

u u

u

11 12 1

21 22 23 2

33 3

1

n

n

n

n nn

a a a

a a a a

a a

a a

36


11 12 1 11 12 1

1 1 1

21 22 23 2 22 23 2

33 3 33 3

1 1

0

n n

n n

n n

n nn n nn

a a a a a a

a a a a a a a

a a a a

a a a a

=


• Using a multiplier: 2121

11

al

a=

PIVOT

37


11 12 1 11 12 1

1 1 1

21 22 23 2 22 23 2

1 1

31 33 3 33 3

1 1

0

0

n n

n n

n n

n nn n nn

a a a a a a

a a a a a a a

a a a a a

a a a a

=


• Using now multiplier: 31

31

11

al

a=

38


11 12 1 11 12 1

1 1 1

21 22 23 2 22 23 2

1 1

33 3 33 3

1

0

0

0

n n

n n

n n

n nn nn

a a a a a a

a a a a a a a

a a a a

a a a

=


• Using now multiplier: 11

11

nn

al

a=

39


11 12 1

1 1 1

22 23 2

1 1

33 3

0

0 0

0 0 0 0

n

n

n

nn

a a a

a a a

U a a

a

=


• Finally:

Upper Triangular

1

11 1

1 1

n

nnnn n

n n

al

a

−

−− −

− −

=

Lower Triangular21

31 32

1 1

1

1

1

1n nn

l

L l l

l l −

=

40



• System solution from LU decomposition:

• Real Live: Permutations

– Troubles when pivot ≈ 0

·

1.

2.

A LU

Ax b LUx b

Ux y

Ly b

=

= ⎯⎯→ =

=

=

41



• Row permutations:

11 12 1

1 1 1

22 23 2

1 1

33 3

1

2

0

0

0

n

n

n

k

nn

a a a

a a a

a a

a

a

42


• Row permutations as matrix product

• Finally:

1 2 1

1 0 0 0

0 1

0 0 1

...0

1 0 0 0

0 0 0 0 0 1

k n

ij ij ij ijP P P P P −

= ⎯⎯→ =


· ·

1.

2.

T

T

P A LU

Ax b LUx P b

Ux y

Ly P b

=

= ⎯⎯→ =

=

=

43


• >> A=rand(6)

• A =

0.8147 0.2785 0.9572 0.7922 0.6787 0.70600.9058 0.5469 0.4854 0.9595 0.7577 0.03180.1270 0.9575 0.8003 0.6557 0.7431 0.27690.9134 0.9649 0.1419 0.0357 0.3922 0.04620.6324 0.1576 0.4218 0.8491 0.6555 0.09710.0975 0.9706 0.9157 0.9340 0.1712 0.8235

• >> [L,U,P]=lu(A)

L =

1.0000 0 0 0 0 00.1068 1.0000 0 0 0 00.8920 -0.6711 1.0000 0 0 00.9917 -0.4726 0.5368 1.0000 0 00.1390 0.9491 -0.0517 -0.2586 1.0000 00.6923 -0.5883 0.5947 0.8836 0.0470 1.0000

MATLAB-LU

U =

0.9134 0.9649 0.1419 0.0357 0.3922 0.04620 0.8676 0.9006 0.9302 0.1293 0.81850 0 1.4349 1.3846 0.4156 1.21410 0 0 0.6204 0.2068 -0.27890 0 0 0 0.6408 -0.51570 0 0 0 0 0.0953

P =

0 0 0 1 0 00 0 0 0 0 11 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 0 1 0

44


MATLAB-LU

b =

123456

>> y=L\P*b;

x=U\y;

A*x-b

ans =

1.0e-014 *

-0.7105

-0.7772

0

0

0.3553

0.7105

45


LU applications

46


Application: Inverse of a Matrix

• Inverse as n-linear systems:

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 33 3

1 1

1 0 0

0 1 0 0

0 1

0

0 0 1

n n

n n

n n

n nn n nn

a a a x x x

a a a a x x x x

a a x x

a a x x

=

A 1A− Id

47



• Inverse as n-linear systems:

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 33 3

1 1

1 0 0

0 1 0 0

0 1

0

0 0 1

n n

n n

n n

n nn n nn

a a a x x x

a a a a x x x x

a a x x

a a x x

=

A 1A− Id

48



• The cost is NOT n times a linear system:

– LU decomposition ONLY ONCE!!

– n times: solution of two triangular systems

iLy e

Ux y

=

=

·A LU=

0

1

0

iwhere e

=

49


Application: Determinant of a Matrix

• Determinant from LU decomposition:

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 33 3

1 1 1

1

1 0

1 · 0

0

1 0 0

n n

n n

n n

n nn n nn nn

a a a u u u

a a a a l u u u

a a l l u u

a a l l u−

=

50


Application: Determinant of a Matrix

• Determinant from LU decomposition:

From prop. det(AB) = det(A) · det(B)

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 33 3

1 1 1

1

1 0

1 · 0

0

1 0 0

n n

n n

n n

n nn n nn nn

a a a u u u

a a a a l u u u

a a l l u u

a a l l u−

=

51

det( 𝐴) = det( 𝐿) · det( 𝑈)/ det( 𝑃) = (−1)𝑝ෑ

1

𝑛

𝑢𝑖𝑖

𝑃 ∗ 𝐴 = 𝐿 ∗ 𝑈


Cholesky

52


• If A is a symmetric positive definite matrix

(i)

(ii)

(all principal minors >= 0)

• LU can be obtained without pivoting :

• Cost

LU: Cholesky Decomposition

0Tx Ax

TA A= A2

A3

·TA R R=

53

A4

>> R = chol(A)


QR - decomposition

54


Matrix Decomposition-QR

• Appropriate for non-square systems (overdetermined systems)

·

=

A Q

mxn mxm mxn

R

55


QR Decomposition

• Matrix A as a product of an Orthogonal matrix and an upper triangular one.

11 12 1 11 12 1 11 12 1

21 22 23 2 21 22 23 2 22 23 2

33 3 33 3 33 3

1 1

0

· 0

0

0 0

n n n

n n n

n n n

n nn n nn nn

a a a q q q r r r

a a a a q q q q r r r

a a q q r r

a a q q r

=

1TQ Q−= R

56



• Solution “only” a triangular system

·

T

A Q R

Ax b QRx b

Rx Q b

=

= ⎯⎯→ =

=

57



• Application: Regression line (least square sol.)

: 1 0r ax by+ + =

x xxx

xx

xx

xx xx

xx

xx

xx

1 1

2 2

1 0

1

·

1

i i

n n

ax by

x y

x y

a

b

x y

+ + =

−

=

−

58


SVD - decomposition

59


· .

=

Singular Value Decomposition-SVD

• , with U, V orthogonals and D diagonal

A U v

TA UDV=

mxn mxm mxn nxn

D

60



• Solution from SVD:

• >>svd

1

1

T

T

T T

T T

T

A UDV

Ax b UDV x b

DV x U b

V x D U b

x VD U b

−

−

=

= ⎯⎯→ =

=

=

=

61



• Singular values:

• rank(A) is the number of σi ≠ 0

• Norm

• Decomposition

1 2( , ,......., )nD diag=

1 2 ...... n

1 =

( ) ( ) ( )1 1 1 2 2 2· · ..... ·T T T

n n nu v u v u v

= + + +

·n

T

i i i

i

u v=

=

62



• Singular values:

• rank(A) is the number of σi ≠ 0

• Norm

• Decomposition

1 2( , ,......., )nD diag=

1 2 ...... n

1 =

( ) ( ) ( )1 1 1 2 2 2· · ..... ·T T T

n n nu v u v u v

= + + +

·n

T

i i i

i

u v=

=

63nxn matrix nxn matrix nxn matrix



• Low rank approximation:

• Compression: kx(m+n) instead of mxn

·k

T

i i i

i

u v

=

= 1k + − =

64



• For SVD we have

• In general singular values

• If A is symmetric and eigenvectors are the columns of V

• For square A

| |i i =

2( )Teig A A =

Av u=

1

| det( ) |n

i

i

A =

=

65



• Stability:

• Condition number:

– Better if

– In general

– Using Norm2

– >> Hilbert Matrix

1

n

=

|| || / || ||sup

|| || / || ||

x x

A A

=

)x x b b ( + )( + = +

· ||−

66


MATLAB-LU

67


Iterative Methods

68


ITERATIVE METHODS

• Compute successive approximations of the solution that “hopefully”converges to the real solution

• Appropriate for large systems (n>1000)

• Faster than direct methods

• Less memory requirements

• Handle special structures (sparse matrices)

(1) (2) ( ), ,....., nx x x

69


ITERATIVE METHODS

• Stationary Methods: Finds a splitting of the matrix , with M invertible

and iterate

converges iff

– Jacobi, Gauss-Seidel, Successive Overrelaxation (SOR)

A M K= +

( 1) 1 ( ) ( )

( )

( )k k k

Ax b

M K x b

Mx Kx b

x M Kx b Rx c+ −

=

+ =

= − +

= − + = +

( 1)kx + ( ) 1R

70


ITERATIVE METHODS

• Krylov subspace methods: use only multiplication by A (or A ) and find solutions in the Krylov subspace generated by

– Conjugate Gradient (CG)

– Generalized Minimal Residual (GMRES)

2 3 1, , , ,....., kb Ab A b A b A b−

T

71


STATIONARY METHODS

• Jacobi Method:

– Idea: to solve ith-unknown for the ith-equation

1 2 3

1 2 3

1 2 3

10 12

10 12

10 12

x x x

x x x

x x x

+ + =

+ + = + + =

( 1) ( ) ( )

1 2 3

( 1) ( ) ( )

2 1 3

( 1) ( ) ( )

3 1 2

1( 12)

10

1( 12)

10

1( 12)

10

k k k

k k k

k k k

x x x

x x x

x x x

+

+

+

= − − +

= − − +

= − − +

72


STATIONARY METHODS

• Jacobi: Split the original matrix11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 3

1 1 1

0 0 0 0

0 0 0 0

0 0 0

0 0

0 0 0 0

n n

n n

n n

n nn n nn nn

a a a a a a

a a a a a a a a

a a a a a

a a a a a−

= + +

( 1) 1 ( )

1 ( )

,

( )

( ( ) )

L D U

D L U

k k

k

D L U

A A A A

M A K A A

x M Kx b

A A A x b

+ −

−

= + +

= = +

= − + =

= − + +

A M K= +

73


STATIONARY METHODS

• Jacobi : Split the original matrix

>>AL=tril(A,-1)

>>AU=triu(A,1)

>>AD=diag(diag(A))

>>Xkp1=inv(AD)*(-(AL+AU)*Xk+b)

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 3

1 1 1

0 0 0 0

0 0 0 0

0 0 0

0 0

0 0 0 0

n n

n n

n n

n nn n nn nn

a a a a a a

a a a a a a a a

a a a a a

a a a a a−

= + +

A M K= +

74


STATIONARY METHODS

• Gauss-Seidel Method:

– Idea: to solve ith-unknown for the ith-equation using the previous updated unknowns

1 2 3

1 2 3

1 2 3

10 12

10 12

10 12

x x x

x x x

x x x

+ + =

+ + = + + =

( 1) ( ) ( )

1 2 3

( 1) ( 1) ( )

2 1 3

( 1) ( 1) ( 1)

3 1 2

1( 12)

10

1( 12)

10

1( 12)

10

k k k

k k k

k k k

x x x

x x x

x x x

+

+ +

+ + +

= − − +

= − − +

= − − +

75(Gauss-Seidel)


STATIONARY METHODS

• Gauss-Seidel: Split the original matrix11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 3

1 1 1

0 0 0

0 0 0

0 0

0 0

0 0 0

n n

n n

n n

n nn n nn nn

a a a a a a

a a a a a a a a

a a a a a

a a a a a−

= +

( 1) 1 ( )

1 ( )

,

( )

( )

LD U

LD U

k k

k

LD U

A A A

M A K A

x M Kx b

A A x b

+ −

−

= +

= =

= − + =

= − +76


STATIONARY METHODS

• Gauss-Seidel: Split the original matrix

>>AL=tril(A,-1)

>>AU=triu(A,1)

>>AD=diag(diag(A))

>>Xkp1=inv(AD+AL)*(-AU*Xk+b)

11 12 1 11 12 1

21 22 23 2 21 22 23 2

33 3 31 32 3

1 1 1

0 0 0

0 0 0

0 0

0 0

0 0 0

n n

n n

n n

n nn n nn nn

a a a a a a

a a a a a a a a

a a a a a

a a a a a−

= +

77


STATIONARY METHODS

• Successive Overrelaxation Method (SOR): The Gauss-Seidel step is extrapolated by a factor

where is the Gauss-Seidel iterate.

• If Gauss-Seidel

• If Overrelaxation

• If Underrelaxation.

=

78


STATIONARY METHODS

• Convergence:– If A is strictly row diagonal dominant

Jacobi and G-S converges.

– If A is symmetric positive definite G-S and SOR converges for

– In general is hard to choose, if the spectral radius of the Jacobi iteration matrix then the optimal is

– G-S can be twice as fast as Jacobi

| | | |ii ij

i j

a a

( )JR

2

1 1 ( )JR =

+ −

79


Conjugate Gradient

80


CONJUGATE GRADIENTS METHOD

• Optimization Problem:

– Solving the system is equivalent to

minimize the quadratic function:1

2

( ) T Tx x Ax x b = −

0Ax b− =

81



• Optimization Problem:

– The minimization can be done by line searches where is minimized along search direction ( )nx

np

82



– The that minimizes

is with residual

– The residual is also the Gradient of

– Simple approach: set the search direction to the negative gradient

1( )n n nx p ++

1

T

n nn T

n n

p r

p Ap + = n nr b Ax= −

1n +

( )nx

'( )n n nx Ax b r = − = −

np

nr83

1

2

( ) T Tx x Ax x b = −



The optimization procedure can be improved by better search directions:

Let the search directions be A-conjugate:

Then the algorithm converges in at most n steps.

0T

i kp Ap =

84



85

Only few storage vectors needed

Finds the best solution in norm


PRECONDITIONING

• The idea is to modify the initial system using a non-singular preconditioner matrix

• Convergence properties based on

• Trade-off between the cost of applying and the improvement of the convergence properties.

• Very usual:

Ax b=

1 1M Ax M b− −=1M A−

1M −

( )M diag A=

86


PRECONDITIONING

87

𝐩0 ≔ 𝑀−1𝐫0𝐳0: = 𝐩0

𝐳𝑘+1 = 𝑀−1𝐫𝑘+1

𝛽𝑘 ≔𝐫𝑘+1𝑇 𝐳𝑘+1

𝐫𝑘𝑇𝐫𝑘

𝐩𝑘+1 ≔ 𝐳𝑘+1 + 𝛽𝑘𝐩𝑘

Just change these few lines


CG code and Examples

88

function x = conjgrad (A, b, x, tol)

r = b - A * x;

p = r;

rsold = r' * r;

for i = 1:length(b)

Ap = A * p;

alpha = rsold / (p' * Ap);

x = x + alpha * p;

r = r - alpha * Ap;

rsnew = r' * r;

if sqrt(rsnew) < tol

break

end

beta=rsNew / rsOld;

p = r + beta * p;

rsold = rsnew;

end

end



89

Simple test example:

A = [5,-2,0;

-2, 5,1;

0, 1,5];

b = [20; 10; -10];

tol = 1.e-10;

x0 = zeros(size(A,2),1);

x = conjgrad (A, b, x0, tol)



90


Block Tridiagonal Matrix

91

Poisson matrix

𝐴𝑥 = 𝑏 with 𝑏 = ℎ2𝑓(𝑥𝑖 , 𝑦𝑖)where (𝑥𝑖 , 𝑦𝑖) are the nodes in the rectangular division NxN

Mètodes Numèrics: A First Course on Finite Elements

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