定義
收斂速度以收斂階衡量, 亦可以收斂因子描述; 依計算方法的不同, 有下述兩種收斂階及收斂階.
商收斂因子及商收斂階
![收斂速度](/img/4/c31/wZwpmL4gjM0IDO3cTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL3UzLzQzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
一、商收斂因子的定義式如下:
![收斂速度](/img/b/1f0/wZwpmLyUTMyQDO4EjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxYzLxEzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/7/eca/wZwpmLwgDNyAjM4YzM3UzM1UTM1QDN5MjM5ADMwAjMwUzL2MzL3YzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/5/8f8/wZwpmL1QTNwETM5QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czLxczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/e/a94/wZwpmLzADO2QDOwkjN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5YzLxMzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
1、如果 , 則稱 是 Q—超線性收斂於 ; 如果 , 則稱 是 Q—線性收斂於 ; 如果 則稱 是 Q—次線性收斂於 .
![收斂速度](/img/d/f68/wZwpmLyMzN0MzN2ITN0MTN1UTM1QDN5MjM5ADMwAjMwUzLyUzL2gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/0/aed/wZwpmLyYTMxEjNwMzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLzczLygzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/4/0a0/wZwpmLyMTO4ATN1EjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxYzLzIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
2、如果 , 則稱 是 Q—超平方收斂於 ; 如果 , 則稱 是 Q—平方收斂於 ; 如果 , 則稱 是 Q—次平方收斂於 .
注意: Q—線性收斂與Q—平方收斂, 以及Q—次線性收斂與Q—次平方收斂的評判標準有些微差別. “Q—平方收斂”也稱為“Q—二次收斂”.
![收斂速度](/img/1/1b5/wZwpmLycDM4YjN5MjM2EzM1UTM1QDN5MjM5ADMwAjMwUzLzIzLwczLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/0/1b0/wZwpmL4cTMzEzM0MzM3UzM1UTM1QDN5MjM5ADMwAjMwUzLzMzL4IzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
依照Q—平方收斂 (不是Q—線性收斂) 的定義, 可以定義Q—立方收斂 (將 改為 ), Q—四次方收斂等更高Q—收斂階.
![收斂速度](/img/1/951/wZwpmLzMjM5YDNycTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL3UzLyczLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
二、商收斂階的定義式如下:
![收斂速度](/img/d/26c/wZwpmL3YDNxADMwkTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5UzL0EzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![收斂速度](/img/f/de4/wZwpmL1YTM1kTOzQDNzEzM1UTM1QDN5MjM5ADMwAjMwUzL0QzLwUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![收斂速度](/img/c/b15/wZwpmLxQDN0kTO5EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL3AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/1/49a/wZwpmLxcTN1kTNyAzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLwMzL0IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/f/9ec/wZwpmLzYTO0ADO5MDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLzQzL0YzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/1/49a/wZwpmLxcTN1kTNyAzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLwMzL0IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/a/e3f/wZwpmL1AzN1kDO3ATMwEDN0UTMyITNykTO0EDMwAjMwUzLwEzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
根收斂因子及根收斂階
![收斂速度](/img/0/f45/wZwpmL2YTO5ETNxITO2UzM1UTM1QDN5MjM5ADMwAjMwUzLykzL4MzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
一、根收斂因子的定義式如下:
![收斂速度](/img/a/964/wZwpmL1YzM4MjN4AjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLwYzLwczLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/8/ccd/wZwpmL4IDO4cTN2IjN2UzM1UTM1QDN5MjM5ADMwAjMwUzLyYzLyMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/c/d99/wZwpmL3ETO4cDO5QzN2UzM1UTM1QDN5MjM5ADMwAjMwUzL0czLzMzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/8/237/wZwpmL3MTM3ADMzQDO2UzM1UTM1QDN5MjM5ADMwAjMwUzL0gzL4czLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
1、如果 則稱 是 R—超線性收斂於 ; 如果 , 則稱 是 R—線性收斂於 ; 如果 , 則稱 是 R—次線性收斂於 .
![收斂速度](/img/0/3b7/wZwpmL0cDO5ADMzMDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLzQzLyMzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/98a/wZwpmLyYTNwUjMwMzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLzczL1EzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/b/9d0/wZwpmL3UTM4AjMxAzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLwczL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/c/f0b/wZwpmLxUjNyATN0kTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL5EzL4QzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
2、如果 , 則稱 是 R—超平方收斂於 ; 如果 , 則稱 是 R—平方收斂於 ; 如果 , 則稱 是 R—次平方收斂於 .
注意: R—次線性收斂與R—次平方收斂的評判標準有些微差別. “R—平方收斂”也稱為“R—二次收斂”.
![收斂速度](/img/b/8fd/wZwpmLxUjN3YTN2cTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL3EzLwIzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/6/43c/wZwpmL3cDO3EjN4cjMzATN1UTM1QDN5MjM5ADMwAjMwUzL3IzL3gzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
依照R—平方收斂 (不是R—線性收斂) 的定義, 可以定義R—立方收斂 (將 改為 ), R—四次方收斂等更高R—收斂階.
![收斂速度](/img/3/8e9/wZwpmL3gTO1AzM5cjN0MTN1UTM1QDN5MjM5ADMwAjMwUzL3YzL1MzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
二、根收斂階的定義式如下:
![收斂速度](/img/1/6ec/wZwpmL2czMzMDO1EjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxYzL0AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/f/de4/wZwpmL1YTM1kTOzQDNzEzM1UTM1QDN5MjM5ADMwAjMwUzL0QzLwUzLt92YucmbvRWdo5Cd0FmLxE2LvoDc0RHa.jpg)
![收斂速度](/img/c/b15/wZwpmLxQDN0kTO5EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL3AzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/2/0cc/wZwpmL1UjNyIzM1EDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxQzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/1/49a/wZwpmLxcTN1kTNyAzMzEzM1UTM1QDN5MjM5ADMwAjMwUzLwMzL0IzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/f/9ec/wZwpmLzYTO0ADO5MDN0MTN1UTM1QDN5MjM5ADMwAjMwUzLzQzL0YzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
![收斂速度](/img/6/92f/wZwpmL2ITO5kzMzQTMzEzM1UTM1QDN5MjM5ADMwAjMwUzL0EzL3AzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/a/e3f/wZwpmL1AzN1kDO3ATMwEDN0UTMyITNykTO0EDMwAjMwUzLwEzL4UzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
對比根收斂因子的描述, 根收斂階是指求出一個數 (不一定是整數), 使得對於 , 點列 都是R—次 次方收於, 且對於 都是R— 次方收斂. 而這個數 就是點列的根收斂階.
兩種收斂階的聯繫
對於一個收斂點列而言, 其Q—收斂階不大於其R—收斂階, 即
![收斂速度](/img/3/ee3/wZwpmLwEDM2YDM5kDN0MTN1UTM1QDN5MjM5ADMwAjMwUzL5QzLxgzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
有時, 一個數列的 R—收斂階可能很高, 但其 Q—收斂階可能很低. 當然可以證明, 一個 R—收斂階高的點列至少比某些 Q—收斂低的點列收斂得更快.
實例
數列
有如下向量列:
![收斂速度](/img/0/9f8/wZwpmL0IDMzkzNygTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL4UzLzEzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
據上作出計算如下,
![收斂速度](/img/6/73c/wZwpmL4MDN0kDNzAjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLwYzL4YzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
故數列為Q線性收斂; Q收斂階為1;
![收斂速度](/img/f/a99/wZwpmLzQzNzYTM4EjN0MTN1UTM1QDN5MjM5ADMwAjMwUzLxYzLyQzLt92YucmbvRWdo5Cd0FmL0E2LvoDc0RHa.jpg)
故數列為R線性收斂; R收斂階為1.
最佳化算法的疊代點列
![收斂速度](/img/b/cd4/wZwpmLxcTNyUjNwMDO4EDN0UTMyITNykTO0EDMwAjMwUzLzgzLzIzLt92YucmbvRWdo5Cd0FmLwE2LvoDc0RHa.jpg)
![收斂速度](/img/8/d43/wZwpmL4MzN0UDN0cjN0MTN1UTM1QDN5MjM5ADMwAjMwUzL3YzL2czLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
![收斂速度](/img/3/41a/wZwpmL2cTM5gDOyUTN0MTN1UTM1QDN5MjM5ADMwAjMwUzL1UzL3gzLt92YucmbvRWdo5Cd0FmLzE2LvoDc0RHa.jpg)
對於牛頓法,可以證明, 如果牛頓法的目標函式 的二階導數 在其收斂點 處Lipschitz連續, 則滿足不等式
![收斂速度](/img/c/d9a/wZwpmL3MjM0UDM3AzN0MTN1UTM1QDN5MjM5ADMwAjMwUzLwczL3gzLt92YucmbvRWdo5Cd0FmLyE2LvoDc0RHa.jpg)
此說明牛頓法的疊代點列是Q平方收斂; 另言之, 牛頓法的收斂速度是二次的.