The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form.
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| - Element de pivot (ca)
- Pivotelement (de)
- Pivot (matematica) (it)
- 피벗 (ko)
- Pivot element (en)
- 主元 (zh)
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| - ( 다른 뜻에 대해서는 피벗 (동음이의) 문서를 참고하십시오.) 선형대수학에서 피벗(pivot) 또는 피벗 성분(pivot entry,pivot element)는 특정 계산을 수행하기 위한 임의의 알고리즘 (예 : 가우스 소거법, 단순 알고리즘 등)에 의해 먼저 선택된 행렬의 성분(항,원소)이다. (ko)
- 主元（英語：pivot或pivot element）是矩陣、陣列或是其他有限集合的一個演算元素，算法（如高斯消去法、快速排序、单纯形法等等）首先选出主元，用于特定计算。 在矩阵算法中，主元必须是非零元素，甚至是距零最远的元素（绝对值最大）。寻找主元的过程被称为pivoting。随后把主元所在的行交换到固定位置，用于随后的计算。主元所在的列组成列空间的一个基。但实际的算法很少移动矩阵的行，因为这对于大矩阵（含有几千到几百万的行与列）将招致极大的时间花费；替代的办法是仅仅记录矩阵的行的交换信息。 整体上，寻找主元的过程增加了算法的计算量。很多情况下这些额外的计算量是必需的，能使算法正常工作，或者对于保持计算结果的数值稳定性来说是完全有价值的. (zh)
- Das Pivotelement (franz. pivot ‚Dreh-, Angelpunkt‘) ist dasjenige Element einer Zahlenmenge, das als Erstes von einem Algorithmus (z. B. Gaußsches Eliminationsverfahren, Quicksort, Pivotverfahren) ausgewählt wird, um bestimmte Berechnungen durchzuführen. Beim Sortieren mittels Quicksort bezeichnet das Pivotelement jenes Element, das als Aufteilungsgrenze gewählt wird. Quicksort sortiert (rekursiv) alle Elemente links und rechts vom Pivotelement. Optimal ist dabei das Median-Element, das zwei gleich große Teillisten erzeugt. (de)
- The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form. (en)
- In matematica, e più specificamente in algebra lineare, e in informatica, il pivot (in francese perno), elemento di pivot o elemento pivotale di una matrice è l'elemento della matrice che viene scelto per primo da un algoritmo (algoritmo di Gauss, ordinamento quicksort, metodo del simplesso, etc) e che si richiede rispetti determinate proprietà allo scopo di far funzionare correttamente o del tutto l'algoritmo, o più semplicemente per renderne l'esecuzione più efficiente. (it)
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| - Das Pivotelement (franz. pivot ‚Dreh-, Angelpunkt‘) ist dasjenige Element einer Zahlenmenge, das als Erstes von einem Algorithmus (z. B. Gaußsches Eliminationsverfahren, Quicksort, Pivotverfahren) ausgewählt wird, um bestimmte Berechnungen durchzuführen. Damit Matrix-Algorithmen wie etwa das Gaußsche Eliminationsverfahren arbeiten können, ist es oft nötig, dass Elemente ungleich null existieren. Je nach Algorithmus wird gegebenenfalls nicht nur nach einem nicht verschwindenden, sondern auch nach dem (betragsmäßig) größten Element in der jeweiligen Zeile oder Spalte gesucht. Die solchermaßen getroffene Auswahl des Elements nennt man dann Pivotisierung. Die Zeile, in der das Pivotelement steht, nennt man Pivotzeile, die Spalte des Pivotelements heißt Pivotspalte. Vor der Pivotisierung ist gegebenenfalls eine Äquilibrierung durchzuführen um die Konditionszahl zu verbessern. Beim Sortieren mittels Quicksort bezeichnet das Pivotelement jenes Element, das als Aufteilungsgrenze gewählt wird. Quicksort sortiert (rekursiv) alle Elemente links und rechts vom Pivotelement. Optimal ist dabei das Median-Element, das zwei gleich große Teillisten erzeugt. (de)
- The pivot or pivot element is the element of a matrix, or an array, which is selected first by an algorithm (e.g. Gaussian elimination, simplex algorithm, etc.), to do certain calculations. In the case of matrix algorithms, a pivot entry is usually required to be at least distinct from zero, and often distant from it; in this case finding this element is called pivoting. Pivoting may be followed by an interchange of rows or columns to bring the pivot to a fixed position and allow the algorithm to proceed successfully, and possibly to reduce round-off error. It is often used for verifying row echelon form. Pivoting might be thought of as swapping or sorting rows or columns in a matrix, and thus it can be represented as multiplication by permutation matrices. However, algorithms rarely move the matrix elements because this would cost too much time; instead, they just keep track of the permutations. Overall, pivoting adds more operations to the computational cost of an algorithm. These additional operations are sometimes necessary for the algorithm to work at all. Other times these additional operations are worthwhile because they add numerical stability to the final result. (en)
- In matematica, e più specificamente in algebra lineare, e in informatica, il pivot (in francese perno), elemento di pivot o elemento pivotale di una matrice è l'elemento della matrice che viene scelto per primo da un algoritmo (algoritmo di Gauss, ordinamento quicksort, metodo del simplesso, etc) e che si richiede rispetti determinate proprietà allo scopo di far funzionare correttamente o del tutto l'algoritmo, o più semplicemente per renderne l'esecuzione più efficiente. Quando ci si riferisce a matrici a scalini, solitamente nell'ambito dell'eliminazione gaussiana, con pivot di una riga si intende il primo elemento non nullo della riga (se esiste). (it)
- ( 다른 뜻에 대해서는 피벗 (동음이의) 문서를 참고하십시오.) 선형대수학에서 피벗(pivot) 또는 피벗 성분(pivot entry,pivot element)는 특정 계산을 수행하기 위한 임의의 알고리즘 (예 : 가우스 소거법, 단순 알고리즘 등)에 의해 먼저 선택된 행렬의 성분(항,원소)이다. (ko)
- 主元（英語：pivot或pivot element）是矩陣、陣列或是其他有限集合的一個演算元素，算法（如高斯消去法、快速排序、单纯形法等等）首先选出主元，用于特定计算。 在矩阵算法中，主元必须是非零元素，甚至是距零最远的元素（绝对值最大）。寻找主元的过程被称为pivoting。随后把主元所在的行交换到固定位置，用于随后的计算。主元所在的列组成列空间的一个基。但实际的算法很少移动矩阵的行，因为这对于大矩阵（含有几千到几百万的行与列）将招致极大的时间花费；替代的办法是仅仅记录矩阵的行的交换信息。 整体上，寻找主元的过程增加了算法的计算量。很多情况下这些额外的计算量是必需的，能使算法正常工作，或者对于保持计算结果的数值稳定性来说是完全有价值的. (zh)
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