Intlab Matlab For Mac
2021年1月13日Download: http://gg.gg/ntmr5
byJiriRohn
*Intlab Matlab For Mac Pro
*Intlab Matlab For Mac Operating System
*Octave For Mac
*Intlab Matlab For Mac Free
1. INTRODUCTION
Sep 23, 2015 Make sure that you save MAC.m and ModeShapes.mat in the same folder and set that folder as MATLAB working directory or add a path to it. Then: load ModeShapes.mat;MACMatrix=MAC(phi) azade ch. Mar 07, 2016 If you are using R2014a or later, and you already have a license for the toolbox (and the license has made it through the processing to be registered against your account), then you run the MATLAB installer, which will detect which toolboxes are on your license and will give you an option to pick which ones you want to install. The Matlab code INTLAB is used for automatic differentiation. The same approach is used to calculate the sensitivity of C l 2 d w.r.t. Mac Wilkinson, W.T. 至于Matlab工具箱安装中涉及到了Matlab的搜索路径、工作目录、当前路径、用户路径等好多术语，我这里不想多说什么 感兴趣的网友，可以直接查看Matlab的帮助系统，在那里你可以得到最直接的答复，但是你需要一定的英文基础哦 添加工具箱的方法很多，所有方法都是为了达到同一个目的，将工具箱. INTLAB - Fast INTerval LABoratory. Matlab Octave toolbox. Self-validating methods. Interval arithmetic for real and complex data including vectors and full and sparse matrices (very fast). Affine arithmetic including vectors and matrices. Fl-numbers: k-bit floating-point and interval arithmetic. Gfp: Galois field GFp toolbox.
This is a very concise description of basic features of INTLAB aimed at newcomers to this area. Only the very basics of INTLAB are presented; INTLAB itself contains much more.
2. INTLAB’S AUTHOR
INTLAB, a MATLAB toolbox for self-validating algorithms, is a one-man work. It has been created by Prof. Dr. Siegfried M. Rump, Institute for Reliable Computing, TechnicalUniversity of Hamburg-Harburg, Germany.
3. INTLAB’S DOWNLOAD
INTLAB can be downloaded (free for academic use; observe the license) from
http://www.ti3.tu-harburg.de/rump/intlab/
4. INTLAB’S INSTALLATION
Follow the instructions in the file Readme.txt.
5. INTLAB’S DEMOS
Look at demointval.m, demointlab.m. Many important features contained there are summarized below.See also Hargreaves.
6. DEFAULT INITIALIZATIONS
There are two basic ways of displaying intervals:
intvalinit(’displayinfsup’) : display of intervals by endpoints,
intvalinit(’displaymidrad’) : display of intervals by midpoint and radius.
In what follows we use the first option. To invoke it, simply type
>> intvalinit(’displayinfsup’)
You will get the response
> Default display of intervals by infimum/supremum (e.g. [ 3.14 , 3.15 ])
7. INTVAL TYPE
INTLAB uses newly defined MATLAB type ’intval’ for interval quantities.
8. BASIC INTERVAL ARITHMETIC PROPERTY
An operation always uses interval arithmetic if at least one of the operands is of type intval. The output is always rigorous (i.e., it encloses the real result in floating point).
9. INPUT
Given two real (of type double) matrices (vectors, numbers) A, B of the same size, the interval matrix (vector, number) C=[A,B] is inputed simply as C=infsup(A,B).
Example. For real (double) matrices Ad, Ah given by
>> Ad=[2 -2; -1 2]
Ad =
2 -2
-1 2
>> Ah=[4 1; 2 4]
Ah =
4 1
2 4
>> A=infsup(Ad,Ah)
yields
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
A is the interval matrix [Ad,Ah].Alternatively, the same result can be achieved using
>> A=[infsup(2,4)infsup(-2,1);infsup(-1,2)infsup(2,4)]
intval A =
[2.0000,4.0000] [-2.0000,1.0000]
[-1.0000,2.0000] [2.0000,4.0000]Intlab Matlab For Mac Pro
Similarly, A=midrad(C,D) gives the interval matrix [C-D,C+D] (i.e, in the midpoint-radius representation). If you wish a real (double) matrix Ao to be handled as a thin interval matrix (so that the interval arithmetic could apply), useAo=intval(Ao)(or, equivalently, Ao=infsup(Ao,Ao)) :
>> Ao=[1 2;3 4]
Ao =
1 2
3 4
>> Ao=intval(Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 3.0000, 3.0000] [ 4.0000, 4.0000]
10. OUTPUT
Given an interval matrix A (of type intval), its bounds, midpoint and radius can be extracted as follows:
the lower bound is inf(A) or A.inf
the upper bound is sup(A) or A.sup
the midpoint is mid(A) or A.mid
the radius is rad(A) or A.rad .
Example. With the above A, we have
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> inf(A)
ans =
2 -2
-1 2
>> sup(A)
ans =
4 1
2 4
>> mid(A)
ans =
3.0000 -0.5000
0.5000 3.0000
>> rad(A)
ans =
1.0000 1.5000
1.5000 1.0000
11. DIFFERENCE BETWEEN A.inf AND inf(A)
A.inf and inf(A) are not quite equivalent. A.inf allows indexing:
>> A.inf(1,1)
ans =
2
whereas this is not possible for inf(A). The same holds for A.sup and sup(A), etc.
12. ENCLOSURE OF THE INVERSE INTERVAL MATRIX
We shall demonstrate some typical INTLAB features on the function B=inv(A) which computes an enclosure of the interval inverse of A.
12.1. Interval input
>> A
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> B=inv(A)
intval B =
[ -2.9704, 3.6191] [ -3.2729, 3.3810]
[ -3.3810, 3.2729] [ -2.9704, 3.6191]
This means that for each real (double) matrix Ao in A, its exact inverse is guaranteed to exist and to belong to B. The output uses outward rounding, so that the result is rigorous. The enclosure B is generally not optimal.
12.2 Real input
Inverse can also be used to obtain a rigorous output for a real (double) input:
Ao =
1 2
3 4
>> Ao=intval(Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 3.0000, 3.0000] [ 4.0000, 4.0000]
>> B=inv(Ao)
intval B =
[ -2.0001, -1.9999] [ 0.9999, 1.0001]
[ 1.4999, 1.5001] [ -0.5001, -0.4999]
To see better the accuracy of the result, use
>> format long
>> B
intval B =
[ -2.00000000000001, -1.99999999999999] [ 0.99999999999999, 1.00000000000001]
[ 1.49999999999999, 1.50000000000001] [ -0.50000000000001, -0.49999999999999]
>> rad(B)
ans =
1.0e-015 *
0.44408920985006 0.33306690738755
0.22204460492503 0.11102230246252
The very small radius matrix shows the high accuracy achieved.
12.3. Singularity
An attempt to invert a singular matrix
>> format short
Ao=[1 2;5 10]
Ao =
1 2
5 10
>> Ao=infsup(Ao,Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 5.0000, 5.0000] [ 10.0000, 10.0000]
>> B=inv(Ao)
results in an interval matrix of NaN’s:
intval B =
[ NaN, NaN] [ NaN, NaN]
[ NaN, NaN] [ NaN, NaN]
12.4. NaN output
Using NaN’s for indication of an empty output is a typical feature of INTLAB. It enables feasibility of succeeding computations. E.g., in the above example
>> C=inv(Ao)*rand(2,2)
gives
intval C =
[ NaN, NaN] [ NaN, NaN]
[ NaN, NaN] [ NaN, NaN]
so that the computation does not break down despite the singularity of Ao.
13. SYSTEMS OF INTERVAL LINEAR EQUATIONS
A system of interval linear equations A*x=b (A, b of type intval, A square) can be solved (i.e., an enclosure of the solution set can be obtained) using X=verifylss(A,b) .
13.1. Interval data
Example.
>> AIntlab Matlab For Mac Operating System
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> bl=[-2 -2]’; bu=-bl;
>> b=infsup(bl,bu)
intval b =
[ -2.0000, 2.0000]
[ -2.0000, 2.0000]
>> X=verifylss(A,b) % Barth-Nuding 1974 example
intval X =
[ -14.0001, 14.0001]
[ -14.0001, 14.0001]
It is guaranteed that for each Ao in A and bo in b, Ao is nonsingular and the solution of Ao*x=bo is contained in X. The enclosure X is generally not optimal: the optimal enclosure XX would be
intval XX =
[ -4.0000, 4.0000]
[ -4.0000, 4.0000] Octave For MacIntlab Matlab For Mac Free
Wilton sprinkles expiration date. The overestimation is caused by too wide input intervals here.
13.2. Real data
The procedure can be used for solving systems with thin data (without converting them to type intval first, contrary to the procedure ’inv’ above):
Example.
>> A=[ 1 2 3;4 5 6;7 8 10]
A =
1 2 3
4 5 6
7 8 10
>> b=A*ones(3,1) % so that the exact solution is [1 11]’
b =
6
15
25
>> format long
>> X=verifylss(A,b)
intval X =
[ 0.99999999999999, 1.00000000000001]
[ 0.99999999999999, 1.00000000000001]
[ 0.99999999999999, 1.00000000000001]
We can see that the degree of guaranteed accuracy is surprising.
14. DATA CONVERSION
Most numbers (like 0.1) cannot be exactly represented in binary finite precision arithmetic. To handle errors created by data conversion, INTLAB enables us to use
>> a=intval(’0.1’)%observe the apostrophes
intval a =
[ 0.09999999999999, 0.10000000000001]
>> rad(a)
ans =
1.387778780781446e-017
(’rigorous representation’).
15. SOME ADDITIONAL INTERVAL FUNCTIONS
For x, y of type intval:
intersect(x,y) intersection of x, y
hull(x,y) union of x, y
abss(x) absolute value of x
mig(x) mignitude of x
in(x,y) x is included in y (logical array)
in0(x,y) x is included in the interior of y (logical array)
16. YOUR TURN NOW
Now, it is your turn to explore INTLAB, this wonderful tool created for us by Siegfried Michael Rump. (Thanks also for his remarks on this text.)
Download: http://gg.gg/ntmr5 https://diarynote.indered.space
byJiriRohn
*Intlab Matlab For Mac Pro
*Intlab Matlab For Mac Operating System
*Octave For Mac
*Intlab Matlab For Mac Free
1. INTRODUCTION
Sep 23, 2015 Make sure that you save MAC.m and ModeShapes.mat in the same folder and set that folder as MATLAB working directory or add a path to it. Then: load ModeShapes.mat;MACMatrix=MAC(phi) azade ch. Mar 07, 2016 If you are using R2014a or later, and you already have a license for the toolbox (and the license has made it through the processing to be registered against your account), then you run the MATLAB installer, which will detect which toolboxes are on your license and will give you an option to pick which ones you want to install. The Matlab code INTLAB is used for automatic differentiation. The same approach is used to calculate the sensitivity of C l 2 d w.r.t. Mac Wilkinson, W.T. 至于Matlab工具箱安装中涉及到了Matlab的搜索路径、工作目录、当前路径、用户路径等好多术语，我这里不想多说什么 感兴趣的网友，可以直接查看Matlab的帮助系统，在那里你可以得到最直接的答复，但是你需要一定的英文基础哦 添加工具箱的方法很多，所有方法都是为了达到同一个目的，将工具箱. INTLAB - Fast INTerval LABoratory. Matlab Octave toolbox. Self-validating methods. Interval arithmetic for real and complex data including vectors and full and sparse matrices (very fast). Affine arithmetic including vectors and matrices. Fl-numbers: k-bit floating-point and interval arithmetic. Gfp: Galois field GFp toolbox.
This is a very concise description of basic features of INTLAB aimed at newcomers to this area. Only the very basics of INTLAB are presented; INTLAB itself contains much more.
2. INTLAB’S AUTHOR
INTLAB, a MATLAB toolbox for self-validating algorithms, is a one-man work. It has been created by Prof. Dr. Siegfried M. Rump, Institute for Reliable Computing, TechnicalUniversity of Hamburg-Harburg, Germany.
3. INTLAB’S DOWNLOAD
INTLAB can be downloaded (free for academic use; observe the license) from
http://www.ti3.tu-harburg.de/rump/intlab/
4. INTLAB’S INSTALLATION
Follow the instructions in the file Readme.txt.
5. INTLAB’S DEMOS
Look at demointval.m, demointlab.m. Many important features contained there are summarized below.See also Hargreaves.
6. DEFAULT INITIALIZATIONS
There are two basic ways of displaying intervals:
intvalinit(’displayinfsup’) : display of intervals by endpoints,
intvalinit(’displaymidrad’) : display of intervals by midpoint and radius.
In what follows we use the first option. To invoke it, simply type
>> intvalinit(’displayinfsup’)
You will get the response
> Default display of intervals by infimum/supremum (e.g. [ 3.14 , 3.15 ])
7. INTVAL TYPE
INTLAB uses newly defined MATLAB type ’intval’ for interval quantities.
8. BASIC INTERVAL ARITHMETIC PROPERTY
An operation always uses interval arithmetic if at least one of the operands is of type intval. The output is always rigorous (i.e., it encloses the real result in floating point).
9. INPUT
Given two real (of type double) matrices (vectors, numbers) A, B of the same size, the interval matrix (vector, number) C=[A,B] is inputed simply as C=infsup(A,B).
Example. For real (double) matrices Ad, Ah given by
>> Ad=[2 -2; -1 2]
Ad =
2 -2
-1 2
>> Ah=[4 1; 2 4]
Ah =
4 1
2 4
>> A=infsup(Ad,Ah)
yields
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
A is the interval matrix [Ad,Ah].Alternatively, the same result can be achieved using
>> A=[infsup(2,4)infsup(-2,1);infsup(-1,2)infsup(2,4)]
intval A =
[2.0000,4.0000] [-2.0000,1.0000]
[-1.0000,2.0000] [2.0000,4.0000]Intlab Matlab For Mac Pro
Similarly, A=midrad(C,D) gives the interval matrix [C-D,C+D] (i.e, in the midpoint-radius representation). If you wish a real (double) matrix Ao to be handled as a thin interval matrix (so that the interval arithmetic could apply), useAo=intval(Ao)(or, equivalently, Ao=infsup(Ao,Ao)) :
>> Ao=[1 2;3 4]
Ao =
1 2
3 4
>> Ao=intval(Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 3.0000, 3.0000] [ 4.0000, 4.0000]
10. OUTPUT
Given an interval matrix A (of type intval), its bounds, midpoint and radius can be extracted as follows:
the lower bound is inf(A) or A.inf
the upper bound is sup(A) or A.sup
the midpoint is mid(A) or A.mid
the radius is rad(A) or A.rad .
Example. With the above A, we have
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> inf(A)
ans =
2 -2
-1 2
>> sup(A)
ans =
4 1
2 4
>> mid(A)
ans =
3.0000 -0.5000
0.5000 3.0000
>> rad(A)
ans =
1.0000 1.5000
1.5000 1.0000
11. DIFFERENCE BETWEEN A.inf AND inf(A)
A.inf and inf(A) are not quite equivalent. A.inf allows indexing:
>> A.inf(1,1)
ans =
2
whereas this is not possible for inf(A). The same holds for A.sup and sup(A), etc.
12. ENCLOSURE OF THE INVERSE INTERVAL MATRIX
We shall demonstrate some typical INTLAB features on the function B=inv(A) which computes an enclosure of the interval inverse of A.
12.1. Interval input
>> A
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> B=inv(A)
intval B =
[ -2.9704, 3.6191] [ -3.2729, 3.3810]
[ -3.3810, 3.2729] [ -2.9704, 3.6191]
This means that for each real (double) matrix Ao in A, its exact inverse is guaranteed to exist and to belong to B. The output uses outward rounding, so that the result is rigorous. The enclosure B is generally not optimal.
12.2 Real input
Inverse can also be used to obtain a rigorous output for a real (double) input:
Ao =
1 2
3 4
>> Ao=intval(Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 3.0000, 3.0000] [ 4.0000, 4.0000]
>> B=inv(Ao)
intval B =
[ -2.0001, -1.9999] [ 0.9999, 1.0001]
[ 1.4999, 1.5001] [ -0.5001, -0.4999]
To see better the accuracy of the result, use
>> format long
>> B
intval B =
[ -2.00000000000001, -1.99999999999999] [ 0.99999999999999, 1.00000000000001]
[ 1.49999999999999, 1.50000000000001] [ -0.50000000000001, -0.49999999999999]
>> rad(B)
ans =
1.0e-015 *
0.44408920985006 0.33306690738755
0.22204460492503 0.11102230246252
The very small radius matrix shows the high accuracy achieved.
12.3. Singularity
An attempt to invert a singular matrix
>> format short
Ao=[1 2;5 10]
Ao =
1 2
5 10
>> Ao=infsup(Ao,Ao)
intvalAo =
[ 1.0000, 1.0000] [ 2.0000, 2.0000]
[ 5.0000, 5.0000] [ 10.0000, 10.0000]
>> B=inv(Ao)
results in an interval matrix of NaN’s:
intval B =
[ NaN, NaN] [ NaN, NaN]
[ NaN, NaN] [ NaN, NaN]
12.4. NaN output
Using NaN’s for indication of an empty output is a typical feature of INTLAB. It enables feasibility of succeeding computations. E.g., in the above example
>> C=inv(Ao)*rand(2,2)
gives
intval C =
[ NaN, NaN] [ NaN, NaN]
[ NaN, NaN] [ NaN, NaN]
so that the computation does not break down despite the singularity of Ao.
13. SYSTEMS OF INTERVAL LINEAR EQUATIONS
A system of interval linear equations A*x=b (A, b of type intval, A square) can be solved (i.e., an enclosure of the solution set can be obtained) using X=verifylss(A,b) .
13.1. Interval data
Example.
>> AIntlab Matlab For Mac Operating System
intval A =
[ 2.0000, 4.0000] [ -2.0000, 1.0000]
[ -1.0000, 2.0000] [ 2.0000, 4.0000]
>> bl=[-2 -2]’; bu=-bl;
>> b=infsup(bl,bu)
intval b =
[ -2.0000, 2.0000]
[ -2.0000, 2.0000]
>> X=verifylss(A,b) % Barth-Nuding 1974 example
intval X =
[ -14.0001, 14.0001]
[ -14.0001, 14.0001]
It is guaranteed that for each Ao in A and bo in b, Ao is nonsingular and the solution of Ao*x=bo is contained in X. The enclosure X is generally not optimal: the optimal enclosure XX would be
intval XX =
[ -4.0000, 4.0000]
[ -4.0000, 4.0000] Octave For MacIntlab Matlab For Mac Free
Wilton sprinkles expiration date. The overestimation is caused by too wide input intervals here.
13.2. Real data
The procedure can be used for solving systems with thin data (without converting them to type intval first, contrary to the procedure ’inv’ above):
Example.
>> A=[ 1 2 3;4 5 6;7 8 10]
A =
1 2 3
4 5 6
7 8 10
>> b=A*ones(3,1) % so that the exact solution is [1 11]’
b =
6
15
25
>> format long
>> X=verifylss(A,b)
intval X =
[ 0.99999999999999, 1.00000000000001]
[ 0.99999999999999, 1.00000000000001]
[ 0.99999999999999, 1.00000000000001]
We can see that the degree of guaranteed accuracy is surprising.
14. DATA CONVERSION
Most numbers (like 0.1) cannot be exactly represented in binary finite precision arithmetic. To handle errors created by data conversion, INTLAB enables us to use
>> a=intval(’0.1’)%observe the apostrophes
intval a =
[ 0.09999999999999, 0.10000000000001]
>> rad(a)
ans =
1.387778780781446e-017
(’rigorous representation’).
15. SOME ADDITIONAL INTERVAL FUNCTIONS
For x, y of type intval:
intersect(x,y) intersection of x, y
hull(x,y) union of x, y
abss(x) absolute value of x
mig(x) mignitude of x
in(x,y) x is included in y (logical array)
in0(x,y) x is included in the interior of y (logical array)
16. YOUR TURN NOW
Now, it is your turn to explore INTLAB, this wonderful tool created for us by Siegfried Michael Rump. (Thanks also for his remarks on this text.)
Download: http://gg.gg/ntmr5 https://diarynote.indered.space
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