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LATIHAN 3

Algoritma Metode Iterasi Jacobi dalam bentuk software Matlab

Image representing Iterasi as depicted in Crun...

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Penggunaan algoritma Metode Iterasi Jacobi dalam bentuk matlab. Matlab merupakan program pengolahan data numerik.

INPUT :

n, A, b, dan hampiran awal Y=(y1 y2 y3…yn)T , batas toleransi T, dan maksimum iterasi N

OUTPUT :

X=(x1 x2 x3…xn)T, vektor galat hampiran g, dan H yang merupakan matriks dengan baris vektor-vektor hampiran selama iterasi.
H=X0′
n=length (b)
X=X0
for k:=1 until N

for i:=i until n,

S = b (i) – A (i,[1:i-1,i+1:n]) * X0 (1:i-1,i+1:n](
X(i) = S / A (i,i)
end
g = abs (X-X0)
err = norm (g)
relerr = err / (norm (X)+eps)
X0 = X
H = [H;X0′]
if (err<T)|(relerr<T), break, end
end

Dimension

The columns of a matrix.

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The dimension of the column space is called the rank of the matrix. The rank is equal to the number of pivots in the reduced row echelon form, and is the maximum number of linearly independent columns that can be chosen from the matrix. For example, the 4 × 4 matrix in the example above has rank three.

Because the column space is the image of the corresponding matrix transformation, the rank of a matrix is the same as the dimension of the image. For example, the transformation R4 → R4 described by the matrix above maps all of R4 to some three-dimensional subspace.

The nullity of a matrix is the dimension of the null space, and is equal to the number of columns in the reduced row echelon form that do not have pivots.[3] The rank and nullity of a matrix A with n columns are related by the equation:

\text{rank}(A) + \text{nullity}(A) = n.\,

This is known as the rank-nullity theorem.

RANGE

Naofuncao1

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In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f(x) = x2. Some books say that range of this function is its codomain, the set of all real numbers, reflecting that the function is real-valued. These books call the actual output of the function the image. This is the current usage for range in computer science. Other books say that the range is the function’s image, the set of non-negative real numbers, reflecting that a number can be the output of this function if and only if it is a non-negative real number. In this case, the larger set containing the range is called the codomain.[1] This usage is more common in modern mathematics. Read the rest of this entry

TRIANGLE INEQUALITY

Generalization for arbitrary triangles, green area

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In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

In Euclidean geometry and some other geometries the triangle inequality is a theorem about distances. In Euclidean geometry, for right triangles it is a consequence of Pythagoras’ theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either R2 or R3. The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.

In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in [0, π]) with those endpoints.

The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (p ≥ 1), and inner product spaces.

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Integrals of differential forms

HauptsatzDerInfinitesimalrechnung-f grad5

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A differential form is a mathematical concept in the fields of multivariable calculus, differential topology and tensors. The modern notation for the differential form, as well as the idea of the differential forms as being the wedge products of exterior derivatives forming an exterior algebra, was introduced by Élie Cartan.

We initially work in an open set in Rn. A 0-form is defined to be a smooth function f. When we integrate a function f over an m-dimensional subspace S of Rn, we write it as

\int_S f\,dx^1 \cdots dx^m.

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NUMERICAL QUADRATURE

Aproximació de la integral indefinida d'una fu...

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The integrals encountered in a basic calculus course are deliberately chosen for simplicity; those found in real applications are not always so accommodating. Some integrals cannot be found exactly, some require special functions which themselves are a challenge to compute, and others are so complex that finding the exact answer is too slow. This motivates the study and application of numerical methods for approximating integrals, which today use floating-point arithmetic on digital electronic computers. Many of the ideas arose much earlier, for hand calculations; but the speed of general-purpose computers like the ENIAC created a need for improvements.

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