1

1.1 Basics

2

程式架構 (I) program {name}

{declarations}

{other statements}

stop

end program {name}

紅色部分為 Fortran 90的做法

3

程式架構 (II) we now show a program that computes

the surface area of a sphere from the formula A= : program area real r, Aarea

c this program reads a real number r c and prints the surface area of a sphere that c has radius r.

read(*,*) r Aarea = 4. * 3.14159 * r * r write(*,*) Aarea stop end

24 r

4

Column position rules(Fortran77) I

Columns 1 : : 空白

1 : 數字 , 代表行數 c : 當成註解

Columns 6 :第六行若為非零以外字元 (即 &), 表示本行接續上一行程式 EX. a=b-c &

& -d+e

此為 a=b-c-d+e

5

Columns 7-72 : 程式敘述區 (開頭需空六格 ) 程式敘述……… ..

Columns 73-80: 不使用 ,超過部份不被當成程式碼

Fortran 90無以上 columns的規定

Column position rules(Fortran77) II

6

Fortran 77 vs. 90

註解 :- c 以後的字元當成註解 (Fortran 77)- ! 以後的字元當成註解 (Fortran 90)

儲存檔案 :*.for(Fortran 77)*.f90(Fortran 90)

7

Read and Write Read :

read( * , * ) {variables}

代表 UNIT(輸出位置 ) FMT(輸出格式 )

Write :write(UNIT= ,FMT= ) {variables}

包裝字串 : 使用‘ ’單引號 (Fortran 77), “ ”雙引號 (Fortran90)

8

Type and Declaration 整數 :Integer {list-of-variables} 複數 :Complex {list-of-variables} 字元 :Character {list-of-variables} 邏輯變數 :Logical {list-of-variables} 浮點數 (單精準 ):Real {list-of-variables} 浮點數 (雙精準 ):Double precision {list-of-variables}

變數名稱可由 a~z, A~Z, 0~9組成 ,

但須注意 , 要以字母開頭且變數長度不超過 6 個

9

Integer Variables

In a typical computer might be allocated for each variable x with 32bits , , and then

x =

Ex.

3110 ,....,, bbb

}1,0{ , ).....()1( 201293031 ibbbbbb

13

2021212021

0110143210

2

)(

10

Floating Point Variables Numbers that are stored in real or double

precision variables are represented in Floating point style.

A typical 32-bit FP number x might have a 24-bit mantissa m and an 8-bit exponent e ,

64-bits computer has 56bits for mantissa.

2221023 )...... ()1( bbbbm

2243031 )....()1( bbbe emx 2

11

Arithmetic Assignment add(+) , subtract(-), multiply(*) ,

divide(/) :Ex. 7*4-6/3

exponential :Ex. 2**3 (2 的 3次方 =8)

build-function :Ex. sqrt(4) (4開平方 =2.000000)

12

String Manipulation I

宣告 :character * 7 str

字串 str長度為 7(初始設定為 1)

假設 s1 = ‘abcde’ , s2=‘xyz’ :- 連結 : s=s1//s2 s=‘abcdexyz’

- 指定 : s1(2:4)=‘123’ s1=‘a123e’- 抽出 : s=s1(2:5) s=‘bcde’

13

String Manipulation II 內建函數 ( 當 s=‘abcdabcd’) :

- 計算長度 (length): Ex. length(s)=8- 比對內文 (index): Ex. index(s,’da’) => s=abcdabcd => 4

index(s,’dc’) => s=abcdabcd => 0- 輸入數字 , 對照電腦使用的字元集 (char): Ex. char(72)=H- 輸入字元 , 對照電腦使用的字元集 (ichar): Ex. ichar(H)=72

Fortran 90可使用 len求字串長度

14

Type conversion

c-charr-reald-double precisioni-integerx-real,double precision,

integer

Ex. If ksum is an integer and sum is real then sum = real (ksum) converts ksum to a floating representation(real), and stores the result in sum.

15

Parameter Statement 在程式中 ,有時需要固定不變的資料常數 ,如圓周率 ,

重力加速度…等 .為了簡化程式 ,可直接宣告成 parameter.

Ex. program areareal r, area1 ,fourpiparameter (fourpi = 12.566e0)read(*,*) rarea1 = fourpi * r * rwrite(*,*) area1stopend

16

1.2 LOGICAL OPERATIONS

17

Logical Variables

Declaration : logical a,b

Assigned either the value .TRUE. Or .FALSE.Ex. a = .TRUE.

It could be show like “T” to .TRUE.“F” to .FALSE.

It could be also assigned to .true. or .false. with the lower case.

18

Logical Expressions I

We define some relational operators :.LT. Less than.LE. Less than or equal.EQ. Equal.NE. Not equal.GT. Greater than.GE. Greater than or equal

Ex. How to test if i between 3 and 7 ? Ans: a =(3 .LE. i).AND.(i .LT. 10)

19

Logical Variables II

Truth table (And ,Or , Not) :

20

“IF” Constructs I

The simplest ‘if’ statement :if ({logical expression}) {executable

statement}

Ex. The following statement prints count if count is positive.

if (count .GT. 0) write(*,*) count

21

“IF” Constructs II

if ( {logical expression} ) then{statement}

endif

Ex. The following statement prints count if it is positive

if (count .GT. 0) then

write(*,*) count

endif

22

“IF” Constructs III

The most general form of ‘if’ :

if ( {logical expression} ) then {statement}

elseif ( {logical expression} ) then {statement}

: elseif ( {logical expression} ) then

{statement} endif

23

“IF” Constructs IV A program reads three distinct integers and prints the

median:program medianinteger x,y,z,median1read(*,*)x,y,zif ( (x-z)*(x-y) .LT. 0 ) then

median1 = xelseif ( (y-x)*(y-z) .LT. 0 ) then

median1 = yelse

median1 = zendifwrite(*,*) median1stopend

24

Stylistic Considerations

Make the program statements involves indentation to enhances the readability of if-then-else constructs.

An example :

25

1.3 LOOPS

26

While-Loops I

{label} if ( { logical expression }) then{statements}

goto {label}endif

All the labels in a program must be unique, except in two program ,like between main program and subprogram.

The label is a number between 1 and 99999 and must be situated within columns 2 through 5.

27

While-Loops II

Example:Prints Fibonacci numbers that are strictly less than 200.

program fibon integer x, y, z x = 1 y = 1    10    if ( x .LT. 200 ) then write(*,*)x z = x + y y = x x = z goto 10 endif stop end

28

Until-Loops {label} continue

{statements} if ( { logical expression }) goto {label}

Example:Prints Fibonacci numbers that are strictly less than 200.

program fibonacci integer x, y, z x = 1 y = 1

10  continue z = x + y y = x x = z write(*,*)x if ( x .LE. 200 ) goto 10 stop end

29

Do-Loops I

do {label} {var} = { exp.1 },{ exp.2 },{ exp.3 }{statements}

{label} continue

var is the loop index.exp.1 is an initial value of var.exp.2 is the terminating bound.exp.3 defines the increment(or decrement).

label is code used to assign the loop range.

30

Do-Loops II

Example:Prints Fibonacci numbers from x1 to x10.

program fibonacci integer x, y, z x = 1 y = 1

do 10 count = 1, 10 z = x + y y = x x = z write(*,*)x

10 continue stop end

31

Do-Loops III

Example: compute s = n(n-1)…….(n-k).

s = n do 20 j = n-1, n-k, -1 s = s*j

20 continue

The do statement here says “step from n-1 to n-k in steps of –1 ”.

32

Nesting Do-Loops

Example:Prints all integer triplets (i,j,k) with the property that

.

do 10 i = 1, 100 do 5 j = 1, 100 k = int( sqrt( real( i*i + j*j ) ) ) if ( k*k .EQ. i*i + j*j ) then write(*,*)i, j, k endif

5 continue 10 continue

222 i k and 100ji1 j

33

More on “goto” and “continue”

The ‘no operation’ continue statement is useful for specifying the target of the goto. For Example 10 continue

{statement}

goto 10

cause control to return up to statement 10 and continue (downward) form there.

The goto statement should be used only as last resort since the presence of goto’s in a program makes for difficult reading.

A do-loop must be entered ‘from the top’.Never jump into the body of a do-loop.

34

Loops, Indentation, and Labels

Indentation :The body of a loop should be uniformly indented to highlight the program structure and it should be more

readability. Labels :

Labels should be assigned so that the sequence of numbers is increasing,top to bottom.It’s good practice to leave gaps between the value of consecutive label and the subsequence program will be easily accommodated.

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1.4 ARRAYS

36

One-dimensional Arrays I

Declaration{type} {name} (size)

Ex. real a(100)- one-dimensional- real array of size 100

37

One-dimensional Arrays II

Contentthe content of cell 1 is denoted by a(1)

Declaration

{name} ({first_index}:{last_index})

Ex. Claim a real array of size 100, with index range –41 to 58:

real b(-41:58)

38

Two-dimensional Arrays I

Declaration{name} (number of row, number of column)

Ex. integer A(4,5),identify a two-dimensional integer array of size 4x5:

(1,1) (1,2) (1,3) (1,4) (1,5)(2,1) (2,2) (2,3) (2,4) (2,5)(3,1) (3,2) (3,3) (3,4) (3,5)(4,1) (4,2) (4,3) (4,4) (4,5)

39

Two-dimensional Arrays II

Declaration

The total size of the array is(last_index1 - first_index1 + 1)*(last_index2 - first_index2 + 1)

integer k(20,100) integer k(1:20,1:100)

It’s the programmer’s responsibility to ensure that the index values are meaningful.Do not assume that array entries are automatically initialized to zero by the compiler.

x2}){last_inde:ex2}{first_indx1},{last_inde:dex1}({first_in {name}

40

Space Allocation for Two-dimensional Arrays I

Fortran stores a two-dimensional array as a contiguous, linear sequence of cells, by column.For example: 4x5 array

41

Space Allocation for Two-dimensional Arrays II

Note the storage of the 3-by-3 times table matrix in the 4-by-5 array results in unused array space:(the following matrix elements are not contiguous in memory)

42

Space Allocation for Two-dimensional Arrays III

Address of an array element isaddr[A(i,j)]=addr[A(1,1)]+(j-1)*adim+(i-1)

where A is an m-by-n matrix stored in array A that has row dimension adim.

43

Space Allocation for Two-dimensional Arrays IV

Physical address:Example:A cell may be 4 bytes long for integer arrays and 8 bytes long for double precision arrays.

Then,the physical address is addr[A(i,j)]=addr[A(1,1)]+[(j-1)*adim+(i-1)]*(4 or 8)byte

program frank integer F(15, 15), n, i, j do 25 n = 2, 6, 2 do 10 i = 1, n do 5 j = 1, n if ( i .LE. j ) then F(i, j) = n-j+1 elseif ( i .EQ. j+1 ) then F(i, j) = n-j else F(i, j) = 0 endif 5 continue10 continue do 15 i = 1, n write(*,*)(F(i, j), j=1, n)15 continue do 20 i = 1, n write(*,*)(F(j, i), j=1, n)20 continue25 continue stop end

45

Packed Storage

Packed form:It’s a useful data structure when dealing with

symmetric and triangular matrices,and only store the lower triangular portion in column-by-column fashion in a one-dimensional array.Ex.

k = 1

do 10 j = 1, n do 5 i = j, n c c a(k) = A(i, j) c

a(k) = i*j k = k+1 5 continue

10 continue

16

12

9

8

6

4

4

3

2

1

a

1

A

161284

12963

8642

432

46

Arrays of Higher Dimension It’s possible to manipulate arrays of dimension up to

seven.real A( 2, 3, 6, 2, 8, 7, 4)

means that there are 2x3x6x2x8x7x4=16128 memory locations are reserved.

Ex. Assign (abcd)2 to hcube(a,b,c,d) [binary to decimal] do 40 a = 0 , 1 do 30 b = 0 , 1 do 20 c = 0 , 1 do 10 d = 0 , 1

hcube(a,b,c,d)=8*a+4*b+2*c+d10 continue20 continue30 continue40 continue

47

1.5 SUBPROGRAMS

48

Built-in Functions

SpecificA specific function requires arguments of a particular type and returns a predefined type.

Ex. char(72) and we will have a string “H” Generic

A generic function accepts arguments of any appropriate type ;the return value is determined

by the types of the arguments.Ex. x = cos(y)

49

Functions I

Declaration{type} function {name} ({list-of-variables})

{Declaration}{statements}returnend

50

Functions II

Rules for user-defined function -The value produced by the function is returned to the

calling program via a variable whose name is the function’s name. Ex. f(x) is returned in the variable f

-A function has a type and it must be declared in the calling program.

-The body of a function resembles the body of a main program and the same rules apply.

-Execution begins at the top and flows to the bottom.Control is passed back to calling program when return statement is encountered the subroutine is exited.

-A reference to a defined function can appear anywhere in an expression,subject to type consideration.

51

Functions III

Solve the problem

program min1integer kreal m greater clarity m=-3.0e0do 10 k=1,10 x = real(k) m= min(m,((x-2.0)*x-7.0)*x-3.0)

10 continuewrite(*,*) mstopend

372)(

)}(),....0(min{),(23

xxxxf

nffnfm

real function f(x)real xf = ((x-2.0)*x-7.0)*x-3.0)returnend

program min2integer kreal m ,fm=f(0.0e0)do 10 k=1,10m= min(m,f(real(k) ) )

10 continuewrite(*,*) mstopend

52

Functions with Other Functions as Arguments

Solve the problem

program min3integer kreal m1,m2 m1=f1(0.0e0)do 10 k=1,10 greater clarity m1= min(m1,f1(real(k))

10continue m2=f2(0.0e0)do 20 k=1,20 m2= min(m2,f2(real(k))

20continuewrite(*,*) 'm(f1,10)=',m1,'m(f2,20)=',m2stopend

)20,( and )10,(

)}(),....0(min{),(

21 fmfm

nffnfm

real function fmin(f,n)integer nreal f fmin = f(0.0e0)do 10 k = 1,n fmin = min(fimn,f(real(k)))10 continuereturnend

program min4real m1,m2,f1,f2external f1,f2m1 =fmin(f1,10)m2 =fmin(f2,20)write(*,*) ‘m(f1,10)=’,m1,’m(f2,20)=’,m2stopend

53

Common I

When we want to solve problem for arbitrary n and arbitrary cubic ,we couldn't use fmin to compute m(f,n).The reason is that fmin expect a function with a single argument,not a function of the form f(x,a,b,c,d).So by placing variables “in common” they can be shared between the main program and one or more subprograms.

dcxbxaxxf 23)(

54

Common II

program min5

integer n

real m,a,b,c,d,f

common /coeff/ a,b,c,d

read (*,*)n,a,b,c,d

m = fmin(f,n)

write(*,*) m

stop

end

55

Common III The syntax for a common statement is as follows:

common /{ name } / { list-of-variables }different common block must have different names.

It’s legal for a variable to belong to more than one common block.

The variables listed in a common block are shared by all subprograms that list the block.

The common statement should be placed before any executable statements and declared in every subprogram using the common block.

The common variables don’t have to be named the same in each such routine, but they need to have same type and in the same order.

56

Subroutines I

Declarationsubroutine {name} ({argument list})

declaration{statements}end

A subroutine is called by a statement of the formcall { name } ({ argument list })

Unlike functions, subroutines do not have a type.The name of a subroutine does not return a value.

57

Subroutines II

subroutine fmin(f,n,value,point)integer n, pointreal f, valueinteger ireal temppoint = 0 value = f(0.0e0)do 10 i = 1,n temp = f(real(i)) if (temp .LT. value) then

point = i value = temp

endif 10 continue

returnend

program min6external finteger n,mptreal mvalread(*,*) n call fmin( f, n, mval, mpt)write(*,*)'m(f,n)=',mval,'mpt=',mpt stopend

58

Functions versus Subroutines I Every function can be put into “equivalent” subroutine form.

Ex. Prints program printreal a, b, c, d, x0, value, fread (*,*) a, b, c, d, x0value = f (a, b, c, d, x0)write (*,*) valuestopend

real function f(a, b, c ,d,x)real a, b, c, d, xf = ((a*x + b)*x + c)*x +dreturn end

dcxbxaxxf 23)(

59

Functions versus Subroutines II Change to “subroutine language”

program printreal a, b, c, d, x0, valueread (*,*) a, b, c, d, x0call f(a, b, ,c, d, x0, value)write (*,*) valuestopend

subroutine f(a, b, c ,d, x, value)real a, b, c, d, x, valuevalue= ((a*x + b)*x + c)*x +dreturn end

dcxbxaxxf 23)(

60

Save Ordinarily, the local variable values are lost when control

passed back to calling program.However, we can retain the value if it is named in a save statement.Ex. subroutine print(k)

integer kinteger lastksave lastkif (k .EQ. 0) then write (*,*) k lastk = kelseif (k .NE. lastk) then write (*,*) k lastk = k endifreturnend

61

Further Rules and Guidelines

Local variables exist only in the subprogram except they are named in a save statement.

Subprograms can be invoked by other subprogram as well as by the main program.

All functions used by a subprogram should be declared.This includes all specific build-in function.

To enhance readability ,there should be only one return in a subprogram.

Minimize the use of common.

62

1.6 ARRAYS AND

SUBPROGRAMS

63

Subprogram with Array Arguments I

A subroutine that performs matrix-vector multiplication

subroutine matvec(p, q, c, cdim, v, w ) integer p, q, cdim real v(*),w(*), C(cdim,*) integer i, j do 10 i = 1, p w(i) = 0.0e010 continue do 30 j = 1, q

do 20 i = 1, p w(i)=w(i)+c(i,j)*v(j)

20 continue30 continue return end

64

Subprogram with Array Arguments II

A main program that calls matvec:

integer idim, jdim parameter (idim = 50, jdim=40) integer i,j,m,n real A(idim,jdim), x(jdim),y(idim)

call matvec(m, n, A, idim, x, y)

stop end

65

Subprogram with Array Arguments III

The last dimension of an F77 array is not needed for address computation, so asterisks suffice.

Ex. The above subroutine , we claim some variables ,like

real v(*),w(*), C(cdim,*)

means v and w is one dimension array and C is two dimension array.

66

Different Dimensions I

The dimension of a passed array does not have to conform to its dimension in the calling program.

It’s perfectly legal to pass a two-dimensional array to a subprogram and then to treat it as one-dimensional in the subprogram.

67

Different Dimensions II

Ex.integer function prod(n,x,y)integer n,x(*),y(*)integer iprod = 0.do 10 i = 1, n prod = prod +x(i)*y(i)

10 continue return

end

Then calculate the AtA with A is an m*n matrix by function prodlength = m * nfrob2A = prod( length, A, A)

First of all ,We must sure m and adim have the same value.

68

Passing Submatrices

Ex.

then

161284

151173

141062

13951

A

1511

1410 )4:3 , 3:2(A

69

Stride

subroutine scale2(n , c, v, incv)integer n, incvreal c, v(*)integer j, k k=1do 10 j = 1, n

v(k)= c*v(k)k = k+incv

10 continuereturnend

suppose c and v(1,8) are initialized,then

call scale2(4 ,c ,v ,2) call scale2(4 ,c ,v(2) ,2) call scale2(2 ,c ,v(3) ,4)

scaled v(7) v(3),

scaled v(8)v(6),v(4), v(2),

scaled v(7)v(5),v(3), v(1),

70

A Note on Index Ranges

Declarationsubroutine sub2(x, p1, p2, A, q1, q2, r1, r2 ….)

integer p1, p2, q1, q2, r1, r2

real x(p1:p2), A(q1:q2,r1:r2)

Locationaddr[A(i, j)] = addr[A(q1, r1)] + (j-r1)*(q2-q1+1) +(i-q1)

71

Passing Arrays in Common Blocks Common blocks provide another way to communicate to

subprograms.[Like the global variable in C]

real function f(x)real u(100) , v(100)integer ncommon /fdate/ u, v, nreal s integer ks = 0.

do 10 k = 1 , n s = s + (u(k)*x – v(k)) ** 2

10 continuef = s- 1 returnend

common /fdate/ u, v, n

n integer

v(100) , u(100) real

mainprogram

72

1.7 INPUT AND OUTPUT

73

“ read” and “write” Statement

Declarationread ({unit number} , {format number}) {list-of-variables}write ({unit number} , {format number}) {list-of-variables}

The first argument indicates where the data is coming from or going to.

The second argument indicates the format of the data.

The asterisk(*) invokes certain convenient default options and it is at this simple level that we begin our discussion.

74

List-directed “read”

Declarationread ({unit number} , {format number}) {list-of-variables}

The format of data is “directed”( ) by the list of variables.Ex. read (*,*) i, j, m, x

read (*,*) y, z

if we set 10 , 5 , 6 , 1.0 -1.0 , 2.0

data items 10, 5 , 6 (i, j, m)are integer.data items 1.0, -1.0, 2.0(x, y, z) are real.

75

List-directed “write”

Declarationwrite ({unit number} , {format number}) {list-of-variables}

We can use single quotes(' ' ) to list the names of the variables along with their values.

Ex. suppose i=1,j=2,x=3,y=4,then when we usewrite(*,*) 'i=',i,',j=',j,',x=',x,',y=',y

and shows like this in the screeni = 1 , j = 2, x = 3 , y = 4

76

Formatted “read” read(*,100) i, j, k

100 format(I6, I6, I6)

There are three integer fields per line,each of the form “I6”.

“6” means six spaces wide. The spaces is includes sign bit.It means that we can’t show

-123456 with ‘I6’.

read(*,100) x, y100 format( E9.1, E10.3)

“E9.1” designates a floating number with length 9 bit including 1 bit mantissa.

It will show asterisk(*) when we don’t allow enough spaces.

77

Formatted “write”

write(* , 101)

write(* , 101)

101 format (1x,//, ' Upon termination : ')

102 format (1x,//, 2x, 'x= ', F8.4, 2x, 'y= ',

& F.4, 2X, 'i= ', I5, 2x, 'j= ', I5)

The slash(/) cause a blank line to be printed. Single quotes are used for names. ‘iX’ indicates that i spaces are to be skipped’

78

Location of “format” Statements

Collecting all the format statements in the program and placing them at the end just before and the end statement.

We recommend labels for all the format statements that appear in a program and encourage the reader interested in the formats to look for their specification at the end of the program.

79

Some Shortcuts

100 format (2x, I2, 2x, I2, 2x, I2)

100 format (3 (2x, I2) )

100 format (I5, I5, /E10.2/, /E10.2/)

100 format (2I5, 2(/E10.2), /)

80

The “data” Statement I

Declarationdata {list-of-variables /{list-of -values}/,…..

Ex. Show the following assignment:n=100, m=-5, x=2.0 ,y=2.0 ,z=2.0

data n/100/, m/-5/, x/2.0/, y/2.0/, z/2.0/data n,m/100,-5/, x,y,z/3*2.0/

81

The “data” Statement II

Ex. Write a code to assign 4-by-3 matrix of ones to A,set the first row of B to [1,2], assign a 4-vector of

zeros to c, and sets the second component of d to 1.

integer idim, jdim, kdim

parameter (idim = 4, jdim = 3, kdim = 2)

real A(idim, jdim), B(kdim, kdim), c(idim), d(kdim)

data A/12 *1.0/, B(1,1)/1.0/

data B(1,2)/2.0/, c/4*0.0/ , d(2)/1.0

A/12 *1.0/ means that matrix A has 12 ones in it.

82

Input and Output of Arrays It’s convenient to read in a m*n matrix :

read (*,*) ((A(i,j), i=1,m) , j=1,n)Ex. read (*,10) ((A(i,j), i=1,5) , j=1,6) 10 format (5 I3)

The example is the same with:j= 1,6 i=1,5

write (*, 10) A10 format (6 F7.4)

which results in 6 numbers per line.If A /1,2,3,4,5,6,7,8,9/ , it shows 1 2 3 4 5 6 -line1

7 8 9 -line2

83

1.8 COMPLEX ARITHMETIC

84

Declaring Complex Variables

Just as the complex number z = x+iy (i2 = -1), and ordered pair (x,y) of real numbers.The declaration

complex z Complex arrays are also possible.

Because complex variables take up twice as much space as real variables, memory constraints sometimes pose a problem when large complex array are involved,

85

Useful Built-in Functions

In a typical situation one often has to ‘set up’ a complex number from two real numbers.

z = complex (x,y)

assigns x+iy to z.Likewise,

x = real (z)

y = aimag(z)

86

Using Complex Variables

Complex variable can be manipulated just as can real and double precision variables.Ex. p = z**2 +z

is equivalent to x = real(z)

y = aimag(z)u = x*x – y*y + xv = 2.0e0*x*y+ yp = complex(u,v)

87

Input/Output

If z is complex write(*,100) z

100 format(‘real(z)=’, f10.7, ‘imag(z)=’, f10.7)

is equivalent tox = real(z)

y = aimag(z)

100 format(‘real(z)=’, f10.7, ‘imag(z)=’, f10.7)

where x and y are real.

88

Avoiding Complex Arithmetic Consider the following function that compute the absolute value

of the large root of the real quadratic equation x2+2bx+c = 0.real function maxrt( b, c ) real b, c d = b*b - c if (d .GE. 0.0e0) then maxrt = abs ( -b + sign(sqrt(d),-b)) else maxrt = sqrt ( b*b - d) endif

return end

this is preferable to the coded= csqrt( complex (b*b +cc,0.0e0)) maxrt = max (cabs(-b+d),cabs(-b-d))

csqrt(a) compute the square root of a,and return complex value.

89

1.9 PROGRAMMING TIPS

90

Intermediate Variables for Clarity and Efficiency I

Ex. If d2 = [r sin(b1) cos(a1) - r sin(b2)cos(a2)]2 +

[r sin(b1) sin(a1) - r sin(b2)sin(a2)]2 +

[r cos(b1) - r cos(b2)]2

Here are three ways to compute d. First: a straight encoding of the formula

d = sqrt (

& ( r*sin(b1)*cos(a1) - r*sin(b2)*cos(a2) )**2 +

& ( r*sin(b1)*sin(a1) - r*sin(b2)*sin(a2) )**2 +

& ( r*cos(b1) - r*cos(b2) )**2 )

91

Intermediate Variables for Clarity and Efficiency II

Second : Use some common subexpressions.

s1 = sin(b1)

s2 = sin(b1)

xdist = s1 * cos(a1) - s2*cos(a2)

ydist = s1 * sin(a1) - s2*sin(a2)

zdist = cos(b1) - cos(b2)

d = r*sqrt ( xdist **2 + xdist**2 + xdist**2)

92

Intermediate Variables for Clarity and Efficiency III

Third : Use some trigonometric identities.

t = cos(a1-a2)

d = r*sqrt (1. – t*cos(b1-b2) -

& cos(b1)*cos(b2)*(1. - t))

The third method is more difficult to read than first and second, and it could be addressed with sufficient comments.It is readability and efficiency ,but is difficult to reconcile.

93

Intermediate Variables for Clarity and Efficiency IV

A temporary array can lead to a more efficient computation.

For example, to set up n*n matrix F with fij = exp(-i –0.5j)

do 10 j=1,n

x(j) = exp(-float(j))

y(j) = exp(-.5 *float(j))

10 continue

do 20 j = 1,n

do 15 i = 1,n

F(i,j) = x(i) * y(j)

15 continue

20 continue

94

Checking Input Parameters

Make sure that the range is permissible.mmult(A, adim, ma, na, BB, bdim, mb, nb, C,

cdim)

if ( na .NE. mb) then

write(*,*) 'Product AB not defined'

return

elseif ( cdim .LT. ma)

write(*,*) 'Array C not big enough'

return

endif

95

Test Most Likely Conditions First

Depend on the comparison permutation

do 10 j=1,n do 5 i= 1,n

if (i .LT. j)then A(i,j)=-1elseif (i .GT. j)then

A(i,j)=0elseif (i .EQ. j)

A(i,j)=1 endif 5 continue10 continue

Avoid all ‘i-j’ comparison do 10 j = 2,n do 5 i =1,j-1 A(i,j) = -1 5 continue 10 continue do 20 j= 1,n A(j,j) = 1 20 continue do 30 j = 1,n-1 do 25 i = j+1,n A(i,j) = 0 25 continue 30 continue

Arrange the conditions in decreasing order of likelihood to minimize execution time.

96

Integer Arithmetic In order to reduce the amount of subscripting, we could do

two.1. Build the running sum in a scalar variable to save space.2. Let alphabets substitute into some fixed variable.

do 30 k =0 ,n-1 s(k) = 0. do 10 i = 0,n-k+1

s(k) = s(k) + x(i)*y(i+k) 10 continue

do 20 i = n-k, n-1 s(k) = s(k) +x(i)*y(i-n+k)

20 continue 30 continue

do 30 k =0 ,n-1 t = 0.

do 10 i = 0,n-k+1 t = t + x(i)*y(i+k)

10 continue kmn = k - n do 20 i = n-k, n-1

t = t +x(i)*y(i+kmn)

20 continue

s(k) = t 30 continue

97

Generality versus Efficiency I Efficient case :

real function dot1(n,x,y) integer n real x(*), y(*) integer i real s s = 0 do 10 i=1,n s = s +x(i)*y(i)

10 continue dot1=s return end

98

Generality versus Efficiency II General case :

real function dot2(n, x, incx, y incy) integer n, incx, incy real x(*), y(*) integer i, ix, iy real s ix = 1 iy = 1 s = 0 do 10 i = 1, n s = s + x(ix)*y(iy) ix = ix + incx iy = iy + incy

10 continue dot2 = s return end

99

Generality versus Efficiency III

Clear case :real function dot2(n, x, incx, y incy) integer n, incx, incy , i, ix, iy real x(*), y(*) ,s s = 0 if ( incx .EQ. 1 .AND. incy .EQ. 1 ) then do 5 i = 1, n

s = s + x(ix)*y(iy) continue else ix = 1 iy = 1 do 10 i = 1,n

s = s + x(ix)*y(iy) ix = ix + incx iy = iy + incy

continue endif

dot3 = s

return

end

100

Machine-independent Testing For Roundoff Noise I The unit roundoff u in a floating point system that

has t-bit mantissas is defined by u=21-t.We have some method to check the precision of the machine.

1.

But it only works in 59-bit floating point system.

endif

else

y)then*if(x.LT.u

55))*(*2uparameter(

101

Machine-independent Testing For Roundoff Noise II

2. A better solution is :

endif

else

y)then .GT. if(z

yxz

102

Termination Criteria I

real function exp1(x,tol)real x, tolreal s, terminteger ks = 1term = xdo 10 k = 1,30

s = s + term if ( abs(term) .LT. tol*abs(s) ) exp1 = s return endif term = term*(x/real(k))

10 continueexp1 = sreturnend

103

Termination Criteria II Some things wrong with above program.

1.’indefinite termination’ is better to use the while construct

2.’tol’ may underlie machine precision.

10 if ( k .LE. kmax .AND. big ) then

s = snew

k = k+1

term = term*(x/float(k))

snew = s +term

big = term .GE. tol*s .AND. snew .NE. s

goto 10

endif

exp2 = s

return

end

real function exp1(x,tol)

real x, tol , s, snew

integer kamx, k

parameter (kmax = 30)

logical big

s = 1.

k = 0

term = x

snew = s + term

big = term .GE. tol*s .AND. snew .NE. s

104

Computing Small Corrections

It’s better to compute the midpoint m of a and b,withm = a-(b-a)/2

rather than with the formulam = (a+b)/2

where a and b are nearby floating point numbers.

105

Pay Attention to the Innermost Loops Ex. We want to evaluate the polynomial

p(x) = a0+a1x2+a2x4+…+anx2n

using Horner’s rule.

s= a(n)do 10 k = n-1:-1:0 s = s*z*z + a(k)

10 continue

we add a number ‘w’ by getting the ‘z*z’ computation outside the loop

s= a(n)w = z*zdo 10 k = n-1:-1:0 s = s*w + a(k)

10 continue

106

Guarding against Overflow I Consider the computation of a cosine-sine pair(c,s) such

that –sx+cy=0, where x and y are given real numbers. A naïve solution:

d = sqrt ( x**2 + y **2 ) if d .GT. 0 c = x/d

s = y/d else c = 1 s = 0 endif

Overflow might occur in the computation of d

107

Guarding against Overflow II

A scaled versions of x and y:m = abs(x) + abs(y)

if m .NE. 0 then

x = x/m

y = y/m

d = sqrt( x*x + y*y )

c = x/d

s = y/d

else

c = 1

s = 0

endif

108

Guarding against Overflow III

Simplify scaled versions of x and y:if abs(y) .GT. abs(x) t = x/y s = 1./sqrt(1 + t*t) c = s*t elseif (abs(x) .GT. 0.) t = y/x c = 1./sqrt(1. + t*t) s = t*c else c = 1 s = 0 endif

109

Column-oriented versus Row-oriented Algorithm

Consider the matrix-vector multiplication y=Ax,where A is an m*n matrix and x is an n-vector. Compute y as follows

do 10 i = 1,m

y(i) = 0.

10 continue

do 30 j = 1,n

do 20 i = 1,m

y(i) = y(i) + A(i,j)*x(j)

20 continue

30 continue

The inner loop is row index , because arrays are stored by column in Fortran

110

2.1 BOOKKEEPING OPERATIONS

111

SCOPY and SSWAP SCOPY

call SCOPY (n, x, incx, y, incy)

ex. call SCOPY(15, x(10),1,y(20),1)y(19+1*i) x(9+1*i) i= 1:15

SSWAPcall SSWAP (n, x, incx, y, incy)

ex. call SSWAP(10, x(20),3,y(2),2)x(17+3*i) <--> y(2*i) i= 1:20

n : the loop from 1 to n x(index) : a vector starts from indexincx,incy : increment number

112

Example 2.1-1 MATRIX COPY

subroutine MATCOP ( m, n, B, bdim, A, adim )

integer m, n, bdim, adim

real B(bdim,*), A(adim,*)

integer k

do 10 k = 1,n

call SCOPY( m, B(1,k), 1, A(1,k), 1)

10 continue

return

end

113

Example 2.1-2 MATRIX TRANSPOSE

subroutine TRANSP( n, A, adim )

integer n, adim

real A(admin,*)

integer k

do 10 k= 1, n-1

call SSWAP( n-k, A(k+1,k), 1, A(k,k+1), adim)

10 continue

return

end

114

Example 2.1-3 CIRCULANT MATRIX

subroutine CIRCUL( n, C, cdim, v )

integer n, cdim

real C(cdim,*), v(*)

integer k

call SCOPY (n,v(1),1,C(1,1),1)

do 10 k = 2,n

call SCOPY( n-1, C(2,k-1), 1, C(1,k), 1)

C(n,k) = C(1,k-1)

10 continue

return

end

115

2.2 VECTOR OPERATIONS

116

Scaling Vectors SSCAL

To product of the form a*x, where a is a scalar and x is a vector:

call SSCAL (n, a, x, incx )

ex. call SSCAL (50, a, x(2),2)

x(2i) a*x(2i) i= 1:50

117

Adding a Multiple of One Vector to Another

SAXPYPerform the operation y a*x + y,where a is a scalar and x and y are vectors:

call SAXPY (n, a, x, incx, y, incy)

ex. call SAXPY (5, a, x(3), 2, y(10), 4) y(6+4i) a*x(1+2i) + y(6+4i)

i= 1:5

118

Inner Products SDOT

To compute the value of inner product:w = SDOT (n, a, x, incx, y, incy)

ex. W = call SDOT (50, x, 1, y, 1) w = x(1)y(1) + … + x(50)y(50)

ex. W = call SDOT (50, x, 2, y, 1) w = x(1)y(1) + x(3)y(2) + … + x(50)y(50)

+2

119

Example 2.2-1 Univariate least squares subroutine LS( m, a, b, x ) integer m real a(*), b(*), xreal SDOTx = SDOT( m, a, 1, a, 1 ) if ( x .NE. 0.0) then

x = SDOT( m, a, 1, a, 1 ) call SAXPY( m, -x, a, 1, b, 1) call SSCAL( m, x, a, 1)

endifreturn end

120

Example 2.2-2 Skew and symmetric Parts

subroutine SYMSKU( n, A, adim ) integer n,adim real A(adim, *) real half integer k,kpl,nmk parameter( half = .50e0) do 10 k = 1, n-1

kp1 = k + 1nmk = n – kcall SAXPY( nmk, 1., A(kp1,k), 1, A(k,kp1), adim )call SSCAL( nmk, half, A(k,kpl),adim ) call SAXPY( nmk, -1., A(k,kp1), adim, A(kp1,k), 1 )

10 continue return end

121

Example 2.2-3 Matrix-Vector Multiplicationsubroutine MMULT( m, n, A, adim, x, y, job ) integer m, n, adim, job real A(adim,*), x(*), y(*) real SDOT integer i,j if (job .EQ. 0) then do 10 i = 1,m

y(i) = 0.0 10 continue

do 20 j = 1,n call SAXPY( m, x(j), A(1,j), 1, y(1), 1)

20 continue else

do 30 j = 1,n y(j) = SDOT ( m, A(1,j), 1, x(1), 1)

30 continue endif return end

122

Example 2.2-4 Krylov Matrix

subroutine KRYLOV( n, A, adim, K, kdim, v, j )

integer n, adim, kdim, j

real A(adim,*), K(kdim,*), v(*)

integer p

call SCOPY( n, v(1), 1, K(1,1), 1 )

do 10 p = 2,j

call MMULT( n, n, A, adim, K(1,p-1), k(1,p), 0 )

10 continue

return

end

123

2.3 NORM COMPUTATIONS

124

1-Norms and 2-Norms SNRM2 & SASUM

The 2-norm of a real n-vector x :sw = SNRM2 (n, x, incx)

The 1-norm of a real n-vector x :sw = SASUM (n, a, x, incx)

ex. sw = SNRM2 (20, x, 1) w sqrt( |x(1)|2 + … + |x(20)|2 )

ex. sw = SASUM (20, x, 1) w |x(1)| + … + |x(20)|

125

Maximal Entry ISAMAX Find the index of element having max. absolute value:

imax = ISAMAX(n, x, incx)

ex. If x = (2, -1, 0, 6, -7)

imax = ISAMAX(5, x, 1)

imax = 5

imax = ISAMAX(2, x(2), 2)

imax = 2

126

Example 2.3-1 Vector Infinity-Norm

real function NORMI(x,n)

integer n

real x(*)

integer ISAMAX

real abs

NORMI = abs( x( ISMAX( n, x, 1) ) )

return

end

127

Example 2.3-2 Frobenius Matrix Norm

real function SNORMF( m, n, A, adim ) integer m, n, adim real A(adim,*)real v(2), SNRM2 integer k SNORMF = 0.0 do 10 k = 1,n

v(1) = SNORMF v(2) = SNRM2( m, A(1,k), 1 ) SNORMF =SNRM2 ( 2, v(1), 1 )

10 continue return

end

128

Example 2.3-3 Matrix 1-Norm

real function SNORM1( m, n, A, adim ) integer m, n, adim real A(adim,*) real s, SASUM integer k SNORM1 = 0.0 do 10 k = 1,n s = SASUM( m, A(1,k), 1 ) SNORM1 = max( s, SNORM1)

10 continue return end

129

Example 2.3-4 Householder Vector

subroutine HOUSE( n, x )

integer n

real x(*)

real alfa,beta,x1,SNRM2, abs

if (alfa .NE. 0.0) then

x1 = x(1)

x(1) = x1 + sign(alfa,x1)

beta = sqrt( 2.0 * alfa * (alfa + abs(x1)) )

call SSCAL( n, 1.0/beta, x(1), 1 )

else

x(1) = 1

endif

return

end

130

2.4 GIVENS ROTATIONS

131

Constructing a Givens Rotation I

SROTGIt can be used to construct a Givens rotation for the purpose of zeroing the second component of a prescribed 2-vector.

call SROTG ( a, b, c, s )

a, c ,and s :

b : if (a>b) then b=sesleif (b>a and c<>0) b=1/celse b=1

1sc0

r

b

a

cs

sc22

132

Constructing a Givens Rotation II

Consider an example: x= ( 1, 4, 7, 3 )We have : call SROTG( x(2), x(4), c, s)

5

3s

5

4c

5

34x52x

ba5

3sb

a5s3c4r5

4c

5

3s

1sc

0c3s4

rs3c4

1sc0

r

3

4

cs

sc

22

22

,,)(,)(

)(

)(

,

負不合

133

Reconstructing a Givens Rotation

subroutine ZTOCS( z, c, s ) real z,c,s if ( abs(z) .LT. 1 ) then

s = z c = sqrt(1. - s*s)

elseif ( abs(z) .GT. 1 ) then c = 1./z s = sqrt(1. - c*c)

else c = 0. s = 1.

endif return end

134

Applying a Givens Rotation SROT

call SROT(n, x, incx, y, incy, c, s)

means

call SROT(n, A(p,1), adim, A(q,1), adim, c, s)

call SROT(m, A(1,p), 1, A(1,q), 1, c, s)

cs

scyxyx

AscqpJA ),,,(

),,,( scqpAJA

135

Example 2.4-1 Zeroing an n-Vector

subroutine ZERO(n,x)

integer n

real x(*)

real c,s

integer k

do 10 k = n-1,1,-1

call SROTG(x(k),x(k+1),c,s)

10 continue

return

end

136

Example 2.4-2 Factored Q Times Vector

subroutine APPLY( n, x, y )

integer n

real x(*), y(*)

real c, s

integer k

do 10 k = n-1, 1, -1

call ZTOCS( x(k+1), c, s )

call SROT( 1, y(k), 1, y(k+1), 1, c, s )

10 continue

return

end

137

Example 2.4-3 Hessenberg QR Factorization

subroutine HESS( n, A, adim)integer n, adimreal A(adim,*)

real c, s integer j, jp1 do 20 j = 1, n-1

jp1 = j + 1call SROTG( A(i,j), A(jp1, j), c, s )

call SROT( n-j, A(j,jp1), adim, & A(jp1,jp1), adim, c, s)20 continue

return end

138

2.5 DOUBLE PRECISION ANDCOMPLEX VERSIONS

139

Copying a Vector call SCOPY(n, sx, incx, sy, incy)

call DCOPY(n, dx, incx, dy, incy)call CCOPY(n, cx, incx, cy, incy)call ZCOPY(n, zx, incx, zy, incy)

s : real single precisiond : real double precisionc : complex single precisionz : double precision

140

Interchanging Vectors call SSWAP(n, sx, incx, sy, incy)

call DSWAP(n, dx, incx, dy, incy)call CSWAP(n, cx, incx, cy, incy)call ZSWAP(n, zx, incx, zy, incy)

s : real single precisiond : real double precisionc : complex single precisionz : double precision

141

Multiplying a Vector by a Scalar

call SSCAL(n, sa, sx, incx)call DSCAL(n, da, dx, incx)call CSCAL(n, ca, cx, incx)call ZSCAL(n, za, zx, incx)

Complex casecall CSSCAL(n, sa, cx, incx)call ZDSCAL(n, da, zx, incx)

142

Inner Product sw = SDOT(n, sx, incx, sy, incy)

dw = DDOT(n, dx, incx, dy, incy)

complex casefor xHy :

cw = CDOTC(n, cx, incx, cy, incy)zw = ZDOTC(n, zx, incx, zy, incy)

for xTy : cw = CDOTU(n, cx, incx, cy, incy)zw = ZDOTU(n, zx, incx, zy, incy)

143

Adding a Multiple of One Vector to Another

call SAXPY(n, sa, sx, incx, sy, incy)

call DAXPY(n, da, dx, incx, dy, incy)

call CAXPY(n, ca, cx, incx, cy, incy)

call ZAXPY(n, za, zx, incx, zy, incy)

144

2-Norm sw = SNRM2(n, sx, incx)

dw = DNRM2(n, dx, incx)

sw = SCNRM2(n, cx, incx)

dw = DZNRM2(n, zx, incx)

145

1-Norm sw = SASUM(n, sx, incx)

dw = DASUM(n, dx, incx)

sw = SCASUM(n, cx, incx)

dw = DZASUM(n, zx, incx)

146

Maximal Entry imax = ISAMAX(n, sx, incx)

imax = IDAMAX(n, dx, incx)

imax = ICAMAX(n, cx, incx)

imax = IZAMAX(n, zx, incx)

147

Constructing a Givens Rotation

call SROTG( sa, sb, sc, ss)

call DROTG( da, db, dc, ds)

148

Applying a Givens Rotation call SROT(n, sx, incx, sy, incy, sc, ss)

call DROT(n, dx, incx, dy, incy, dc, ds)

call CSROT(n, cx, incx, cy, incy, sc, ss)

call ZDROT(n, zx, incx, zy, incy, dc, ds)

149

Example 2.5-1 Complex Rotation

subroutine SCROTG( ca, cb, sc1, ss1, sc2, ss2)complex ca,cbreal sc1,ss1,sc2,ss2 real a1, a2, b1, b2

real real aimag a1 = real(ca)a2 = aimag(ca)b1 = real(cb)b2 = aimag(cb) call SROTG( a1, a2, sca, ssa )

call SROTG( b1, b2, scb, ssb )call SROTG( a1, b1, sc1, ss1 )

sc2 = sca*scb + ssa*ssb ss2 = ssa*scb - sca*ssb return end

150

3.1 TRIANGULAR SYSTEMS

151

Solving Triangular Systems I STRSL

It can be used to solve triangular systems of the form Tx=b and Ttx=b,where T is either upper or lower triangular.

call STRSL( T, tdim, n, b, job, info )

[Input]T : The triangular matrix T.tdim : The row dimension of the array T.n : The dimension of matrix T.b : The right-hand side.job : Indicate the type of system to be solved.

[Output]b : If T is nonsingular.Otherwise unchanged.info : Return a singularity flag.

152

Solving Triangular Systems II [job]

00 solve Tx=b, where T is lower triangular.

01 solve Tx=b, where T is upper triangular.10 solve Ttx=b, where T is lower triangular.11 solve Ttx=b, where T is upper triangular.

[info]1. T is nonsingular, info is zero.2. T is singular , info returns the smallest k such that tkk is zero.

153

Solving Triangular Systems III

EX: To solve an upper Triangular system

program ANSWER1 real T(3,3),B(3,1) DATA T /1,0,0,2,1,0,3,-1,2 / DATA B /14,-1,6 /

integer info call strsl(T,3,3,B,01,info) ! info = singular flag & B = T**(-1)*b

write(*,10) info,B10 format (1X,I1,/, G10.3, G10.3, G10.3)

stop end

6

1

14

x

x

x

200

110

321

3

2

1

154

Condition Estimation STRCO

It can predict that a matrix may be close to a singular matrix. If so, it will return a ‘unmeaning and sufficiently small value’ rcond.

call STRCO( T, tdim, n, rcond, z, job )

[Input]T : The triangular matrixtdim : The row dimension of the array T.n : The dimension of matrix T.job : Indicate whether T is upper or lower

triangular( 0=lower, 1=upper ) .

[Output]rcond : Return estimate of 1/k1(T).

z : Return 1-norm z so ||Tz||1 = rcond||z||1.

155

Determinant and Explicit Inverse I

STRDITo solve the inverse or determinant of triangular matrix.

call STRDI( T, tdim, n, d, job, info )

[Job]It contains three-digit integer ABC.A :if A=1, We'll compute the determinant of T. A=0, No determinant computation.B :if B=1, compute the T-1

B=0, No inverse computation.C :if C=1, T is upper triangular. if C=0, T is lower triangular.

156

Determinant and Explicit Inverse II

EX: Solve the Determinant and inverse of

program teststrdi real T(2,2),d(2,1) DATA T /3,0,4,2 /

integer info call strdi(T,2,2,d,111,info)

write(*,10) info ! singular flagwrite(*,20) d(1,1)*10**d(2,1)!determinantwrite(*,30) T ! inverse of T

10 format (1X,I2)20 format (1X,G10.3)30 format (1X,G10.3, G10.3, G10.3, G10.3)

stop end

20

43

157

Example3.1-1 Multiple Right-hand Sides

subroutine TINVB( T, tdim, n, B, bdim, p, info, rcond, z )integer tdim, n, bdim, p, inforeal T(tdim,*), B(bdim,*), rcond, z(*)integer jreal rcond1call STRCO( T, tdim, n, rcond, z, 1 )rcond1 = 1.0e0 + rcondif ( rcond1 .EQ. 1.0e0 ) then

info = 1else

info = 0do 10 j = 1, p

call STRSL( T, tdim, n, B(1,j), 01, info )10 continue

endifreturnend

158

3.2 General SYSTEMS

159

LU Factorization I SGECO

It computes the factorization PA=LU via Gaussian elimination with partial pivoting.

call SGECO( A, adim, n, ipvt, rcond, z )

[Input]A : n*n matrix.adim : The row dimension of A.n : The dimension of the matrix A.

[Output]A : Return L and U.ipvt : In order to compute P.rcond : Return estimate of 1/k1(T).

z : Return 1-norm z so ||Tz||1 = recond||z||1.

160

LU Factorization II SGEFA

If the condition is not required ( factorization PA=LU), we can use SGEFA instead of SGECO to save time.

call SGEFA( A, adim, n, ipvt, info )

[info] = 0 : No zero pivots encountered. Use SGESL ， SGEDI safely.= k : Ukk=0, k is the address of the last zero element.

161

Back-substitution/Forward-elimination

SGESLTo solve either the system Ax=b or Atx=b.

call SGESL( A, adim, n, ipvt, b, job )

[Input]A : Contain A’s LU decomposition from SGECO or SGEFA.ipvt : The pivot vector from SGECO.b : The right-hand side.job := 0 , solve Ax=b. = 1 , solve Atx=b.

[Output]b : Return solution to requested system.

162

Explicit Inverse and Determinate I

SGEDICalculate inverse and determinant of A.

call SGEDI( A, adim, n, ipvt, d, work, job )

[Input]

job : = 01 , Only inverse. = 10 , Only determinant. = 11 , inverse and determinant.

[Output]A : if job=0, A=A. Otherwise A= A-1.d : if job=x1,return a two-vector determinant

d=d(1)*10.0**d(2).

163

Explicit Inverse and Determinate II

EX: Solve the inverse of the matrix( use SGECO , SGEDI)

program testLU real A(3,3),rcond,z(3,1),work(3,1),d(2,1)

integer ipvt(3,1) DATA A /1.0, 1.0, 1.0, 3.0, 3.0, 4.0, 3.0, 4.0, 3.0/ call sgeco(A,3,3,ipvt,rcond,z) !PA = LU

call sgedi(A,3,3,ipvt,d,work,11)write(*,10) ((A(i,j),j=1,3),i=1,3) !inverse of Awrite(*,20) ipvtwrite(*,30) d(1,1)*10**d(2,1) !determinant of A

10 format (1X,G20.3, G20.3, G20.3,/,G20.3, G20.3, G20.3,/, G20.3, G20.3, G20.3)

20 format (1X,I1, I1, I1)30 format (1X,G10.3) stop end

341

431

331

164

Example3.2-1 Ax=b Solver with Condition Estimate

subroutine SGSOL( A, adim, n, ipvt, rcond, z, b, info)integer adim, n, ipvt(*), info real A(adim,*), b(*), rcond, z(*) real rcond1 call SGECO( A, adim, n, ipvt, rcond, b ) rcond1 = rcond + 1.0e0 if ( rcond1 .GT. 1.0e0 ) then

info = 0 call SGESL( A, adim, n, ipvt, b, 0 )

else info = 1

endif return end

165

3.3 SYMMETRIC SYSTEMS

166

Cholesky Factorization I SPOCO

To compute the Cholesky factorization A=RtR of a symmetric positive definite matrix.

CALL SPOCO (A, adim, n, rcond, z, info)

[Input]info := 0 , R is found.

> 0 , A is not positive definite.

167

Cholesky Factorization II SPOFA

If the condition is not required(symmetric positive definite matrix), we can use SPOFA instead of SPOCO to save time.

CALL SPOFA (A, adim, n, info)

168

Cholesky Factorization III SPOSL

The output of either SPOCO or SPOFA can be used(info=0) to solve Ax=b. Here we can use SPOSL to solve it.

CALL SPOSL (A, adim, n, b)

[Input]A : Either the matrix of SPOCO or SPOFA.info : = 0 , R is found.

> 0 , A is not positive definite.

169

Example 3.3-1 Symmetric Positive Definite System Solver

subroutine SPOSOL( A, adim, n, b, rcond, z, info )integer n, adim, info real A(adim,*), b(*), rcond, z(*) real rcond1 call SPOCO( A, adim, n, rcond, z, info ) rcond1 = rcond + 1.0e0 if ( info .EQ. 0 .AND. rcond, z, info ) then call SPOSL( A, adim, n, b ) elseif( info .EQ. 0 .AND. rcond1 .GT. 1.0e0) then

info = -1 endif return end

170

Diagonal Pivoting Method I SSICO

To solve symmetric indefinite systems with A =UDUt.

CALL SSICO(A, adim, n, ipvt, rcond, z) [Input]

A : the symmetric matrix. [Output]

A : U and D.ipvt : the pivot information.rcond : 1/k1(A) estimate.

171

Diagonal Pivoting Method II SSIFA

If the condition estimate is not desired then use :

CALL SSIFA( A, adim, n, ipvt, rcond, k)

[Output]k : = 0, A is nonsingular.

> 0, kth diagonal block of D is singular.

172

Diagonal Pivoting Method III SSISL

After the output of either SSICO or SSIFA can be solved the symmetric indefinite systems Ax=b.

CALL SSISL( A, adim, n, ipvt, b)

[Input]A : from SSICO or SSIFA.b : the right-hand side.

[Output]ipvt : the pivot information from SSICO or SSIFA.b : A-1b.

173

Example3.3-2 Symmetric Indefinite System Solver

subroutine SSISOL( A, adim, n, ipvt, rcond, z, b, info)integer adim, n , ipvt(*), info real A(adim,*), b(*), rcond, z(*) real rcond1 call SSICO( A, adim, n, ipvt, rcond, z ) rcond1 = 1.0e0 + rcondif(rcond1 .GT. 1.0e0) then

info = 0 call SSISL( A, adim, n, ipvt, b )

else info = -1

endif return end

174

Inverses,Determinant,Inertia I

SPODIAfter the Cholesky factorization has been calculated via SPOCO or SPOFA, then we can use SPODI to solve the inverse and/or determinant of a symmetric positive matrix.

call SPODI ( A, adim, n, det, job )

[Input] job : =11, inverse and determinant .

=10, determinant . =1, inverse . [Output]

A : If job=1 or 11 , A will return the upper triangular matrix.

175

Inverses,Determinant,Inertia II

SPIDILike the SPODI, but in an indefinite case.

call SPIDI ( A, adim, n, kpvt, det, inert, work, job )

[Output]inert : an integer three-vector

inert (1) = the number of positive eigenvalues.

inert (2) = the number of zero eigenvalues. inert (3) = the number of negative

eigenvalues.

176

Packed Storage We could store upper triangular matrix column by

column . See 1.4

Packed-storage version of SPOFA, SPODI, SSIFA, and SSIFI also exist.

Packed version Conventional version

SPPCO( AP, n, rcond, z, info)SPPSL( AP, n, b)SSPCO( AP, n, ipvt, rcond, z)SSPSL( AP, n, ipvt, b)

SPOCO( A, adim, n, rcond, z, info)SPOSL( A, adim, n, b)SSICO( A, adim, n, ipvt, rcond, z)SSISL( A, adim, n, ipvt, b)

177

Cholesky with Pivoting SCHDC

Decomposition PAPt = GGt, where A is symmetric positive semidefinite and P is a permutation matrix.

call SCHDC ( A, adim, n, work, ipvt, job, info )

[Input]job : =0 without change pivot =1 otherwise

178

Updating and Downdating the Cholesky Factorization

SCHUD . SCHDD . SCHEXSuppose the Cholesky factorization A = GGt and x is n-vector.We can use a fast subroutine SCHUD(or SCHDD) to compute Cholesky factorization.

call SCHUD (R,adim,P,X,Z,LDZ,NZ,Y,RHO,c,s)call SCHDD

(R,adim,P,X,Z,LDZ,NZ,Y,RHO,C,S,INFO)

SCHUD -> A+=A+XXt

SCHUD -> A-=A-XXt

The subroutine SCHEX is frequently used in conjunction with SCHUD and SCHUD.

179

3.4 BANDED SYSTEMS

180

Band Gaussian Elimination Here are the corresponding banded versions(Full

general case in 3.2):

Full : SGECO( A, adim, n, ipvt, rcond, z )Banded: SGBCO( A, adim, n, p, q, ipvt, rcond, z )

Full : SGEFA( A, adim, n, ipvt, info )Banded: SGBFA( A, adim, n, p, q, ipvt, info)

Full : SGESL( A, adim, n, ipvt, b, job )Banded: SGBSL( A, adim, n, p, q, ipvt, b, job )

Full : SGEDI( A, adim, n, ipvt, b, job )Banded: SGBDI( A, adim, n, p, q, ipvt, b, job )

181

Band Cholesky Here are the banded analogs(Full positive

definite system case in 3.3):

Full : SPOCO( A, adim, n, rcond, z, info )Banded: SPBCO ( A, adim, n, q, rcond, z, info )

Full : SPOFA( A, adim, n, info )Banded: SPBFA( A, adim, n, q, info )

Full : SPOSL( A, adim, n, b )Banded: SPBSL( A, adim, n, q, b )

Full : SPODI( A, adim, n, det, job )Banded: SGECO( A, adim, n, q, det )

182

Tridiagonal Systems I SGTSL

Consider the tridiaginal system:

we can invoke SGTSL to solve linear system by using Gaussian elimination with partial pivoting.

call SGTSL( n, c, d, e, b, info )

n

1n

2

1

n

1n

2

1

nn

1n1n1n

22

11

b

b

b

b

x

x

x

x

dc

edc

dc

ed

183

Tridiagonal Systems II SPTSL

If the matrix A is symmetric positive define, then we use SPTSL.

call SPTSL( n, d, e, b)

b is overwritten with the solution to Ax = b.

184

Example 3.4-1 Symmetric Positive Definite Band Solver

subroutine SPBSOL ( A, adim, n, q, rcond, z, b, info )

integer n, q, adim, info

real A(adim,*), b(*), z(*), rcond

real rcond1

call SPBCO( A, adim, n, rcond, z, info )

rcond1 = rcond + 1.0e0

if ( info .EQ. 0 .AND. rcond1 .GT. 1.0e0 ) then

call SPBSL( A, adim, n, q, b )

else

info = -1

endif

return

end

185

3.5 THE QR FACTORIZATION

186

SQRDCTo compute the QR factorization with no column pivoting :call SQRDC( A, adim, m, n, qraus, ipvt, work, job )

SQRSLcall SQRSL( A, adim, m, k, qraus, b, Qb,

QTb, x, rsd, Ax, job, info )

187

Example3.5-1 Full Rank Least Squares Solver

subroutine LS( A, adim, m, n, b, x, rcond, z, info, qraux )

integer adim, m, n, info

real A(adim,*), b(*), x(*), rcond, z(*), qraux(*)

real rcond1

call SQRDC( A, adim, m, n, qraux, 1, z, 0 )

call STRCO( A, adim, n, rcond, z, 1 )

rcond1 = rcond + 1.0e0

if ( rcond .GT. 1.0e0 ) then

call SQRSL( A,adim,m,n,qraux,b,z,b,x,b,z,110,info )

else

info = -1

endif

return

end

188

Example3.5-2 Underdetermined System Solver

subroutine UND(A,adim,m,n,b,x,rcond,z,info,qraux,ipvt)integer adim, m, n, info, ipvt(*)real A(adim,*), b(*), x(*), z(*), qraux(*), rcondreal rcond1integer kcall SQRDC( A, adim, m, n, qraux, ipvt, z, 1 )call STRCO( A, adim, m, rcond, z, 1 )rcond1 = rcond + 1.0e0if (rcond1 .GT. 1.0e0) then

call SQRSL(A,adim,m,m,qraux,b,z,b,b,b,z,100,info)

do 30 k = 1, nx(k) = 0if (ipvt(k) .LE. m) x(k) = b(ipvt(k))

30 continueelse info = -1endifreturnend

189

3.6THE SINGULAR VALUE

DECOMPOSITION

190

SSVDCThe singular value decomposition A = USVt of an m-by-n matrix A can be computed using the subroutine SSVDC :call SSVDC( A, adim, m, n, s, e, U, udim,

V,vdim, work, job, info )

191

Job Identification

A U V

Effect

0000010110112020212121

A A AA U UA A VA A AA U AA U VA U AA A AA U VA U AA A V

No U or V.No U or V. A saved.Matrix V returned in V.Matrix V returned in top of A.Matrix U returned in U. A saved.Matrix U and V returned in U and V. A saved.Matrix U1 returned in U. A savedMatrix U1 returned in A.Matrix U1 and V returned in U and V. A saved.Matrix U1 and V returned in U and A. A saved.Matrix U1 and V returned in A and V.

192

Example3.6-1 Rank Deficient Least Squaressubroutine LSSVD(a,adim,m,n,s,e,V,vdim,b,tol,x,info)integer adim, m, n, vdim, inforeal A(adim,*), s(*), e(*), V(vdim,*), b(*), tol, x(*)integer i, jreal ujtbcall SSVDC( a,asim,m,n,s,e,A,adim,V,vdim,x,21,info )if (info .NE. 0) then

do 5 i = 1, nx(i) = 0

5 continue do 50 j = 1, n if (s(j) .GE. tol) then

ujtb = SDOT( m, a(1,j), 1, b(1), 1 )call SAXPY( n,ujtb/s(j),V(1,j),a,x(a),1 )

info = info – 1 endif50 continue endif return end

193

3.7 DOUBLE PRECISION AND COMPLEX

VERSIONS

194

Triangular SOLVE

STRSL( SA, adim, n, sb, job, info )DTRSL( DA, adim, n, db, job, info )CTRSL( CA, adim, n, cb, job, info )ZTRSL( ZA, adim, n, zb, job, info )

  Condition

STRCO( SA, adim, n, srcond, sz, job )DTRCO( DA, adim, n, drcond, dz, job )CTRCO( CA, adim, n, crcond, cz, job )ZTRCO( ZA, adim, n, zrcond, zz, job )

195

General I Factor and Condition

SGECO( SA, adim, n, ipvt, srcond, sz ) DGECO( DA, adim, n, ipvt, drcond, dz ) CGECO( CA, adim, n, ipvt, crcond, cz )

ZGECO( ZA, adim, n, ipvt, zrcond, zz )  Factor

SGEFA( SA, adim, n, ipvt, info ) DGEFA( DA, adim, n, ipvt, info ) CGEFA( CA, adim, n, ipvt, info ) ZGEFA( ZA, adim, n, ipvt, info )

196

General II Solve

SGESL( SA, adim, n, ipvt, sb, job ) DGESL( DA, adim, n, ipvt, db, job ) CGESL( CA, adim, n, ipvt, cb, job ) ZGESL( ZA, adim, n, ipvt, zb, job )

Inverse and DeterminantSGEDI( SA, adim, n, ipvt, sdet, swork, job )

DGEDI( DA, adim, n, ipvt, ddet, dwork, job ) CGEDI( CA, adim, n, ipvt, cdet, cwork, job ) ZGEDI( ZA, adim, n, ipvt, zdet, zwork, job )

197

General Banded I Factor and Condition

SGBCO( SA, adim, n, p, q, ipvt, srcond, sz) DGBCO( DA, adim, n, p, q, ipvt, drcond, dz) CGBCO( CA, adim, n, p, q, ipvt, crcond, cz) ZGBCO( ZA, adim, n, p, q, ipvt, zrcond, zz)   Factor

SGBFA( SA, adim, n, p, q, ipvt, info ) DGBFA( DA, adim, n, p, q, ipvt, info ) CGBFA( CA, adim, n, p, q, ipvt, info ) ZGBFA( ZA, adim, n, p, q, ipvt, info )

198

General Banded II Solve

SGBSL( SA, adim, n, p, q, ipvt, sb, job ) DGBSL( DA, adim, n, p, q, ipvt, db, job ) CGBSL( CA, adim, n, p, q, ipvt, cb, job ) ZGBSL( ZA, adim, n, p, q, ipvt, zb, job )

Inverse and DeterminantSGBDI( SA, adim, n, p, q, ipvt, sdet, swork, job )DGBDI( DA, adim, n, p, q, ipvt, ddet, dwork, job )

CGBDI( CA, adim, n, p, q, ipvt, cdet, cwork, job ) ZGBDI( ZA, adim, n, p, q, ipvt, zdet, zwork, job )

199

General Tridiagonal SGTSL( n, sc, sd, se, sb, info )

DGTSL( n, dc, dd, de, db, info )

CGTSL( n, cc, cd, ce, cb, info )

ZGTSL( n, zc, zd, ze, zb, info )

200

Hermitian Positive Definite I Factor and Condition

SPOCO( SA, adim, n, srcond, sz, info ) DPOCO( DA, adim, n, drcond, dz, info ) CPOCO( CA, adim, n, crcond, cz, info ) ZPOCO( ZA, adim, n, zrcond, zz, info )   Factor

SPOFA( SA, adim, n, info ) DPOFA( DA, adim, n, info ) CPOFA( CA, adim, n, info ) ZPOFA( ZA, adim, n, info )

201

Hermitian Positive Definite II Solve SPOSL( SA, adim, n, sb ) DPOSL( DA, adim, n, db ) CPOSL( CA, adim, n, cb ) ZPOSL( ZA, adim, n, zb )

Inverse and DeterminantSPODI( SA, adim, n, sdet, job )

DPODI( DA, adim, n, ddet, job ) CPODI( CA, adim, n, cdet, job ) ZPODI( ZA, adim, n, zdet, job )

202

Banded Hermitian Positive Definite I

Factor and Condition SPBCO( SA, adim, n, q, srcond, sz, info )

DPBCO( DA, adim, n, q, drcond, dz, info ) CPBCO( CA, adim, n, q, crcond, cz, info ) ZPBCO( ZA, adim, n, q, zrcond, zz, info )   Factor

SPBFA( SA, adim, n, q, info ) DPBFA( DA, adim, n, q, info ) CPBFA( CA, adim, n, q, info ) ZPBFA( ZA, adim, n, q, info )

203

Banded Hermitian Positive Definite II

Solve SPBSL( SA, adim, n, q, sb )

DPBSL( DA, adim, n, q, db ) CPBSL( CA, adim, n, q, cb ) ZPBSL( ZA, adim, n, q, zb )

Inverse and DeterminantSPBDI( SA, adim, n, q, sdet, job )

DPBDI( DA, adim, n, q, ddet, job ) CPBDI( CA, adim, n, q, cdet, job ) ZPBDI( ZA, adim, n, q, zdet, job )

204

Hermitian Positive Definite Tridiagonal

SPTSL( n, sd, se, sb, info )

DPTSL( n, dd, se, db, info )

CPTSL( n, cd, se, cb, info )

ZPTSL( n, zd, se, zb, info ) 

205

Hermitian Indefinite I Factor and Condition

SSICO( SA, adim, n, ipvt, srcond, sz ) DSICO( DA, adim, n, ipvt, drcond, dz ) CSICO( CA, adim, n, ipvt, crcond, cz ) ZSICO( ZA, adim, n, ipvt, zrcond, zz )  Factor SSIFA( SA, adim, n, ipvt, info ) DSIFA( DA, adim, n, ipvt, info ) CSIFA( CA, adim, n, ipvt, info ) ZSIFA( ZA, adim, n, ipvt, info )

206

Hermitian Indefinite II Solve

SSISL( SA, adim, n, ipvt, sb ) DSISL( DA, adim, n, ipvt, db ) CSISL( CA, adim, n, ipvt, cb ) ZSISL( ZA, adim, n, ipvt, zb )

Inverse and DeterminantSSIDI( SA, adim, n, ipvt, sdet, inert, swork, job )

DSIDI( DA, adim, n, ipvt, ddet, inert, dwork, job ) CSIDI( CA, adim, n, ipvt, cdet, inert, cwork, job ) ZSIDI( ZA, adim, n, ipvt, zdet, inert, zwork, job )

207

QR Factorization Factor

SQRDC( SA, adim, m, n, sqraux, ipvt, swork, job ) DQRDC( DA, adim, m, n, dqraux, ipvt, dwork, job ) CQRDC( CA, adim, m, n, cqraux, ipvt, cwork, job ) ZQRDC( ZA, adim, m, n, zqraux, ipvt, zwork, job )  SolveSQRSL(SA,adim,m,n,sqraux,sb,sQb,SQTb,sx,srsd,aAx,job,info)DQRSL(DA,adim,m,n,dqraux,db,dQb,dQTb,dx,drsd,dAx,job,inf

o)CQRSL(CA,adim,m,n,cqraux,cb,cQb,cQTb,cx,crsd,cAx,job,info)ZQRSL(ZA,adim,m,n,zqraux,zb,zQb,zQTb,zx,zrsd,zAx,job,info)

208

Singular Value Decomposition

SSVDC( SA,adim,m,n,ss,se,SU,udim,SV,vdim,swork,job,info)

DSVDC( DA,adim,m,n,ds,de,DU,udim,DV,vdim,dwork,job,info)

CSVDC( CA,adim,m,n,cs,ce,CU,udim,CV,vdim,cwork,job,info)

ZSVDC( ZA,adim,m,n,zs,ze,ZU,udim,SV,vdim,zwork,job,info)

209

4.1 BASICS

210

Setting Up Matrices Assignment a matrix

A = [ {row 1}; {row 2}; ……; {row m} ]

Ex: To set up a 4x4 matrix A ,like

A = [1 1 1 1;1 2 3 4;1 3 6 10;1 4 10 20]

It’s also legal to make input ‘look like’ output.A = [1 1 1 1

1 2 3 41 3 6 101 4 10 20 ]

201041

10631

4321

1111

211

Simple Output If you enter A = [1 2;3 4] without a

semicolon ,then the system responds with

If you already assignment A wit a semicolon, but now you want to display it .You could only type A to display.

43

21A

212

Types and Dimension Complex syntax

c = complex(a,b), it means c = a + bi

Dimension is handled automatically in MATLAB.Suppose you set B = [1 2 3; 4 5 6 ], and then set B = [1 0; 0 1].The system ‘knows enough’ to recognize that B has changed from 2-by-3 matrix to a 2-by-2 matrix.

213

Subscripts The subscripts in MATLAB must be positive. If you type A(1.2 , 3.5) = 10, you will get a error

message . ?? Subscript indices must either be real positive integers or logicals

214

Vectors and Scalars Vector and scalar are ‘special matrices’.

The commands, x = [ 1 ; 2 ; 3 ] ; or x = [1 2 3]’ ; establish x as a column vector while

Assignment an 1-by-1 matrix with c = [3], and it equals to c = 3 .

3

2

1

x

215

Size and LengthIf we declare A = [ 1 2 ; 3 4 ; 5 6 ], v = [7 8 9 10]

Size & Length syntaxsize(A)

ans = [3 2]length(v)

ans = 4

216

Continuation Sometimes it is not possible to put an entire

MATLAB instruction on a single line. We can break the instruction in a ‘natural’ place.

There are some ways to express a instruction, and they are the same.

A = [1 2 3; 4 5 6 ;....... 7 8 9]

A = [1 2 3; 4 5 6 ; 7 8 9]

A = [1 2 3; 4 5 6 ; 7 8 9]

217

Variable Names Variable names in MATLAB must consist of

nineteen or fewer characters.

The name must begin with a letter.Any letter, number, underscore may follow, e.g., sym_part1.

Upper and lower cases are distinguished in MATLAB.

218

Addition, Multiplication, Exponentiation

If A and B are matrices the C = A+B sets C to be the sum while C = A*B is the product.

To raise a square A to power r enterC = A^r

C = A^(-1) assigns the inverse of A to C.

f = c1Ab + c2A2b + c3A3b

f = A*(A*(c(3)*A*b+c(2)*b)c(1)*b)

f = ( c(3)*A^3+ c(2)*A^2+c(1)*A )*b

219

Transpose The conjugate transpose of a matrix can be

obtained using a single quote:Set B = At

B = A’

220

The “Colon Method” for Setting Up Vectors

Colons can be used to prescribe vectors whose components differ by a fixed increment.

v = 1:3 v = [1 2 3] v = 3:-1:0 v = [3 2 1 0] v = 1:2:10 v = [1 3 5 7 9]

221

Logarithmically Spaced Vectors

LOGSPACElogspace(a,b,n) generates n points between decades 10^a and 10^b.

If w = logspace(d1,d2,n), thenw(i) = 10 ^ [ d1 +(i-1)(d2-d1)/(n-1) ]

w = logspace(0,3,4) w = [1 10 100 1000]

222

Generation of Special Matrices

MATLAB has a number of build-in functions that can be used to generate frequently occurring matrices:

A = eye(m , n ) A m-by-n, ones on diagonal, zeros

elsewhere.A = zeros(m , n )

A m-by-n, zeros everywhere.A = ones(m , n )

A m-by-n, ones everywhere.

223

Random Matrices It’s possible to generate matrices and vectors

of random numbers:

A = rand(m , n ) A m-by-n, random element between 0

and 1.

224

Complex Matrices If A and B are matrices with the same

dimension and i2=-1, then the complex matrix C = A+iB can be generated as follows:

C = A + sqrt(-1) * B ; or C = A + i * B ;

225

Pointwise Operations Element operations between two matrices of

the same size are also possible:

C = A.*B cij = aij * bij

C = A.\B cij = bij / aij

C = A./B cij = aij / bij

C = A.^B cij = aijbij

C = A.’ cij = aji

If we have a=[1 2], then type a.*[3 4]We get the answer = [1*3 2*4 ]= [3 8].

226

More On Input/Output I The format for printed output can be controlled

using the following commands:format short 5-digit fixed point

styleformat short e 5-digit floating point

styleformat long 15-digit fixed point

styleformat long e 15-digit floating

point styleformat hex hexadecimal

227

More On Input/Output II format hex hexadecimal

To represent a number into double precise floating point(with hexadecimal form).

Ex. To show 1 in format ‘ hex ’ in matlab1 = + 1.0 * 2^0

sign exponent+1023 mantissa

0011 1111 1111 0000 0000 0000 ..... 0000 3 f f 0 0 0 ..... 0

sign(1 bit)

exponent(11 bit)

mantissa(52 bit)

0 01111111111 000000000.....00+1023

228

Machine Precision At the start of MATLAB session, the variable eps

contains the effective machine precision,i.e.,the smallest floating point number such that if x = 1 + eps then x>1 .

eps = 2.220446049250313e-016

229

Flops This is an obsolete function.

It can estimate the count of the program.

For complex operands, addition and subtraction count as two flops while multiplications and divisions each count as six flops.

230

4.2 LOOPS AND CONDITIONALS

231

For-Loops Syntax

for {var} = {row vector of counter values}{statements}

end

Ex: for i = 1:3 x(i) = i;end

is equivalent to x = [1;2;3];

232

Relations Relation operators

== equals

~= not equals< less than<= less than or equal to> greater than>= greater than or equal to

Ex: A=[1 2;3 4]; B=[1 2;2 4];T = A==BThen, the output is T = 1 1

0 1

233

‘and’, ’or’, ’not’ And(&), or(|), not(~)

The operations are possible between 0-1 matrices of equal dimension.

If T1 = [1 1 ; 0 1] ,T2 = [1 0 ; 0 0]T=T1&T2, T= 1 0

0 0T=T1|T2, T= 1 1

0 1T=~T1, T= 0 0

1 0

234

‘if’ Constructions Syntax

if {relation} {statements}end

Ex:Set a upper triangular matrix that has 1’s on the diagonal and –1’s above the diagonal.for i = 1 : n for j = 1 : n

if i < j A(i,j) = -1; elseif i > j A(i,j) = 0; else A(i,j) = 1; end

end end

235

The ‘any’ and ‘all’ Functions Any

If any component of the vector is nonzero, it returns a “1”.

AllIf all the entries are nonzero, it returns a “1”.

a = [ 0 1 1; 0 0 1; 0 0 1]any(a) ans =[0 1 1]all(a) ans =[0 0 1]

236

While-Loops Syntax

while {relation} {statements}

end

Print a sequence{Ai} of random 2-by-2 with the property that Ai+1>Ai:

A = zeros(2)

B = rand(2,2)

while(B>A)

A=B

B=rand(2,2)

end

237

The ‘break’ Command Break

It can be used to terminate a loop.

Print all Fibonnaci number less than 1000:fib(1) = 1

fib(2) = 1

for j = 3:1000

z = fib(j-1) + fib(j-2)

if z >= 1000

break

end

fib(j) = z

end

238

4.3 WORKING WITH SUBMATRICES

239

Setting Up Block Matrices We can set up the block matrix as follows:

A=[A11 A12 ; A21 A22 ; A31 A32 ]

If we want to make a 6-by-6 matrix with random 2-by-2 block diagonal, like

A = rand(2,2)

for i = 2:3

[m,m] = size(A);

A = [ A zeros(m,2) ; zeros(2,m) rand(2,2) ]

end

lk0000

ji0000

00hg00

00fe00

0000dc

0000ba

240

The Empty Matrix The empty matrix [] is often useful for

initialization of certain matrix computations.B=[] B is empty

To set up a column-reversed version of above example:

B = []

for j = 6:-1:1

B = [ B A(:,j) ];

end

0000kl

0000ij

00gh00

00ef00

cd0000

ab0000

241

Designating Submatrices If A is m-by-n matrix and i,j,p,q are integers.

Then, B= A( i:j , p:q ) B=

B = A ( : , p:q ), the range ( : ) means from 1:n, if the dimension of matrix is n.

jqjp

iqip

aa

aa

242

Assignment to Submatrix

If A = , we can assign the column

3 to [3 3 3]’ by using A(:,3)=3 or A ([7 8 9])=[3 3 3].

Then A =

021

021

021

321

321

321

243

Kronecker Product The Kronecker product of two matrices A and B can

be calculated using the build-in function kron: C = kron(A,B) C = (aijB)

C = []; [m,n] = size(A); for j = 1:n ccol = []; for i = 1:m ccol = [ ccol ; A(i,j)*B ]; end

C = [C ccol] end If C = kron( [ 1 2 ] , [ 1 2 ; 3 4 ] ) then C = 1 2 2 4 3 4 6 8

244

Turning Matrices into Vectors and Vice Versa

If A is a m-by-n matrix and we want to set v is an mn-by-1 vector.

v = A(:)

Set a transpose of a 2-by-2 matrix A = [ 1 2 ; 3 4 ]x = A(:);

x( [2 3] ) = x( [3 2] );

A(:) = x;

245

4.4 BUILD-IN FUNCTIONS

246

‘abs’, ‘real’, ‘imag’, ‘conj’ B = abs(A) B = ( |aij| )

B = real(A) B = ( real(aij) )

B = imag(A) B = ( imag(aij) )

B = conj(A) B = real(A)-i*imag(A), i2=-1

If A=[1-2i 2-i;3-4i 4-3i]; conj(A);Then, ans = 1+2i 2+i

3+4i 4+3i MATLAB’s build-in functions are in lower case,

unless user define.Otherwise, it’s will show ‘Capitalized internal function XXXX; Caps Lock may be on.’

247

Norms t = norm(A) t = || A ||2

t = norm(A,1) t = || A ||1

t = norm(A,’inf’) t = || A ||t = norm(A,'fro') t = || A ||F

If A = [1 2 3 4]norm(A)

ans = 5.47722557505166sqrt(1+4+9+16)ans = 5.47722557505166

248

Largest and Smallest Entries If v = max(A) v = [ max(A(:,1)), … , max(A(:,n)) ]

[v,i] = max(A)v = [ max(A(:,1)), … , max(A(:,n)) ]i = [the row of the max value

in]

A =[ 1 2 86 9 4

7 5 3];and [v,i]=max(A);

v= 7 9 8i = 3 2 1

249

Sums and Products t = sum(A) t =

t = prod(A) t =

A =[ 1 2 9 6 9 4

7 5 3]sum(A)ans =14 16 15prod(A)ans =42 90 96

) )(:,, .... ),(:, ( nA1A

))(:, , .... ),(:, ( nA1A

250

‘sort’ and ’find’ I Sorts each column of A in ascending order by using

sort(A)

[v,i] = sort(A), returns in sorted v and an integer vector i which indicate the permutation of new from old matrix.

A =[ 3 5 92 6 81 4 7]

[B,i]=sort(A); B =[1 4 7 2 5 8 3 6 9]

i =[ 3 3 3 2 1 2 1 2 1]

251

‘sort’ and ’find’ II The find function can be used to locate nonzero

entries.

A =[1 0 2 3 0]find(A)

ans =[1 3 4]find(A == 0)

ans =[2 5]

252

Rounding, Remainders, and Signs I

There are several conversion-to-integer routines:

round(x) round x to nearest integerfix(x) round x to zerofloor(x) round x to -ceil(x) round x to +

A = [ 1.1 -2.2 -0.9 3.8 ]round(A) ans=[1 - 2 -1 4]fix(A) ans=[1 -2 0 3]floor(A) ans=[1 -3 -1 3]ceil(A) ans=[2 -2 0 4]

253

Rounding, Remainders, and Signs II

t = sign(x) t =

t = rem(x,y) t = x-y*fix(x/y)If y = 0, rem(x,0) is NaN. And x and y must be the same dimension.

rem equals mod(x,y) if x and y are the same sign.Otherwise , mod(x,y) = rem(x,y)+ya = mod(-7,4)a = 1a = rem(-7,4) a = -3

0 if x 1

0 if x0

0 if x1

254

Extracting Triangular and Diagonal Parts I

B = tril (A) bij=aij if i j, zero otherwise.B = triu (A) bij=aij if i j, zero otherwise.B = tril (A,k) bij=aij if i+k j, zero otherwise.

B = triu (A,k) bij=aij if i+k j, zero otherwise.

diag(v) puts v on the main diagonal.v=diag(A) return a vector v with A’s diagonal.

255

Extracting Triangular and Diagonal Parts II

T = toeplitz(c,r) is equivalent to

T = zeros(n);

for k = 1:n

j = n-k;

if k<n

T = T + diag( r(k+1)*ones(j,1) , k);

end

T = T + diag( c(k)*ones(j+1,1) , -k+1);

end

256

Extracting Triangular and Diagonal Parts III

Take c=[6 7 8 9 10];r=[1 2 3 4 5];n=5;

into above two function. We can show

T =[ 6 2 3 4 5 7 6 2 3 4 8 7 6 2 3 9 8 7 6 2 10 9 8 7 6]

257

‘rank’ If A is a m-by-n matrix, rank(A)= min{m,n}.

It Is also possible to base the rank decision on an arbitrary tolerance.If tol is a nonnegative scalar then

r = rank(A , tol)

A = [ 1 2 ; 3 4 ; 5 6 ]rank(A)=min{3,2}=2

A = [1 0.1 ; 3 0.2 ; 5 0.3 ]rank(A,1)=1

258

Condition cond(A) can be computed the 2-norm

condition (the ratio of the largest

singular value of X to the smallest) of a matrix A.

min/ 1

259

Determinant The determinant of a square matrix A can be

computed as follows:det(A)

A =[1 2;3 4];det(A)=1*4-2*3

=-2

260

Elementary Functions Trigonometric

sin() cos() tan()

Inverse trigonometricasin() acos() atan()

Exponentialexp() log() log10()

Hyperbolicsinh() cosh() tanh()

261

Functions of Matrices F = expm(A) F = eA

F = logm(A) A = eF

F = sqrtm(A) F2 = A

262

‘roots’ and ‘poly’ If A is an n-by-n matrix then

c = poly(A) det( ) =

The roots are returned in a column vector by root(A), if we want to get the root.

If A=[2 -1;1 4];r=poly(A); r=[1 –6 9]a=roots(r); a=[3 -3]

AI n

1

1n

2n1n cccc

263

Fourier Transforms y = fft(x) y = Fourier transform of x

y = ifft(x) y = inverse Fourier transform of x

B = fft2(x) B = 2DFourier transform of xB = ifft2(x) B = inverse 2DFourier transform of x

264

4.5 FUNCTIONS

265

An Example I The first line in the function is of the form

function {output variable} = {function name} ({arguments})

Comments begin with the ’%’ sign.All functions often begins with ‘how-to-use’ comments.They will be displayed, as we input the help with the function name.

266

An Example II Write a function return a unit 2-norm vector

in the direction of Ax

function y = proAx(A,x)%% Compute K = [ v , Av , A^2 v , . . . , A^(p-1) v ] D,% where A is n-by-n, v is n-by-1, and D is diagonal% chosen so that the column of K have unit 2-norm.%y = A*x;c = norm(y);if c ~= 0

y = y/c;else

y = [ 1 ; zero( length( x-1,1 ) ) ];disp('Matrix-vector product is zero. First

column..of I returned.')end

267

Functions can call other Functions

Compute K = [v,Av,A^2v,…,A^(p-1)v]D,where A is n-by-n,v is n-by-1,and D is diagonal.

function K = kry(A,v,p) [ m,n ] = size(A); nv = length(v) if m ~= n | n ~= nv disp('Dimension do not agree.') else K = v/norm(v) for j = 2:p-1 K = [ K prodAx(A,K(:,j-1) )]; end end return

268

Multiple Input and Output Arguments I

Computes nonnegative matrices P and N, where A(i,j) > EPS , implies P(i,j) = A(i.j) , N(i,j) = 0 A(i,j) < EPS , implies P(i,j) = 0 , N(i,j) = -A(i.j)

| A(i,j) | <= EPS , implies P(i,j) = N(i,j) = 0 

function [P N] = pn(A)E = EPS*ones(A);

P = A.*( A > E );

N = -A.*( -A > E );

end

269

Multiple Input and Output Arguments II

If one input argument(decided by nargin), tol = EPS.If one output argument (decided by nargout), P only is returned.

function [P N] = pn(A,tol)

if nargin == 1

tol = EPS;

end

E = tol*ones(A);

P = A.*( A > E );

if nargin == 2

N = -A.*( -A > E );

End

Statement Tolerance

Output

P = pn(A)[P,N]=pn(A)P = pn(A,tol)[P,N]=pn(A,tol)

epsepstoltol

Just PP and

NJust PP and

N

270

Passing Function Names It’s possible to pass the function name of another

function,but it requires some text processing with the eval function.

function F = afun(A,f)

[m,n] = size(A);

for j = 1:m

for i = 1:n

F(i,j) = eval([ f ,'(A(i,j))' ]);

end

end F(i,j) = eval([ f ,‘A(i,j)’ ]);

assign f(A(i,j)) to F(i,j). Ex: afun(A,'cos');

[0.54030230586814 -0.41614683654714 -0.98999249660045 -0.65364362086361]

271

Global Variables The values in all global variables can be

accessed whenever inside a function.

Like the common in Fortran, global should be used sparingly.

272

Recursion I Compute the 1-norm of an n-vector x function normL1 = f(x)

n = length(x);

if n == 1

normL1 = abs(x);

else

normL1 = f( x(1:n-1) ) + abs(x(n));

end

Ex: f([1 2 -3 4]) = 10

273

Recursion II Compute A*B with recursion method(Strassen

matrix), where A,B are square matrices.function C = strass(A,B,nmin)

[ n n ] = size(A); if n <= nmin % n small, get C conventionally C = A*B;

else m = n/2; u = 1:m; v = m+1:n; P1 = strass(A(u,u)+A(v,v), B(u,u)+B(v,v), nmin);

P2 = strass(A(v,u)+A(v,v), B(u,u), nmin); P3 = strass(A(u,u), B(u,v)-B(v,v), nmin);

P4 = strass(A(v,v), B(v,u)-B(u,u), nmin); P5 = strass(A(u,u)+A(u,v), B(v,v), nmin); P6 = strass(A(v,u)-A(u,u), B(u,u)+B(u,v), nmin); P7 = strass(A(u,v)-A(v,v), B(v,u)+B(v,v), nmin); C = [ P1+P4-P5+P7 P3+P5 ; P2+P4 P1+P3-P2+P5 ];end

274

4.6 FACTORIZATION

275

Solving Linear Systems It can be computed the solution to the multiple

tight-hand side linear system AX=B.X = A \ B

A less preferable way to solve AX = B isX = inv(A) * B

276

The LU Factorization The factorization A =LU, where U is upper

triangular and L is a row-permuted unit lower triangular matrix,can be computed as follows:

[ L , U ] = lu(A) Another less efficient method for solving AX =

b.[ L , U ] = lu(A)

y = L \ b;

x = U \ y;

277

The Cholesky Factorization It can be computed an upper triangular R

such that A =RHR, where A is an m-by-n Hermitian, positive definite matrix.

R = chol(A)

There is no build-in function for computing the factorization A = LDLH, where L is unit lower triangular and D is diagonal. How to do it?L = chol(A);d = diag(L);L=L';for k =1:n

L(:,k) = L(:,k)/d(k);endD = diag(d.*d);

278

QR Factorization Compute unitary Q(m-by-m) and upper

triangular R(m-by-n) such that A =QR.[Q , R] = qr(A)

Compute the minimizer of ||Ax - b||2.

[Q , R] = qr(A);

[m , n] = size(A);

b = Q'*b;

x = R(1:n,1:n)\b(1:n);

279

Householder and Givens Transformations

Compute a Householder matrix P such that Px is zero below the first component:

[P , R] = qr(x)

280

QR Factorization with Column Pivoting

Syntax[Q,R,P] = qr(A)

where Q is m-by-m unitary, p is n-by-n permutation, and upper triangular m-by-n matrix R.So, QTAP = R =

Compute the solution of Ax = b.[Q,R,P] = qr(A);

[m,n] = size(A);

y= R(1:m , 1:m) \Q'*b;

x= P* [y; zeros(n-m,1)];

00

RR 1211

281

Range and Null Space Compute an orthonormal basis for the range

space.Q = orth(A)

or[m,n]=size(A);[Q,R]=qr(A);Q=Q(:, 1:n);

A null space of a matrix AQ=null(A)

282

The Singular Value Decomposition I

If A is a m-by-n matrix ,we can compute unitary U(m-by-m), unitary V (n-by-n), and diagonal S(m-by-n):

[ U , S , V ] = svd(A)so, UHAV = S.

If m n then the ‘compact’ SVD in which U is m-by-n and S is n-by-n, use a two-argument version of svd:

[ U , S , V ] = svd(A, 0)

283

The Singular Value Decomposition II

A = [1 2 ;4 5;3 4];[ U , S , V ] = svd(A,0)[ U1 , S1 , V1 ] = svd(A,0)

U = [-0.26118 0.92755 -0.26726 -0.76072 -0.36822 -0.53452 -0.59420 0.06370 0.80178]S = [8.41440 0 0 0.444672 0 0 ]V = [-0.60452 -0.796587 -0.79658 0.60452]

U1 = [-0.26118 0.92755

-0.76072 -0.36822

-0.59420 0.06370]

S1 = [8.41440 0

0 0.444672]

V1 =[-0.60452 -0.796587

-0.79658 0.60452]

284

Pseudoinverse and Least Squares

If A is initialized then the pseudoinverse X can be found using pinv:

X = pinv(A)Then, multiple right-hand side least squares problem can be solved with either of the commands

x=A\B;or

X = pinv(A)*B;

285

The Hessenberg Decomposition

If A is a m-by-n matrix ,we can compute unitary Q(n-by-n), and upper Hessenberg H such that QHAQ = H.

H = hess(A)

A =[1 2 4;3 4 2;1 2 4];hess(A);

[1 -3.16227 3.162277 -3.16227 5.2 -1.6 0 -1.6 2.8 ]

286

The Schur Decomposition I Real SCHUR form

If A is real then Q is orthogonal and T = QTAQ is upper quasitriangular.

Imaginary SCHUR formIf the imaginary part of A is nonzero, then Q is unitary and T = QTAQ is upper triangular.

Syntax[Q,T] = schur(A);

It can be computed the complex Schur form from the real Schur form.[Q,T] = schur(A);[Q,T] = rsf2csf(Q,T);

287

The Schur Decomposition II If A=[7 -4 0

8 -5 0 6 -6 3 ];

T = schur(A);T= [3 13.2 6.18448

0 -1 -1.87408 0 0 3 ] A=[3 4;-2 -1];

[Q,T] = schur(A);[Q,T] = rsf2csf(Q,T);T=[ 1+2i -4.47213; 0 1-2i ]

288

The Eigenvector Decomposition

It can be compute a nonsingular X and diagonal D such that X-1AX = D is diagonal.

[X,D] = eig(A);

If A=[7 -4 0 8 -5 0 6 -6 3 ];

e = eig(A);e=[3 1 3]’

289

Generalized Eigenvalues A two-argument version of eig can be used to

solve the generalized eigenvalue problem Ax = Bx, where A and B are square and of the same dimension.

It can be compute a matrix X and a diagonal matrixD such that AX = BXD.

[X,D] = eig( A, B )

290

4.7 MISCELLANEOUS

291

IEEE Arithmetic In MATLAB, the result of a zero divide is a

special value called “inf”.Example, 1/0.

NaN means not a number.Example, 0/0.

Thus, x = inf and y = NaNa=x+z a = infa=1/x a = 0a=y*z a = NaNa=x/y a = NaNa=[x;y] a(1)=inf, a(2)=NaN

292

Timings We can execute the time which we run a

program.The function clock returns the current time in a row vector(year,month,day,hour,minute,and second).The function etime(t1,t2) returns the time in seconds between vectors t1 and t2.

for k =1:500n=k;A=rand(n);x=rand(n,1);t1=clock;y=fft(x);t2=clocktime=time+etime(t2,t1);end

293

Timed Displays A delay can be build into a MATLAB segment

with the pause statement.

Display the n-by-n magic square matrix on the screen for approximately n seconds for n=1:5.

for n=1:5

A=magic(n)

pause(n)

end

294

Getting Input It can be input a value during execution of a

MATLAB segment.

Write a function which displays magic square of chosen dimension by user. while(n>=1)

A=magic(n)

n=input('Enter dimension')

end

295

Converting Numbers to Strings

Num2str can be used to convert a number to its string format.

s = num2str(2) s = ‘2.0000’s = num2str(2) s = ‘-1.000E+2’s = num2str(2) s = ‘1.2345E+7’