PDF / 04-Матричные операции

1

3. ВЕКТОРЫ И МАТРИЦЫ

Пример. Вычислить значение матричного выражения R. Сохраните промежуточные вычисления. Составите программу для вычисления R, используя язык программирования VBA. Проведите расчёт по программе и сравните результаты. Операция sled([матрица]) – произведение диагональных элементов матрицы.

							R =						=		= →		= →
													A B c			+A d
	=		2	1 1				= 1							2		→	2		→	3
	A		=		;			B	= 3						1 ;		c =	;		d =	1
			1	3 2														3
													4		1						2
			=	=	=		2 1								1	1	2	9		6
			RM = A B			= 1 3 2										3 1		= 18 7
																4	1
=	= →		9	6 2				36							=		= →		2	1 1			3	9
RV = RM c		= 18		7	3		= 57								RV1 = A d =				1	3	2		1	= 10
																							2
=	=	=		9	36			=	45						R =		2			452 +672 =80.709
RV1	= RV1+RV =			+													∑rv1i =
			10		57				67								i=1

Программа вычисления значения матричного выражения

Исполняемая процедура

Option Explicit

Option Base 1 Sub main()

Dim A!(2, 3), B!(3, 2), c!(2), D!(3), RM!(2, 2), RV!(2), RV1!(2) Dim i%, j%, l%, k!, s!, R!

'Ввод элементов матрицы A For i = 1 To 2

For j = 1 To 3: A(i, j) = Cells(i + 4, j): Next j Next i

'Ввод элементов матрицы B For i = 1 To 3

For j = 1 To 2: B(i, j) = Cells(i + 4, j + 4): Next j Next i

'Ввод элементов вектора C

For i = 1 To 2: C(i) = Cells(i + 4, 8): Next i 'Ввод элементов вектора D

For i = 1 To 3: D(i) = Cells(i + 4, 10): Next i

2

'Вычисление матрицы RM=A*B For i = 1 To 2

For j = 1 To 2 s = 0

For l = 1 To 3: s = s + A(i, l) B(l, j): Next l RM(i, j) = s

Next j

Next i

'Вычисление вектора RV=RM*C For i = 1 To 2

s = 0

For l = 1 To 2: s = s + RM(i, l) * C(l): Next l RV(i) = s

Next i

'Вычисление вектора RV1=A*D For i = 1 To 2

s = 0

For l = 1 To 3: s = s + A(i, l) * D(l): Next l RV1(i) = s

Next i

'Вычисление RV1=RV1+RV

For i = 1 To 2: RV1(i) = RV1(i) + RV(i): Next i 'Вычисление нормы вектора ||RV1||

s = 0: For i = 1 To 2: s = s + RV1(i) * RV1(i): Next i R = Sqr(s): Cells(2, 5) = R

End Sub

С использованием стандартных (вызываемых) процедур и функций

Option Explicit

Option Base 1 Sub main()

Dim A!(2, 3), B!(3, 2), C!(2), D!(3), RM!(2, 2), RV!(2), RV1!(2) Dim i%, j%, l%, k!, s!, rez!

Call InputMatrix("лист1", 5, 1, 2, 3, A)

Call InputMatrix("лист1", 5, 5, 3, 2, B) Call InputVector("лист1", 5, 8, 2, C) Call InputVector("лист1", 5, 10, 3, D) Call MultMatr(2, 3, 2, A, B, RM)

Call MultMatrVect(2, 2, RM, C, RV)

Call MultMatrVect(2, 3, A, D, RV1) Call AddVect(2, RV, RV1, RV1) Cells(2, 5) = NormaVect(2, RV1) End Sub

'Умножение матриц

Public Sub MultMatr(LeftRow%, Col_Row%, RightCol%, _

LeftMatr() As Single, RightMatr() As Single, ResMatr() As Singl)

3

Dim s As Single, i As Integer, j As Integer, l As Integer For i = 1 To LeftRow Step 1

For j = 1 To RightCol Step 1 s = 0

For l = 1 To Col_Row Step 1

s = s + LeftMatr(i, l) * RightMatr(l, j) Next l

ResMatr(i, j) = s Next j

Next i

End Sub

'Сложение векторов

Public Sub AddVect(Row As Integer, LeftVect() As Single, _ RightVect() As Single, ResVect() As Single)

Dim i As Integer

For i = 1 To Row Step 1

ResVect(i) = LeftVect(i) + RightVect(i) Next i

End Sub

'Умножение матрицы на вектор‘

Public Sub MultMatrVect(Row As Integer, ColRow As Integer, _

Matr() As Single, Vect() As Single, ResVect() As Single) Dim i As Integer, j As Integer, s As Single

For i = 1 To Row Step 1 s = 0

For j = 1 To ColRow Step 1 s = s + Matr(i, j) * Vect(j)

Next j ResVect(i) = s

Next i

End Sub 'Норма вектора

Public Function NormaVect(Row As Integer, Vect() As Single) As Single Dim norma As Single, i As Integer

norma = 0

For i = 1 To Row Step 1: norma = norma + Vect(i) * Vect(i): Next i NormaVect = Sqr(norma)

End Function'

'Ввод матрицы (sh – Лист)

Public Sub InputMatrix(sh As String, UpRow As Integer, _ leftCol As Integer, Row As Integer, _ Col As Integer, Matr() As Single)

For i = 1 To Row Step 1

For j = 1 To Col Step 1

Matr(i, j) = Worksheets(sh).Cells(UpRow - 1 + i, leftCol - 1 + j) Next j

Next i

4

End Sub 'Ввод вектора

Public Sub InputVector(sh As String, UpRow As Integer, _ leftCol As Integer, Row As Integer, _ Vect() As Single)

For i = 1 To Row Step 1

Vect(i) = Worksheets(sh).Cells(UpRow - 1 + i, leftCol) Next i

End Sub

Задания. Вычислите значение матричного выражения R Составите программу, для вычисления R, используя язык программирования VBA.

1) R =

=

=

→

−

→

A B c

d

7

1

=

1 2

8

=

→

5

A =

2

;

B =

4

4 ;

c

= ;

4

3

9

1

4

2) R =

=T

=

=

=T

→

−

→

A A−B B

d

c

=

1 8

5

=

5

4

→

7

A =

2

;

B =

4

3 ;

c

=

;

6

6

2

9

3

3) R =

→T

=

=T

→T

=

c A A − d B

=

2 4

3

=

3

7

→

9

=

A =

;

B =

5

2 ;

c

;

8

3

9

3

8

3

4) R =

→T = =T

→T

= →

d

B B

+ c

A d

1

1

=

3 1

9

=

→

3

A =

3

;

B =

6

1 ;

c

=

;

1

3

5

7

2

3 →d = 56

7 →d = 12

3 →d = 57

7 →d = 94

5

		→T									→	→T												=T									= →
5) R = c c												− d A A d
=	3 7											6								=													8								4					→		5
A =											3	;												B =									7							9 ;						c	= ;
	3											6																																				6
																																	6								2
6) R =								=			=	→										→						+							→
										A B c														d											c
=	4 3											3								=													7								7					→		7
A =											4	;												B =									8								8 ;					c	= ;
	5											1																																				7
																																	6											2
7) R =										=	=	→						−						=									=		→
										A B c														B A d
																																	5											1
=	4 9											1								=																										→		1
A =											4	;												B =									9											7 ;		c	= ;
	7											4																																				9
																																	5											1
						→T = =																→ T														→T →
8) R =																																			− c c
		d B− A d
=		5 6										7								=															3									3		→		3
A =											4					;								B =											1									6 ;		c		= ;
		9										7																																				1
																																			4									1
											=T =																						=T			=
9) R =sled A A																				−												B B
=	6 3											4								=													1											6		→		5
A =											5	;												B =									2											5 ;		c	= ;
	2											1																																				3
																																	3									9

3 →d = 49

8 →d = 95

3 →d = 41

8 →d = 86

3 →d = 42

6

			= =		= =		=T =
10)
	R =sled A B			−sled B A−A				A
=	6	8	1	=	8	9	→	7	→		8
										=
A =		5	;	B =	3	4 ;	c	= ;	d		8
	4		4					4
					3	9					7

			= → T						→									=
11)									c		+							B
	R = A d
=	7		5 8								=								6							3
A =			6	;							B =								4							3 ;
	6			7
																			2							8
			= → T						→							=
12)									d		+							A
	R = B c
=	8		2 5								= 4															6
A =			6	;							B					= 5										2 ;
	9			2
																		1								8
13)	R =		→ →T						+		→ →T
			c d								d	c
=	8		7 2								= 2															9
A =			6	;							B = 5															1 ;
	2			5
																		9								7

→c = 16 ;

→c = 73 ;

→c = 95 ;

3 →d = 33

8 →d = 79

4 →d = 35

7

14)

R =

→ →T T

→

+

=

=

c d

c

A B

9

3

8

=

9 4

8

=

→

8

→

=

A =

;

B =

6

9 ;

c

;

d

= 7

4 7

8

1

8

7

1

15)

R =

→ →T T

→

+

=

=

d c

d

A B

7

6

4

=

1 1

6

=

→

1

→

A =

;

B =

7

8 ;

c

=

;

d

= 2

6 7

2

3

8

6

6

16)

=

=T

→

T

→

+

=

R =

A+B

d

c

A

6

8

8

=

1 6

3

=

→

3

→

A =

;

B =

8

7 ;

c

= ;

d

= 7

8 7

5

4

7

6

2

17)

=

=T

→

T

→

+

=

R =

A−B

d

c

A

4

2

4

=

2 3

9

=

→

6

→

A =

;

B =

9

6 ;

c

= ;

d

= 2

1 8

8

6

6

5

7

8

=	=T		→ T	→	+				=
18) R = B+ A			c	d					B

=	2	9	6	= 2	5	→	8	→	8
A =		8	;	B = 1	5 ;	c	= ;	d	= 6
	3		3				7
				5	5				3

=	=T		→ T	→	+		=
19) R = B−A			c	d			B

=	3		5 4		=	9		8					→	1
A =			8	;	B =	2		3 ;					c	= ;
	5			6										9
						5		4
			=T →		= =	→
20)	R =
			A c +B A d
=	4		2 1		=	7		2					→	4
A =			9	;	B =	3		2 ;					c	= ;
	7			9										1
						4		4
21)	R =		= →	=T =T →
			A d	+B	A	c
=	4		8 7		=	5		5					→	6
A =				;	B =	4		1 ;					c	= ;
	9		9	3										3
						3		3

4 →d = 19

9 →d = 65

4 →d = 11

9

			=T →		= =		→
22)	R =		B d +
					A B c
=	5		5 4			=	3		8								→	8	→		9
																				=
A =				;		B =	5		9 ;								c	= ;	d		5
	2		9 6															4
							2		3												6
23)	R =		= →	=T =T →
			B c	+A B			d
=	6 1 2					=	1		2								→	2	→		4
																				=
A =				;		B =	6		8 ;								c	= ;	d		1
	4 1 9																	6
							1		2												2
24)	R =	→T		=	→T		= →
		c A+			d	B c
=	6		7 8			=	8		5								→	4	→		9
A	=			;		B =	7		7 ;								c	= ;	d	= 5
	6		1 4															7
							1		2												7
25)			→ →T				=	=									=
	R =sled c d					+A B +											B
=	7		4 5			=	6		7								→	6	→		4
A =				;		B =	7		6 ;								c	= ;	d	= 9
	8		1 7															9
							9		1												3

10

26)					→ →T				=	=						=
	R =sled d					c		+B A +								A
=	7			9 2			=		5			1				→		9	→		9
																				=
A =					;		B =		8			5 ;				c	= ;		d		4
	1			2 1														1
									8			1									8
27)	R =			→ →T			= T		=
				c d		+A			+B
=	8			6 9			=		3			4				→		2	→		5
A	=				;		B =		9			4 ;				c	= ;		d	= 9
	3			2 4														2
									7			9									5
28)	R =			→ →T			= T			=
				d c		+B			+A
=	1 8				2		=		2			9				→		1	→		8
																				=
A =					;		B =		8			1 ;				c	= ;		d		3
	1 1				8													5
									3			7									3
29)	R =		→T		→ →T			= T		→
				d c d			+ A			c
=	1 1 9						=		1			6				→		7	→		1
																				=
A =					;		B =		9			9 ;				c	= ;		d		5
	2 1 9																	2
									7			3									5