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北京理工深圳研究院 →Poetry on Marriage
on Marriage
Mary Weston Fordham
die is cast, come weal, come woe
Two lives are joined together,,
For better or for worse, the link
Which naught but death can sever.
The die is cast, come grief, come joy.
Come richer, or come poorer,
If love but binds the mystic tie,
Blest is the bridal hour.
Ogden Nash ()
of countless nations
Have told their love for generations
Till all their memorable phrases
Are common as goldenrod or daisies.
Their girls have glimmered like the moon,
Or shimmered like a summer moon,
Stood like a lily, fled like a fawn,
Now the sunset, now the dawn,
Here the princess in the tower
There the sweet forbidden flower.
Darling, when I look at you
Every aged phrase is new,
And there are moments when it seems
I've married one of Shakespeare's dreams.
The Colour of My Love
Foster and Arthur Janov
paint a sun to warm your heart
Knowing that we'll never part.
I'll draw the years all passing by
So much to learn, so much to try.
paint my mood in a shadow blue,
Paint my soul to be with you.
I'll sketch your lips in shaded tones,
Draw your mouth to my own.
trace a hand to wipe your tears
And trace a look to calm your fears.
A silhouette of dark and light
To hold each other oh so tight.
paint the stars in the evening sky,
Draw the light into your eyes,
A touch of love, a touch of grace,
To softly fall on your moonlit face.
with this ring our lives will start,
Let nothing keep our love apart.
I'll take your hand to hold in mine,
And be together through all time.
From This Day Forward
this day forward,
You shall not walk alone.
My heart will be your shelter,
And my arms will be your home.
William Shakespeare ()
me not to the marriage of true minds
Admit impediments. Love is not love
Which alters when it alteration finds.
Or bends with the remover to remove:
O, no! it is an ever-fixed mark,
That looks on tempests and is not shaken:
It is the star to every wandering bark,
Whose worth's unknown, although his height be taken.
Loves's not time's fool, though rosy lips and cheeks
Within his bending sickle'
Love alters not with his brief hours and weeks,
But bears it out even to the edge of doom,
If this be error, and upon me proved,
I never writ, nor no man ever loved.
Edwin Arnold ()
there waiteth in this world of ours
for one lone soul, another lonely soul -
Each chasing each through all the weary hours,
And meeting strangel
Then blend they - like green leaves with golden flowers,
Into one beautiful and perfect whole -
And life's long night is ended, and the way
Lies open onward to eternal day.
These I Can Promise
cannot promise yo
I cannot promise riches, wealth,
I cannot promise you an easy pathway
That leads away from change or growing old.
I can promise all my heart'
A smile to chase away
A love that's ever t
A hand to hold in yours through each tomorrow.
Yes, I'll Marry You
I'll marry you, my dear,
So I can push you out of bed
When the baby starts to cry,
And if we hear a knocking
And it's creepy and it's late,
I hand you the torch you see,
And you investigate.
I'll marry you, my dear,
You may not apprehend it,
But when the tumble-drier goes
It's you that has to mend it,
You have to face the neighbour
Should our labrador attack him,
And if a drunkard fondles me
It's you that has to whack him.
I'll marry you,
You're virile and you're lean,
My house is like a pigsty
You can help to keep it clean.
That sexy little dinner
Which you served by candlelight,
As I do chipolatas,
You can cook it every night!
you who has to work the drill
and put up curtain track,
And when I've got PMT it's you who gets the flak,
I do see great advantages,
But none of them for you,
And so before you see the light,
I do, I do, I do!
Our Mother
are the mother I received
The day I wed your son.
And I just want to thank you, Mum
For all the things you've done.
given me a gracious man
With whom I share my life.
You are his loving mother and
I his lucky wife.
used to pat his little head,
And now I hold his hand.
You raised in love a little boy
And gave to me a man.
This Day I Married My Best Friend
day I married my best friend
...the one I laugh with as we share life's wonderous zest,
as we find new enjoyments and experience all that's best.
...the one I live for because the world seems brighter
as our happy times are better and our burdens feel much lighter.
...the one I love with every fiber of my soul.
We used to feel vaguely incomplete, now together we are whole.
Today I Married My Best Friend
Elizabeth Cooper (b. 1977)
I married my best friend,
Our bond complete, it hath no end,
We share one soul, we share one heart,
A perfect time - a perfect start.
these rings we share together,
Love so close to last forever,
This special day - two special hearts,
Let nothing keep this love apart.
At My Daughter's Side
was a lifetime away, now suddenly it's here.
How did it happen so quickly? This wedding drawing near.
How can I act so happy? How can I act so gay?
When in such a very short time, I'll give my daughter away.
I wish I could grasp a moment, and make the clock stand still
So I could let my heart catch up, but I know it never will.
All the worries of being a parent, all the battles won,
No one ever warned me about the day the job is done.
Yet, there is another side, where my heart is not as sad.
When I look in my daughter's eyes, I can't help but be joyful and glad.
This day she has dreamt about, for just about all her life.
She's going to be such a beautiful bride and a loving, caring wife.
I'll stand with the congregation as my daughter walks down the aisle
And even though there are tears in my eyes, my face wll bear a smile.
For I know that I was very blessed when God lent this child to me. To
love and care for and nurture, so she would grow up to be This lovely,
bright young woman, who tomorrow will be a bride And as always I'll
be there, with love at my daughter's side.
promise to give you the best of myself
and to ask of you no more than you can give.
promise to respect you as your own person
and to realize that your interests, desires and needs
are no less important than my own.
promise to share with you my time and my attention
and to bring joy, strength and imagination to our relationship.
I promise to keep myself open to you,
to let you see through the window of my world into my innermost fears
and feelings, secrets and dreams.
promise to grow along with you,
to be willing to face changes in order to keep our relationship alive
and exciting.
I promise to love you in good times and bad,
with all I have to give and all I feel inside in the only way I know
how. Completely and forever.
Marriage Advice
Wells (1886)
your love be stronger than your hate and anger.
Learn the wisdom of compromise, for it is better to bend a little than
to break. Believe the best rather than the worst.
People have a way of living up or down to your opinion of them.
Remember that true friendship is the basis for any lasting relationship.
The person you choose to marry is deserving of the courtesies and kindnesses
you bestow on your friends.
Please hand this down to your children and your children's children.
Never Marry But For Love
William Penn ()
but see that thou lovest what is lovely. He that
minds a body and not a soul has not the better part of that relationship,
and will consequently lack the noblest comfort of a married life.
Between a man and his wife nothing ought rule but love. As love ought
to bring them together, so it is the best way to keep them well together.
husband and wife that love one another show their children that they
should do so too. Others visibly lose their authority in their families
by their contempt of one another, and teach their children to be unnatural
by their own examples.
not enjoyment lessen, but augment, it being the basest of
passions to like when we have not, what we slight when we possess.
it is we ought to search out our pleasure, where the field is large
and full of variety, and o sickness, poverty or
disgrace being not able to shake it because it is not under the moving
influences of worldly contingencies.
can be more entire nothing more zealous, affectionate
nothing more contented than such a couple, nor greater
temporal felicity than to be one of them.
The Art Of A Good Marriage
Arlan Peterson
in marriage is not something that just happens.
A good marriage must be created. In marriage the little things are the
big things. It is never being too old to hold hands.
It is remembering to say &I love you& at least once a day.
It is never going to sleep angry. It is at no time taking the other
the courtship should not end with the honeymoon, it should
continue through the years. It is having a mutual sense of values and
common objectives. It is standing together facing the world. It is forming
a circle of love that gathers the whole family.
It is doing things for each other, not in the attitude of duty or sacrifice,
but in the spirit of joy. It is speaking words of appreciation and demonstrating
gratitude in thoughtful ways.
It is not looking for perfection in each other. It is cultivating flexibility,
patience, understanding and a sense of humour.
It is having the capacity to forgive and forget. It is giving each other
an atmosphere in which each can grow old. It is a common search for
the good and the beautiful. It is establishing a relationship in which
the independence is equal, dependence is mutual and the obligation is
reciprocal. It is not only marryi it is being the
right partner.
A Good Wedding Cake
lb. of love.
1 lb. butter of youth.
1/2 lb. of good looks.
l lb. sweet temper.
1 lb. of blindness for faults.
1 lb. of self forgetfulness.
l lb. of pounded wit.
l lb. of good humour.
2 tablespoons of sweet argument.
1 pint of rippling laughter.
1 wine glass of common sense.
1 oz. modesty.
Put the love, good looks and sweet temper into a well furnished house.
Beat the butter of youth to a cream, and mix well together with the
blindness of faults. Stir the pounded wit and good humour into the sweet
argument, then add the rippling
laughter and common sense. Work the whole together until everything
is well mixed, and bake gently for ever.
The Blessing Of The Apaches
you will feel no rain,
For each of you will be shelter to the other.
Now you will feel no cold,
For each of you will be warmth to the other.
Now there is no more lonliness for you,
For each of you will be companion to the other.
Now you are two bodies,
But there is only one life before you.
Go now to your dwelling place,
To enter into the days of your togetherness.
And may your days be good and long upon the earth.
Apache Blessing
the sun bring you new energy by day,
May the moon softly restore you by night,
May the rain wash away your worries
And the breeze blow new strength into your being,
And all the days of your life may you walk
Gently through the world and know its beauty.
Eskimo Love Song
Author Unknown
are my husband (wife)
My feet shall run because of you
My feet shall dance because of you
My heart shall beat because of you
My eyes see because of you
My mind thinks because of you
And I shall love because of you.
Marriage Joins Two People In The Circle Of Its Love
O'Neill (b. 1929)
is a commitment to life, the best that two people can find and bring
out in each other. It offers opportunities for sharing and growth that
no other relationship can equal. It is a physical and an emotional joining
that is promised for a lifetime.
Within the circle of its love, marriage encompasses all of life's most
important relationships. A wife and a husband are each other's best
friend, confidant, lover, teacher, listener, and critic. And there may
come times when one partner is heartbroken or ailing, and the love of
the other may resemble the tender caring of a parent or child.
Marriage deepens and enriches every facet of life. Happiness is fuller,
memories are fresher, commitment is stronger, even anger is felt more
strongly, and passes away more quickly.
understands and forgives the mistakes life is unable to avoid. It encourages
and nurtures new life, new experiences, new ways of expressing a love
that is deeper than life.
two people pledge their love and care for each other in marriage, they
create a spirit unique unto themselves which binds them closer than
any spoken or written words. Marriage is a promise, a potential made
in the hearts of two people who love
each other and takes a lifetime to fulfill.
Our Family
family is a circle of love and strength.
With every birth and every union, the circle grows.
Every joy shared adds more love.
Every obstacle faced together makes the circle stronger.
Why Marriage?
to the depths of me, I long to love one person,
With all my heart, my soul, my mind, my body...
Because I need a forever friend to trust with the intimacies of me,
Who won't hold them against me,
Who loves me when I'm unlikable,
Who sees the small child in me, and
Who looks for the divine potential of me...
I need to cuddle in the warmth of the night
With someone who thanks God for me,
With someone I feel blessed to hold...
Because marriage means opportunity
To grow in love in friendship...
Because marriage is a discipline
To be added to a list of achievements...
Because marriages do not fail, people fail
When they enter into marriage
Expecting another to make them whole...
Because, knowing this,
I promise myself to take full responsibility
For my spiritual, mental and physical wholeness
I create me, I take half of the responsibility for my marriage
Together we create our marriage...
of this understanding
The possibilities are limitless.
On Marriage
Almitra spoke again and said, &And what of Marriage, master?&
And he answered saying: You were born together, and together you shall
be forevermore. You shall be together when white wings of death scatter
your days. Aye, you shall be together even in the silent memory of God.
But let there be spaces in your togetherness, And let winds of the heavens
dance between you. Love one another but make not a bond of love: Let
it rather be a moving sea between the shores of your souls. Fill each
other's cup but drink not from one cup. Give one another of your bread
but eat not from the same loaf. Sing and dance together and be joyous,
but let each one of you be alone, Even as the strings of a lute are
alone though they quiver with the same music. Give your hearts, but
not into each other's keeping. For only the hand of Life can contain
your hearts. And stand together, yet not too near together: For the
pillars of the temple stand apart, And the oak tree and the cypress
grow not in each other's shadow.
Blessing For A Marriage
Dillet Freeman
May your marriage bring you all the exquisite
excitements a marriage should bring, and may life grant you also patience,
tolerance, and understanding.
you always need one another - not so much to fill your emptiness as
to help you know your fullness. A mountain needs a v
the valley does not make the mountain less, and the valley
is more a valley because it has a mountain towering over it. So let
it be with you and you.
you need one another, but not out of weakness.
May you want one another, but not out of lack.
May you entice one another, but not compel one another.
May you succeed in all important ways with one another, and not fail
in the little graces. May you look for things to praise, often say,
&I love you!& and take no notice of small faults.
you have quarrels that push you apart, may both of you hope to have
good sense enough to take the first step back.
you enter into the mystery which is the awareness of one another's presence
- no more physical than spiritual, warm and near when you are side by
side, and warm and near when you are in separate rooms or even distant
cities. May you have happiness, and may you find it making one another
happy. May you have love, and may you find it loving one another!
To Be One With Each Other
greater thing is there for two human souls than to feel that they are
joined together to strengthen each other in all labour, to minister
to each other in all sorrow, to share with each other in all gladness,
to be one with each other in the silent unspoken memories?浸果处理后苯醚甲环唑在柑橘贮藏过程中残留量的变化_图文_百度文库
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你可能喜欢迭代法_百度百科
迭代法也称辗转法，是一种不断用变量的旧值递推新值的过程，跟迭代法相对应的是，即一次性解决问题。迭代法又分为精确迭代和近似迭代。“二分法”和“”属于近似迭代法。迭代算法是用计算机解决问题的一种基本方法。它利用计算机运算速度快、适合做重复性操作的特点，让计算机对一组指令（或一定步骤）进行重复执行，在每次执行这组指令（或这些步骤）时，都从变量的原值推出它的一个新值。
迭代法算法
迭代是中通过从一个初始估计出发寻找一系列近似解来解决问题（一般是解或者）的过程，为实现这一过程所使用的方法统称为迭代法（Iterative Method）。
一般可以做如下定义：对于给定的线性方程组
（这里的x、B、f同为，任意线性方程组都可以变换成此形式），用公式
代表迭代k次得到的x，初始时k=0)逐步带入求近似解的方法称为迭代法（或称一阶定常迭代法）。如果
存在，记为x*，称此迭代法。显然x*就是此的解，否则称为迭代法发散。
跟迭代法相对应的是（或者称为一次解法），即一次性的快速解决问题，例如通过开方解决x +3= 4。一般如果可能，直接解法总是优先考虑的。但当遇到复杂问题时，特别是在未知量很多，方程为非线性时，我们无法找到直接解法（例如五次以及更高次的没有，参见），这时候或许可以通过迭代法寻求方程（组）的近似解。
最常见的迭代法是。其他还包括、共轭迭代法、变尺度迭代法、、、、单纯型法、惩罚函数法、斜率投影法、、等等。
利用迭代算法解决问题，需要做好以下三个方面的工作：
迭代法确定迭代变量
在可以用迭代算法解决的问题中，至少存在一个直接或间接地不断由旧值递推出新值的变量，这个变量就是迭代变量。
迭代法建立迭代关系式
所谓迭代关系式，指如何从变量的前一个值推出其下一个值的公式（或关系）。迭代关系式的建立是解决迭代问题的关键，通常可以顺推或倒推的方法来完成。
迭代法对迭代过程进行控制
在什么时候结束迭代过程？这是编写迭代程序必须考虑的问题。不能让迭代过程无休止地重复执行下去。迭代过程的控制通常可分为两种情况：一种是所需的迭代次数是个确定的值，可以计算出来；另一种是所需的迭代次数无法确定。对于前一种情况，可以构建一个固定次数的循环来实现对迭代过程的控制；对于后一种情况，需要进一步分析出用来结束迭代过程的条件。
迭代法举例
例 1 ：一个饲养场引进一只刚出生的新品种兔子，这种兔子从出生的下一个月开始，每月新生一只兔子，新生的兔子也如此繁殖。如果所有的兔子都不死去，问到第 12 个月时，该饲养场共有兔子多少只？
分析：这是一个典型的递推问题。我们不妨假设第 1 个月时兔子的只数为 u 1 ，第 2 个月时兔子的只数为 u 2 ，第 3 个月时兔子的只数为 u 3 ，……根据题意，“这种兔子从出生的下一个月开始，每月新生一只兔子”，则有
u 1 = 1 ， u 2 = u 1 + u 1 × 1 = 2 ， u 3 = u 2 + u 2 × 1 = 4 ，……
根据这个规律，可以归纳出下面的：
u n = u(n － 1）× 2 (n ≥ 2）
对应 u n 和 u(n － 1），定义两个迭代变量 y 和 x ，可将上面的递推公式转换成如下迭代关系：
让计算机对这个迭代关系重复执行 11 次，就可以算出第 12 个月时的兔子数。参考程序如下：
for i=2 to 12
例 2 ：用简单分裂的方式繁殖，它每分裂一次要用 3 分钟。将若干个阿米巴放在一个盛满营养参液的容器内， 45 分钟后容器内充满了阿米巴。已知容器最多可以装阿米巴 220,220个。试问，开始的时候往容器内放了多少个阿米巴？请编程序算出。
分析：根据题意，阿米巴每 3 分钟分裂一次，那么从开始的时候将阿米巴放入容器里面，到 45 分钟后充满容器，需要分裂 45/3=15 次。而“容器最多可以装阿米巴2^ 20 个”，即阿米巴分裂 15 次以后得到的个数是 2^20。题目要求我们计算分裂之前的阿米巴数，不妨使用倒推的方法，从第 15 次分裂之后的 2^20 个，倒推出第 15 次分裂之前（即第 14 次分裂之后）的个数，再进一步倒推出第 13 次分裂之后、第 12 次分裂之后、……第 1 次分裂之前的个数。
设第 1 次分裂之前的个数为 x 0 、第 1 次分裂之后的个数为 x 1 、第 2 次分裂之后的个数为 x 2 、……第 15 次分裂之后的个数为 x 15 ，则有
x 14 =x 15 /2 、 x 13 =x 14 /2 、…… x n-1 =x n /2 (n ≥ 1）
因为第 15 次分裂之后的个数 x 15 是已知的，如果定义迭代变量为 x ，则可以将上面的倒推公式转换成如下的迭代公式：
x=x/2 （x 的初值为第 15 次分裂之后的个数 2^20）
让这个迭代公式重复执行 15 次，就可以倒推出第 1 次分裂之前的阿米巴个数。因为所需的迭代次数是个确定的值，我们可以使用一个固定次数的循环来实现对迭代过程的控制。参考程序如下：
for i=1 to 15
ps:java中幂的算法是Math.pow（2,20）；返回double，稍微注意一下
例 3 ：验证谷角猜想。日本数学家谷角静夫在研究自然数时发现了一个奇怪现象：对于任意一个自然数 n ，若 n 为偶数，则将其除以 2 ；若 n 为，则将其乘以 3 ，然后再加 1。如此经过有限次运算后，总可以得到自然数 1。人们把谷角静夫的这一发现叫做“谷角猜想”。
要求：编写一个程序，由键盘输入一个自然数 n ，把 n 经过有限次运算后，最终变成自然数 1 的全过程打印出来。
分析：定义迭代变量为 n ，按照谷角猜想的内容，可以得到两种情况下的迭代关系式：当 n 为偶数时， n=n/2 ；当 n 为奇数时， n=n*3+1。用 QBASIC 语言把它描述出来就是：
if n 为偶数 then
这就是需要计算机重复执行的迭代过程。这个迭代过程需要重复执行多少次，才能使迭代变量 n 最终变成自然数 1 ，这是我们无法计算出来的。因此，还需进一步确定用来结束迭代过程的条件。仔细分析题目要求，不难看出，对任意给定的一个自然数 n ，只要经过有限次运算后，能够得到自然数 1 ，就已经完成了验证工作。因此，用来结束迭代过程的条件可以定义为：n=1。参考程序如下：
input &Please input n=&;n
do until n=1
if n mod 2=0 then
rem 如果 n 为偶数，则调用迭代公式 n=n/2
print &—&;n;
print &—&;n;
#include&stdio.h&
#include&math.h&
void main()
double a,x0,x1;
printf(&Input a:\n&);
scanf(&%lf&,&a);//因为a是double型数据，所以要用%lf，而不是%f
printf(&Error!\n&);
x1=(x0+a/x0)/2;
x1=(x0+a/x0)/2;
}while(fabs(x0-x1）&=1e-6）；
printf(&Result:\n&);
printf(&sqrt(%g)=%g\n&,a,x1）；
求的迭代公式：x1=1/2*(x0+a/x0）。
算法：1.先自定一个初值x0，作为a的平方根值，在我们的程序中取a/2作为a的初值；利用迭代公式求出一个x1。此值与真正的a的平方根值相比，误差很大。
⒉把新求得的x1代入x0中，准备用此新的x0再去求出一个新的x1.
⒊利用迭代公式再求出一个新的x1的值，也就是用新的x0又求出一个新的平方根值x1，此值将更趋近于真正的平方根值。
⒋比较前后两次求得的平方根值x0和x1，如果它们的差值小于我们指定的值，即达到我们要求的精度，则认为x1就是a的平方根值，去执行步骤5；否则执行步骤2，即循环进行迭代。
迭代法是用于求或近似根的一种常用的算法设计方法。设方程为f(x)=0，用某种数学方法导出等价的形式x=g(x），然后按以下步骤执行：
⑴ 选一个方程的近似根，赋给变量x0；
⑵ 将x0的值保存于变量x1，然后计算g(x1），并将结果存于变量x0；
⑶ 当x0与x1的差的绝对值还小于指定的精度要求时，重复步骤⑵的计算。
若方程有根，并且用上述方法计算出来的近似根序列收敛，则按上述方法求得的x0就认为是方程的根。上述算法用C程序的形式表示为：
【算法】迭代法求方程的根
{ x0=初始近似根；
x0=g(x1）； /*按特定的计算新的近似根*/
} while (fabs(x0-x1）&Epsilon）；
printf（“方程的近似根是%f\n”，x0）；
迭代算法也常用于求的根，令
X=（x0，x1，…，xn-1）
设方程组为：
xi=gi(X) (I=0，1，…，n-1）
则求方程组根的迭代算法可描述如下：
【算法】迭代法求方程组的根
{ for (i=0;i
x=初始近似根；
for (i=0;i
for (i=0;i
for (delta=0.0,i=0;i
if (fabs(y-x)&delta) delta=fabs(y-x）；
} while (delta&Epsilon）；
for (i=0;i
printf（“变量x[%d]的近似根是 %f”，I，x）；
printf（“\n”）；
具体使用迭代法求根时应注意以下两种的情况：
⑴ 如果无解，算法求出的近似根序列就不会收敛，迭代过程会变成死循环，因此在使用迭代算法前应先考察方程是否有解，并在程序中对迭代的次数给予限制；
⑵ 方程虽然有解，但迭代公式选择不当，或迭代的初始近似根选择不合理，也会导致迭代失败。
递归是设计和描述算法的一种有力的工具，由于它在复杂算法的描述中被经常采用，为此在进一步介绍其他算法设计方法之前先讨论它。
能采用递归描述的算法通常有这样的特征：为求解规模为N的问题，设法将它分解成规模较小的问题，然后从这些小问题的解方便地构造出大问题的解，并且这些规模较小的问题也能采用同样的分解和，分解成规模更小的问题，并从这些更小问题的解构造出规模较大问题的解。特别地，当规模N=1时，能直接得解。
【问题】 编写计算（Fibonacci）的第n项函数fib（n）。
为：0、1、1、2、3、……，即：
fib(n)=fib(n-1）+fib(n-2） （当n&1时）。
int fib(int n)
{ if (n==0) return 0;
if (n==1） return 1;
if (n&1） return fib(n-1）+fib(n-2）；
递归算法的执行过程分递推和回归两个阶段。在递推阶段，把较复杂的问题（规模为n）的求解推到比原问题简单一些的问题（规模小于n）的求解。例如上例中，求解fib(n），把它推到求解fib(n-1）和fib(n-2）。也就是说，为计算fib(n），必须先计算fib(n-1）和fib(n- 2），而计算fib(n-1）和fib(n-2），又必须先计算fib(n-3）和fib(n-4）。依次类推，直至计算fib⑴和fib(0），分别能立即得到结果1和0。在递推阶段，必须要有终止递归的情况。例如在函数fib中，当n为1和0的情况。
在回归阶段，当获得最简单情况的解后，逐级返回，依次得到稍复杂问题的解，例如得到fib⑴和fib(0）后，返回得到fib⑵的结果，……，在得到了fib(n-1）和fib(n-2）的结果后，返回得到fib(n）的结果。
在编写时要注意，函数中的局部变量和参数知识局限于当前调用层，当递推进入“简单问题”层时，原来层次上的参数和局部变量便被隐蔽起来。在一系列“简单问题”层，它们各有自己的参数和局部变量。
由于递归引起一系列的函数调用，并且可能会有一系列的重复计算，递归算法的执行效率相对较低。当某个递归算法能较方便地转换成时，通常按递推算法编写程序。例如上例计算的第n项的函数fib(n）应采用递推算法，即从斐波那契数列的前两项出发，逐次由前两项计算出下一项，直至计算出要求的第n项。
【问题】 组合问题
问题描述：找出从自然数1、2、……、n中任取r个数的所有组合。例如n=5，r=3的所有组合为：⑴5、4、3 ⑵5、4、2 ⑶5、4、1
⑷5、3、2 ⑸5、3、1 ⑹5、2、1
⑺4、3、2 ⑻4、3、1 ⑼4、2、1
分析所列的10个组合，可以采用这样的递归思想来考虑求组合函数的算法。设函数为void comb(int m,int k）为找出从自然数1、2、……、m中任取k个数的所有组合。当组合的第一个数字选定时，其后的数字是从余下的m-1个数中取k-1数的组合。这就将求m 个数中取k个数的组合问题转化成求m-1个数中取k-1个数的组合问题。设函数引入工作数组a[ ]存放求出的组合的数字，约定函数将确定的k个数字组合的第一个数字放在a[k]中，当一个组合求出后，才将a[ ]中的一个组合输出。第一个数可以是m、m-1、……、k，函数将确定组合的第一个数字放入数组后，有两种可能的选择，因还未去顶组合的其余元素，继续递归去确定；或因已确定了组合的全部元素，输出这个组合。细节见以下程序中的函数comb。
# define MAXN 100
int a[MAXN];
void comb(int m,int k)
{ int i,j;
for (i=m;i&=k;i--)
comb(i-1,k-1）；
{ for (j=a[0];j&0;j--)
printf（“%4d”，a[j]);
printf（“\n”）；
void main()
comb（5,3）；
问题描述：有不同价值、不同重量的物品n件，求从这n件物品中选取一部分物品的选择方案，使选中物品的总重量不超过指定的限制重量，但选中物品的价值之和最大。
设n 件物品的重量分别为w0、w1、…、wn-1，物品的价值分别为v0、v1、…、vn-1。采用递归寻找物品的选择方案。设前面已有了多种选择的方案，并保留了其中总价值最大的方案于数组option[ ]，该方案的总价值存于变量maxv。当前正在考察新方案，其物品选择情况保存于数组cop[ ]。假定当前方案已考虑了前i-1件物品，现在要考虑第i件物品；当前方案已包含的物品的重量之和为tw；至此，若其余物品都选择是可能的话，本方案能达到的总价值的期望值为tv。算法引入tv是当一旦当前方案的总价值的期望值也小于前面方案的总价值maxv时，继续考察当前方案变成无意义的工作，应终止当前方案，立即去考察下一个方案。因为当方案的总价值不比maxv大时，该方案不会被再考察，这同时保证函数后找到的方案一定会比前面的方案更好。
对于第i件物品的选择考虑有两种可能：
⑴ 考虑物品i被选择，这种可能性仅当包含它不会超过方案总重量限制时才是可行的。选中后，继续递归去考虑其余物品的选择。
⑵ 考虑物品i不被选择，这种可能性仅当不包含物品i也有可能会找到价值更大的方案的情况。
按以上思想写出递归算法如下：
try（物品i，当前选择已达到的重量和，本方案可能达到的总价值tv)
{ /*考虑物品i包含在当前方案中的可能性*/
if（包含物品i是可以接受的）
{ 将物品i包含在当前方案中；
try(i+1,tw+物品i的重量，tv);
/*又一个完整方案，因为它比前面的方案好，以它作为最佳方案*/
以当前方案作为临时最佳方案保存；
恢复物品i不包含状态；
/*考虑物品i不包含在当前方案中的可能性*/
if （不包含物品i仅是可男考虑的）
try(i+1,tw,tv-物品i的价值）；
/*又一个完整方案，因它比前面的方案好，以它作为最佳方案*/
以当前方案作为临时最佳方案保存；
为了理解上述算法，特举以下实例。设有4件物品，它们的重量和价值见表：
物品 0 1 2 3
重量 5 3 2 1
价值 4 4 3 1
并设限制重量为7。则按以上算法，下图表示找解过程。由图知，一旦找到一个解，算法就进一步找更好的佳。如能判定某个查找分支不会找到更好的解，算法不会在该分支继续查找，而是立即终止该分支，并去考察下一个分支。
按上述算法编写函数和程序如下：
# define N 100
double limitW,totV,maxV;
int option[N],cop[N];
void find(int i,double tw,double tv)
/*考虑物品i包含在当前方案中的可能性*/
if (tw+a.weight&=limitW)
{ for (k=0;k
option[k]=cop[k];
/*考虑物品i不包含在当前方案中的可能性*/
if (tv-a.value&maxV)
{ for (k=0;k
option[k]=cop[k];
maxv=tv-a.
void main()
double w,v;
printf（“输入物品种数\n”）；
scanf（（“%d”，&n);
printf（“输入各物品的重量和价值\n”）；
for (totv=0.0,k=0;k
{ scanf（“%1f%1f”，&w,&v);
a[k].weight=w;
a[k].value=v;
printf（“输入限制重量\n”）；
scanf（“%1f”，&limitV);
for (k=0;k find(0,0.0,totV);
for (k=0;k
if (option[k]) printf（“%4d”，k+1）；
printf（“\n总价值为%.2f\n”，maxv);
作为对比，下面以同样的解题思想，考虑非递归的程序解。为了提高找解速度，程序不是简单地逐一生成所有候选解，而是从每个物品对候选解的影响来形成值得进一步考虑的候选解，一个候选解是通过依次考察每个物品形成的。对物品i的考察有这样几种情况：当该物品被包含在候选解中依旧满足解的总重量的限制，该物品被包含在候选解中是应该继续考虑的；反之，该物品不应该包括在当前正在形成的候选解中。同样地，仅当物品不被包括在候选解中，还是有可能找到比目前临时最佳解更好的候选解时，才去考虑该物品不被包括在候选解中；反之，该物品不包括在当前候选解中的方案也不应继续考虑。对于任一值得继续考虑的方案，程序就去进一步考虑下一个物品。
# define N 100
double limitW;
int cop[N];
struct ele {
void next(int i,double tw,double tv)
double find(struct ele *a,int n)
{ int i,k,f;
double maxv,tw,tv,
for (totv=0.0,k=0;k
totv+=a[k].
next(0,0.0,totv);
While (i&=0)
{ case 1: twv.++;
if (tw+a.weight&=limitW)
{ next(i+1,tw+a.weight,tv);
for (k=0;k
cop[k]=twv[k].!=0;
case 0: i--;
default: twv.=0;
if (tv-a.value&maxv)
{ next(i+1,tw,tv-a.value);
{ maxv=tv-a.
for (k=0;k
cop[k]=twv[k].!=0;
void main()
printf（“输入物品种数\n”）；
scanf（（“%d”，&n);
printf（“输入限制重量\n”）；
scanf（“%1f”，&limitW);
printf（“输入各物品的重量和价值\n”）；
for (k=0;k
scanf（“%1f%1f”，&a[k].weight,&a[k].value);
maxv=find(a,n);
printf（“\n选中的物品为\n”）；
for (k=0;k
if (option[k]) printf（“%4d”，k+1）；
printf（“\n总价值为%.2f\n”，maxv);
迭代法递归的基本概念和特点
程序调用自身的编程技巧称为递归（recursion）。
一个过程或函数在其定义或说明中又直接或间接调用自身的一种方法，它通常把一个大型复杂的问题层层转化为一个与原问题相似的规模较小的问题来求解，递归策略只需少量的程序就可描述出解题过程所需要的多次重复计算，大大地减少了程序的代码量。递归的能力在于用有限的语句来定义对象的无限集合。用递归思想写出的程序往往十分简洁易懂。
一般来说，递归需要有边界条件、递归前进段和递归返回段。当边界条件不满足时，递归前进；当边界条件满足时，递归返回。
⑴ 递归就是在过程或函数里调用自身；
⑵ 在使用递增归策略时，必须有一个明确的递归结束条件，称为递归出口。
企业信用信息}