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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">Go语言圣经</a></li><li class="chapter-item expanded affix "><a href="../preface.html">前言</a></li><li class="chapter-item expanded "><a href="../ch1/ch1.html"><strong aria-hidden="true">1.</strong> 入门</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch1/ch1-01.html"><strong aria-hidden="true">1.1.</strong> Hello, World</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-02.html"><strong aria-hidden="true">1.2.</strong> 命令行参数</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-03.html"><strong aria-hidden="true">1.3.</strong> 查找重复的行</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-04.html"><strong aria-hidden="true">1.4.</strong> GIF动画</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-05.html"><strong aria-hidden="true">1.5.</strong> 获取URL</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-06.html"><strong aria-hidden="true">1.6.</strong> 并发获取多个URL</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-07.html"><strong aria-hidden="true">1.7.</strong> Web服务</a></li><li class="chapter-item expanded "><a href="../ch1/ch1-08.html"><strong aria-hidden="true">1.8.</strong> 本章要点</a></li></ol></li><li class="chapter-item expanded "><a href="../ch2/ch2.html"><strong aria-hidden="true">2.</strong> 程序结构</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch2/ch2-01.html"><strong aria-hidden="true">2.1.</strong> 命名</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-02.html"><strong aria-hidden="true">2.2.</strong> 声明</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-03.html"><strong aria-hidden="true">2.3.</strong> 变量</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-04.html"><strong aria-hidden="true">2.4.</strong> 赋值</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-05.html"><strong aria-hidden="true">2.5.</strong> 类型</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-06.html"><strong aria-hidden="true">2.6.</strong> 包和文件</a></li><li class="chapter-item expanded "><a href="../ch2/ch2-07.html"><strong aria-hidden="true">2.7.</strong> 作用域</a></li></ol></li><li class="chapter-item expanded "><a href="../ch3/ch3.html"><strong aria-hidden="true">3.</strong> 基础数据类型</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch3/ch3-01.html"><strong aria-hidden="true">3.1.</strong> 整型</a></li><li class="chapter-item expanded "><a href="../ch3/ch3-02.html"><strong aria-hidden="true">3.2.</strong> 浮点数</a></li><li class="chapter-item expanded "><a href="../ch3/ch3-03.html" class="active"><strong aria-hidden="true">3.3.</strong> 复数</a></li><li class="chapter-item expanded "><a href="../ch3/ch3-04.html"><strong aria-hidden="true">3.4.</strong> 布尔型</a></li><li class="chapter-item expanded "><a href="../ch3/ch3-05.html"><strong aria-hidden="true">3.5.</strong> 字符串</a></li><li class="chapter-item expanded "><a href="../ch3/ch3-06.html"><strong aria-hidden="true">3.6.</strong> 常量</a></li></ol></li><li class="chapter-item expanded "><a href="../ch4/ch4.html"><strong aria-hidden="true">4.</strong> 复合数据类型</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch4/ch4-01.html"><strong aria-hidden="true">4.1.</strong> 数组</a></li><li class="chapter-item expanded "><a href="../ch4/ch4-02.html"><strong aria-hidden="true">4.2.</strong> Slice</a></li><li class="chapter-item expanded "><a href="../ch4/ch4-03.html"><strong aria-hidden="true">4.3.</strong> Map</a></li><li class="chapter-item expanded "><a href="../ch4/ch4-04.html"><strong aria-hidden="true">4.4.</strong> 结构体</a></li><li class="chapter-item expanded "><a href="../ch4/ch4-05.html"><strong aria-hidden="true">4.5.</strong> JSON</a></li><li class="chapter-item expanded "><a href="../ch4/ch4-06.html"><strong aria-hidden="true">4.6.</strong> 文本和HTML模板</a></li></ol></li><li class="chapter-item expanded "><a href="../ch5/ch5.html"><strong aria-hidden="true">5.</strong> 函数</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch5/ch5-01.html"><strong aria-hidden="true">5.1.</strong> 函数声明</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-02.html"><strong aria-hidden="true">5.2.</strong> 递归</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-03.html"><strong aria-hidden="true">5.3.</strong> 多返回值</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-04.html"><strong aria-hidden="true">5.4.</strong> 错误</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-05.html"><strong aria-hidden="true">5.5.</strong> 函数值</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-06.html"><strong aria-hidden="true">5.6.</strong> 匿名函数</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-07.html"><strong aria-hidden="true">5.7.</strong> 可变参数</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-08.html"><strong aria-hidden="true">5.8.</strong> Deferred函数</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-09.html"><strong aria-hidden="true">5.9.</strong> Panic异常</a></li><li class="chapter-item expanded "><a href="../ch5/ch5-10.html"><strong aria-hidden="true">5.10.</strong> Recover捕获异常</a></li></ol></li><li class="chapter-item expanded "><a href="../ch6/ch6.html"><strong aria-hidden="true">6.</strong> 方法</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch6/ch6-01.html"><strong aria-hidden="true">6.1.</strong> 方法声明</a></li><li class="chapter-item expanded "><a href="../ch6/ch6-02.html"><strong aria-hidden="true">6.2.</strong> 基于指针对象的方法</a></li><li class="chapter-item expanded "><a href="../ch6/ch6-03.html"><strong aria-hidden="true">6.3.</strong> 通过嵌入结构体来扩展类型</a></li><li class="chapter-item expanded "><a href="../ch6/ch6-04.html"><strong aria-hidden="true">6.4.</strong> 方法值和方法表达式</a></li><li class="chapter-item expanded "><a href="../ch6/ch6-05.html"><strong aria-hidden="true">6.5.</strong> 示例: Bit数组</a></li><li class="chapter-item expanded "><a href="../ch6/ch6-06.html"><strong aria-hidden="true">6.6.</strong> 封装</a></li></ol></li><li class="chapter-item expanded "><a href="../ch7/ch7.html"><strong aria-hidden="true">7.</strong> 接口</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch7/ch7-01.html"><strong aria-hidden="true">7.1.</strong> 接口是合约</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-02.html"><strong aria-hidden="true">7.2.</strong> 接口类型</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-03.html"><strong aria-hidden="true">7.3.</strong> 实现接口的条件</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-04.html"><strong aria-hidden="true">7.4.</strong> flag.Value接口</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-05.html"><strong aria-hidden="true">7.5.</strong> 接口值</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-06.html"><strong aria-hidden="true">7.6.</strong> sort.Interface接口</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-07.html"><strong aria-hidden="true">7.7.</strong> http.Handler接口</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-08.html"><strong aria-hidden="true">7.8.</strong> error接口</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-09.html"><strong aria-hidden="true">7.9.</strong> 示例: 表达式求值</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-10.html"><strong aria-hidden="true">7.10.</strong> 类型断言</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-11.html"><strong aria-hidden="true">7.11.</strong> 基于类型断言识别错误类型</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-12.html"><strong aria-hidden="true">7.12.</strong> 通过类型断言查询接口</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-13.html"><strong aria-hidden="true">7.13.</strong> 类型分支</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-14.html"><strong aria-hidden="true">7.14.</strong> 示例: 基于标记的XML解码</a></li><li class="chapter-item expanded "><a href="../ch7/ch7-15.html"><strong aria-hidden="true">7.15.</strong> 补充几点</a></li></ol></li><li class="chapter-item expanded "><a href="../ch8/ch8.html"><strong aria-hidden="true">8.</strong> Goroutines和Channels</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch8/ch8-01.html"><strong aria-hidden="true">8.1.</strong> Goroutines</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-02.html"><strong aria-hidden="true">8.2.</strong> 示例: 并发的Clock服务</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-03.html"><strong aria-hidden="true">8.3.</strong> 示例: 并发的Echo服务</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-04.html"><strong aria-hidden="true">8.4.</strong> Channels</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-05.html"><strong aria-hidden="true">8.5.</strong> 并发的循环</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-06.html"><strong aria-hidden="true">8.6.</strong> 示例: 并发的Web爬虫</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-07.html"><strong aria-hidden="true">8.7.</strong> 基于select的多路复用</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-09.html"><strong aria-hidden="true">8.8.</strong> 并发的退出</a></li><li class="chapter-item expanded "><a href="../ch8/ch8-10.html"><strong aria-hidden="true">8.9.</strong> 示例: 聊天服务</a></li></ol></li><li class="chapter-item expanded "><a href="../ch9/ch9.html"><strong aria-hidden="true">9.</strong> 基于共享变量的并发</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch9/ch9-01.html"><strong aria-hidden="true">9.1.</strong> 竞争条件</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-02.html"><strong aria-hidden="true">9.2.</strong> sync.Mutex互斥锁</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-03.html"><strong aria-hidden="true">9.3.</strong> sync.RWMutex读写锁</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-04.html"><strong aria-hidden="true">9.4.</strong> 内存同步</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-06.html"><strong aria-hidden="true">9.5.</strong> 竞争条件检测</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-07.html"><strong aria-hidden="true">9.6.</strong> 示例: 并发的非阻塞缓存</a></li><li class="chapter-item expanded "><a href="../ch9/ch9-08.html"><strong aria-hidden="true">9.7.</strong> Goroutines和线程</a></li></ol></li><li class="chapter-item expanded "><a href="../ch10/ch10.html"><strong aria-hidden="true">10.</strong> 包和工具</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch10/ch10-01.html"><strong aria-hidden="true">10.1.</strong> 包简介</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-02.html"><strong aria-hidden="true">10.2.</strong> 导入路径</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-03.html"><strong aria-hidden="true">10.3.</strong> 包声明</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-04.html"><strong aria-hidden="true">10.4.</strong> 导入声明</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-05.html"><strong aria-hidden="true">10.5.</strong> 包的匿名导入</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-06.html"><strong aria-hidden="true">10.6.</strong> 包和命名</a></li><li class="chapter-item expanded "><a href="../ch10/ch10-07.html"><strong aria-hidden="true">10.7.</strong> 工具</a></li></ol></li><li class="chapter-item expanded "><a href="../ch11/ch11.html"><strong aria-hidden="true">11.</strong> 测试</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch11/ch11-01.html"><strong aria-hidden="true">11.1.</strong> go test</a></li><li class="chapter-item expanded "><a href="../ch11/ch11-02.html"><strong aria-hidden="true">11.2.</strong> 测试函数</a></li><li class="chapter-item expanded "><a href="../ch11/ch11-03.html"><strong aria-hidden="true">11.3.</strong> 测试覆盖率</a></li><li class="chapter-item expanded "><a href="../ch11/ch11-04.html"><strong aria-hidden="true">11.4.</strong> 基准测试</a></li><li class="chapter-item expanded "><a href="../ch11/ch11-05.html"><strong aria-hidden="true">11.5.</strong> 剖析</a></li><li class="chapter-item expanded "><a href="../ch11/ch11-06.html"><strong aria-hidden="true">11.6.</strong> 示例函数</a></li></ol></li><li class="chapter-item expanded "><a href="../ch12/ch12.html"><strong aria-hidden="true">12.</strong> 反射</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch12/ch12-01.html"><strong aria-hidden="true">12.1.</strong> 为何需要反射?</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-02.html"><strong aria-hidden="true">12.2.</strong> reflect.Type和reflect.Value</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-03.html"><strong aria-hidden="true">12.3.</strong> Display递归打印</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-04.html"><strong aria-hidden="true">12.4.</strong> 示例: 编码S表达式</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-05.html"><strong aria-hidden="true">12.5.</strong> 通过reflect.Value修改值</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-06.html"><strong aria-hidden="true">12.6.</strong> 示例: 解码S表达式</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-08.html"><strong aria-hidden="true">12.7.</strong> 显示一个类型的方法集</a></li><li class="chapter-item expanded "><a href="../ch12/ch12-09.html"><strong aria-hidden="true">12.8.</strong> 几点忠告</a></li></ol></li><li class="chapter-item expanded "><a href="../ch13/ch13.html"><strong aria-hidden="true">13.</strong> 底层编程</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../ch13/ch13-01.html"><strong aria-hidden="true">13.1.</strong> unsafe.Sizeof, Alignof 和 Offsetof</a></li><li class="chapter-item expanded "><a href="../ch13/ch13-02.html"><strong aria-hidden="true">13.2.</strong> unsafe.Pointer</a></li><li class="chapter-item expanded "><a href="../ch13/ch13-03.html"><strong aria-hidden="true">13.3.</strong> 示例: 深度相等判断</a></li><li class="chapter-item expanded "><a href="../ch13/ch13-04.html"><strong aria-hidden="true">13.4.</strong> 通过cgo调用C代码</a></li><li class="chapter-item expanded "><a href="../ch13/ch13-05.html"><strong aria-hidden="true">13.5.</strong> 几点忠告</a></li></ol></li><li class="chapter-item expanded "><a href="../appendix/appendix.html"><strong aria-hidden="true">14.</strong> 附录</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../appendix/appendix-a-errata.html"><strong aria-hidden="true">14.1.</strong> 附录A：原文勘误</a></li><li class="chapter-item expanded "><a href="../appendix/appendix-b-author.html"><strong aria-hidden="true">14.2.</strong> 附录B：作者译者</a></li><li class="chapter-item expanded "><a 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<!-- 头部 -->
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<li><em>KusonStack一站式可编程配置技术栈: <a href="https://github.com/KusionStack/kusion">https://github.com/KusionStack/kusion</a></em></li>
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<li><em>凹语言™: <a href="https://github.com/wa-lang/wa">https://github.com/wa-lang/wa</a></em></li>
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<hr>
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<h2 id="33-复数"><a class="header" href="#33-复数">3.3. 复数</a></h2>
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<p>Go语言提供了两种精度的复数类型：complex64和complex128，分别对应float32和float64两种浮点数精度。内置的complex函数用于构建复数，内建的real和imag函数分别返回复数的实部和虚部：</p>
|
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<pre><code class="language-Go">var x complex128 = complex(1, 2) // 1+2i
|
||
var y complex128 = complex(3, 4) // 3+4i
|
||
fmt.Println(x*y) // "(-5+10i)"
|
||
fmt.Println(real(x*y)) // "-5"
|
||
fmt.Println(imag(x*y)) // "10"
|
||
</code></pre>
|
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<p>如果一个浮点数面值或一个十进制整数面值后面跟着一个i，例如3.141592i或2i，它将构成一个复数的虚部，复数的实部是0：</p>
|
||
<pre><code class="language-Go">fmt.Println(1i * 1i) // "(-1+0i)", i^2 = -1
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</code></pre>
|
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<p>在常量算术规则下，一个复数常量可以加到另一个普通数值常量（整数或浮点数、实部或虚部），我们可以用自然的方式书写复数，就像1+2i或与之等价的写法2i+1。上面x和y的声明语句还可以简化：</p>
|
||
<pre><code class="language-Go">x := 1 + 2i
|
||
y := 3 + 4i
|
||
</code></pre>
|
||
<p>复数也可以用==和!=进行相等比较。只有两个复数的实部和虚部都相等的时候它们才是相等的（译注：浮点数的相等比较是危险的，需要特别小心处理精度问题）。</p>
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<p>math/cmplx包提供了复数处理的许多函数，例如求复数的平方根函数和求幂函数。</p>
|
||
<pre><code class="language-Go">fmt.Println(cmplx.Sqrt(-1)) // "(0+1i)"
|
||
</code></pre>
|
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<p>下面的程序使用complex128复数算法来生成一个Mandelbrot图像。</p>
|
||
<p><u><i>gopl.io/ch3/mandelbrot</i></u></p>
|
||
<pre><code class="language-Go">// Mandelbrot emits a PNG image of the Mandelbrot fractal.
|
||
package main
|
||
|
||
import (
|
||
"image"
|
||
"image/color"
|
||
"image/png"
|
||
"math/cmplx"
|
||
"os"
|
||
)
|
||
|
||
|
||
func main() {
|
||
const (
|
||
xmin, ymin, xmax, ymax = -2, -2, +2, +2
|
||
width, height = 1024, 1024
|
||
)
|
||
|
||
img := image.NewRGBA(image.Rect(0, 0, width, height))
|
||
for py := 0; py < height; py++ {
|
||
y := float64(py)/height*(ymax-ymin) + ymin
|
||
for px := 0; px < width; px++ {
|
||
x := float64(px)/width*(xmax-xmin) + xmin
|
||
z := complex(x, y)
|
||
// Image point (px, py) represents complex value z.
|
||
img.Set(px, py, mandelbrot(z))
|
||
}
|
||
}
|
||
png.Encode(os.Stdout, img) // NOTE: ignoring errors
|
||
}
|
||
|
||
func mandelbrot(z complex128) color.Color {
|
||
const iterations = 200
|
||
const contrast = 15
|
||
|
||
var v complex128
|
||
for n := uint8(0); n < iterations; n++ {
|
||
v = v*v + z
|
||
if cmplx.Abs(v) > 2 {
|
||
return color.Gray{255 - contrast*n}
|
||
}
|
||
}
|
||
return color.Black
|
||
}
|
||
</code></pre>
|
||
<p>用于遍历1024x1024图像每个点的两个嵌套的循环对应-2到+2区间的复数平面。程序反复测试每个点对应复数值平方值加一个增量值对应的点是否超出半径为2的圆。如果超过了，通过根据预设置的逃逸迭代次数对应的灰度颜色来代替。如果不是，那么该点属于Mandelbrot集合，使用黑色颜色标记。最终程序将生成的PNG格式分形图像输出到标准输出，如图3.3所示。</p>
|
||
<p><img src="../images/ch3-03.png" alt="" /></p>
|
||
<p><strong>练习 3.5：</strong> 实现一个彩色的Mandelbrot图像，使用image.NewRGBA创建图像，使用color.RGBA或color.YCbCr生成颜色。</p>
|
||
<p><strong>练习 3.6：</strong> 升采样技术可以降低每个像素对计算颜色值和平均值的影响。简单的方法是将每个像素分成四个子像素，实现它。</p>
|
||
<p><strong>练习 3.7：</strong> 另一个生成分形图像的方式是使用牛顿法来求解一个复数方程，例如$z^4-1=0$。每个起点到四个根的迭代次数对应阴影的灰度。方程根对应的点用颜色表示。</p>
|
||
<p><strong>练习 3.8：</strong> 通过提高精度来生成更多级别的分形。使用四种不同精度类型的数字实现相同的分形：complex64、complex128、big.Float和big.Rat。（后面两种类型在math/big包声明。Float是有指定限精度的浮点数；Rat是无限精度的有理数。）它们间的性能和内存使用对比如何？当渲染图可见时缩放的级别是多少？</p>
|
||
<p><strong>练习 3.9：</strong> 编写一个web服务器，用于给客户端生成分形的图像。运行客户端通过HTTP参数指定x、y和zoom参数。</p>
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