THE HEART SUTRA---《般若波罗蜜多心经》英文版 by Edward Conze

本文深入探讨了普贤菩萨通过依靠智慧之法而达到无思无虑境界的过程,阐述了空性的本质及其在五蕴中的体现。进一步解释了所有法门皆为空性的核心观念,并强调了这一理念如何引领菩萨走向涅槃。最后引用了“空门”咒语,完整地阐述了完美智慧之心的精髓。

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找了半天,留作备份,改天再详读。

Om Homage to the Perfection of Wisdom the Lovely, the Holy !

Avalokita, the Holy Lord and Bodhisattva, was moving in the deep course of the Wisdom which has gone beyond.

He looked down from on high, He beheld but five heaps, and He saw that in their own-being they were empty.

Here, O Sariputra,

form is emptiness and the very emptiness is form ;

emptiness does not differ from form, form does not differ from emptiness, whatever is emptiness, that is form,

the same is true of feelings, perceptions, impulses, and consciousness.

Here, O Sariputra,

all dharmas are marked with emptiness ;

they are not produced or stopped, not defiled or immaculate, not deficient or complete.

Therefore, O Sariputra,

in emptiness there is no form nor feeling, nor perception, nor impulse, nor consciousness ;

No eye, ear, nose, tongue, body, mind ; No forms, sounds, smells, tastes, touchables or objects of mind ; No sight-organ element, and so forth, until we come to :

No mind-consciousness element ; There is no ignorance, no extinction of ignorance, and so forth, until we come to : There is no decay and death, no extinction of decay and death. There is no suffering, no origination, no stopping, no path.

There is no cognition, no attainment and no non-attainment.

Therefore, O Sariputra,

it is because of his non-attainmentness that a Bodhisattva, through having relied on the Perfection of Wisdom, dwells without thought-coverings. In the absence of thought-coverings he has not been made to tremble,

he has overcome what can upset, and in the end he attains to Nirvana.

All those who appear as Buddhas in the three periods of time fully awake to the utmost, right and perfect Enlightenment because they have relied on the Perfection of Wisdom.

Therefore one should know the prajnaparamita as the great spell, the spell of great knowledge, the utmost spell, the unequalled spell, allayer of all suffering, in truth -- for what could go wrong ? By the prajnaparamita has this spell been delivered. It runs like this :

gate gate paragate parasamgate bodhi svaha.

( Gone, gone, gone beyond, gone altogether beyond, O what an awakening, all-hail ! -- )

This completes the Heart of perfect Wisdom.

资源下载链接为: https://pan.quark.cn/s/9e7ef05254f8 行列式是线性代数的核心概念,在求解线性方程组、分析矩阵特性以及几何计算中都极为关键。本教程将讲解如何用C++实现行列式的计算,重点在于如何输出分数形式的结果。 行列式定义如下:对于n阶方阵A=(a_ij),其行列式由主对角线元素的乘积,按行或列的奇偶性赋予正负号后求和得到,记作det(A)。例如,2×2矩阵的行列式为det(A)=a11×a22-a12×a21,而更高阶矩阵的行列式可通过Laplace展开或Sarrus规则递归计算。 在C++中实现行列式计算时,首先需定义矩阵类或结构体,用二维数组存储矩阵元素,并实现初始化、加法、乘法、转置等操作。为支持分数形式输出,需引入分数类,包含分子和分母两个整数,并提供与整数、浮点数的转换以及加、减、乘、除等运算。C++中可借助std::pair表示分数,或自定义结构体并重载运算符。 计算行列式的函数实现上,3×3及以下矩阵可直接按定义计算,更大矩阵可采用Laplace展开或高斯 - 约旦消元法。Laplace展开是沿某行或列展开,将矩阵分解为多个小矩阵的行列式乘积,再递归计算。在处理分数输出时,需注意避免无限循环和除零错误,如在分数运算前先约简,确保分子分母互质,且所有计算基于整数进行,最后再转为浮点数,以避免浮点数误差。 为提升代码可读性和可维护性,建议采用面向对象编程,将矩阵类和分数类封装,每个类有明确功能和接口,便于后续扩展如矩阵求逆、计算特征值等功能。 总结C++实现行列式计算的关键步骤:一是定义矩阵类和分数类;二是实现矩阵基本操作;三是设计行列式计算函数;四是用分数类处理精确计算;五是编写测试用例验证程序正确性。通过这些步骤,可构建一个高效准确的行列式计算程序,支持分数形式计算,为C++编程和线性代数应用奠定基础。
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