2019独角兽企业重金招聘Python工程师标准>>> 
在使用Breeze 库时，需要导入相关包：
import breeze.linalg._
import breeze.numerics._
Breeze创建函数
//全0矩阵
DenseMatrix.zeros[Double](3,2)
res0: breeze.linalg.DenseMatrix[Double] =
0.0 0.0
0.0 0.0
0.0 0.0
//全0向量
DenseVector.zeros[Double](2)
res1: breeze.linalg.DenseVector[Double] = DenseVector(0.0, 0.0)
//全1向量
DenseVector.ones[Double](2)
res2: breeze.linalg.DenseVector[Double] = DenseVector(1.0, 1.0)
//按数值填充向量
DenseVector.fill[Double](3, 2)
res3: breeze.linalg.DenseVector[Double] = DenseVector(2.0, 2.0, 2.0)
//生成随机向量
DenseVector.range(1, 9, 2)
DenseVector.rangeD(1, 9, 2)
DenseVector.rangeF(1, 9, 2)
res4: breeze.linalg.DenseVector[Int] = DenseVector(1, 3, 5, 7)
res5: breeze.linalg.DenseVector[Double] = DenseVector(1.0, 3.0, 5.0, 7.0)
res6: breeze.linalg.DenseVector[Float] = DenseVector(1.0, 3.0, 5.0, 7.0)
//单位矩阵
DenseMatrix.eye[Double](4)
res7: breeze.linalg.DenseMatrix[Double] =
1.0 0.0 0.0 0.0
0.0 1.0 0.0 0.0
0.0 0.0 1.0 0.0
0.0 0.0 0.0 1.0
//对角矩阵
diag(DenseVector(3.0, 4.0, 5.0))
res8: breeze.linalg.DenseMatrix[Double] =
3.0 0.0 0.0
0.0 4.0 0.0
0.0 0.0 5.0
//按照行创建矩阵
DenseMatrix((4.0, 5.0, 6.0), (7.0, 8.0, 9.0))
res9: breeze.linalg.DenseMatrix[Double] =
4.0 5.0 6.0
7.0 8.0 9.0
//按照行创建向量
DenseVector((4.0, 5.0, 6.0, 7.0, 8.0, 9.0))
res10: breeze.linalg.DenseVector[(Double, Double, Double, Double, Double, Double)] = DenseVector((4.0,5.0,6.0,7.0,8.0,9.0))
//向量转置
DenseVector((4.0, 5.0, 6.0, 7.0, 8.0, 9.0)).t
res11: breeze.linalg.Transpose[breeze.linalg.DenseVector[(Double, Double, Double, Double, Double, Double)]] = Transpose(DenseVector((4.0,5.0,6.0,7.0,8.0,9.0)))
//从函数创建向量
DenseVector.tabulate(5)(i => i*i)
DenseVector.tabulate(0 to 5)(i => i*i)
res12: breeze.linalg.DenseVector[Int] = DenseVector(0, 1, 4, 9, 16)
res13: breeze.linalg.DenseVector[Int] = DenseVector(0, 1, 4, 9, 16, 25)
//从函数创建矩阵
DenseMatrix.tabulate(3, 4){ case (i, j) => i*i+j*j }
res14: breeze.linalg.DenseMatrix[Int] =
0 1 4 9
1 2 5 10
4 5 8 13
//从数组创建向量
new DenseVector[Double](Array(2.0, 5.0, 8.0))
res15: breeze.linalg.DenseVector[Double] = DenseVector(2.0, 5.0, 8.0)
//从数组创建矩阵
new DenseMatrix[Double](3, 2, Array(1.0, 4.0, 7.0, 3.0, 6.0, 9.0))
res16: breeze.linalg.DenseMatrix[Double] =
1.0 3.0
4.0 6.0
7.0 9.0
//0 到 1的随机向量
DenseVector.rand(9, Rand.uniform)
DenseVector.rand(9, Rand.gaussian)
res17: breeze.linalg.DenseVector[Double] = DenseVector(0.30960687979350654, 0.5779984012083466, 0.4880956198283952, 0.1013947992922748, 0.19635570812305936, 0.8533170989347008, 0.6619843996111201, 0.03131533370356321, 0.5430592884856604)
res18: breeze.linalg.DenseVector[Double] = DenseVector(0.48361471134641176, -1.734778260551877, -0.7319505628964431, 0.19971267958211184, -1.033191008131693, -1.7961545888066046, 0.2364555601503527, 0.22843047924270285, 1.7288956723034343)
//0 到 1的随机矩阵
DenseMatrix.rand(3, 2, Rand.uniform)
DenseMatrix.rand(3, 2, Rand.gaussian)
res19: breeze.linalg.DenseMatrix[Double] = 0.11270960774886585 0.19871332589909851
0.5581898434134047 0.8295064603050235
0.8692650535288642 0.4015512971620494
res20: breeze.linalg.DenseMatrix[Double] = 0.712041684728872 2.7007736007506216
0.053520407807479485 0.19044772577405517
-0.7370909025873376 -1.024737052742153
Breeze元素访问
val a = new DenseVector[Int](Array(10 to 20: _*))
a: breeze.linalg.DenseVector[Int] = DenseVector(10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
//指定位置
a(0)
res21: Int = 10
//向量子集
a(1 to 4)
res22: breeze.linalg.DenseVector[Int] = DenseVector(11, 12, 13, 14)
//按照指定步长取子集
a(5 to 0 by -1)
res23: breeze.linalg.DenseVector[Int] = DenseVector(15, 14, 13, 12, 11, 10)
//指定开始位置至结尾
a(1 to -1)
res24: breeze.linalg.DenseVector[Int] = DenseVector(11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
//最后一个元素
a(-1)
res25: Int = 20
val m = DenseMatrix((1.0, 2.0, 3.0), (4.0, 5.0, 6.0))
m: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0
4.0 5.0 6.0
//指定位置
m(0, 1)
res26: Double = 2.0
//矩阵指定列
m(::, 1)
res27: breeze.linalg.DenseVector[Double] = DenseVector(2.0, 6.0, 10.0)
Breeze元素操作
//调整矩阵形状
m.reshape(4, 3)
res28: breeze.linalg.DenseMatrix[Double] =
1.0 6.0 11.0
5.0 10.0 4.0
9.0 3.0 8.0
2.0 7.0 12.0
//矩阵转成向量
m.toDenseVector
res29: breeze.linalg.DenseVector[Double] = DenseVector(1.0, 5.0, 9.0, 2.0, 6.0, 10.0, 3.0, 7.0, 11.0, 4.0, 8.0, 12.0)
//复制下三角
lowerTriangular(m)
res30: breeze.linalg.DenseMatrix[Double] =
1.0 0.0 0.0
5.0 6.0 0.0
9.0 10.0 11.0
//复制上三角
upperTriangular(m)
res31: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0
0.0 6.0 7.0
0.0 0.0 11.0
//矩阵复制
m.copy
res32: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0 4.0
5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
//取对角线元素
diag(upperTriangular(m))
res33: breeze.linalg.DenseVector[Double] = DenseVector(1.0, 6.0, 11.0)
//子集赋数值
a(1 to 4) := 5
a
res34: breeze.linalg.DenseVector[Int] = DenseVector(5, 5, 5, 5)
res35: breeze.linalg.DenseVector[Int] = DenseVector(10, 5, 5, 5, 5, 15, 16, 17, 18, 19, 20)
//子集赋向量
a(1 to 4) := DenseVector(1, 2, 3, 4)
a
res36: breeze.linalg.DenseVector[Int] = DenseVector(1, 2, 3, 4)
res37: breeze.linalg.DenseVector[Int] = DenseVector(10, 1, 2, 3, 4, 15, 16, 17, 18, 19, 20)
//矩阵赋值
m(1 to 2,1 to 2) := 0.0
m
res38: breeze.linalg.DenseMatrix[Double] =
0.0 0.0
0.0 0.0
res39: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0 4.0
5.0 0.0 0.0 8.0
9.0 0.0 0.0 12.0
//矩阵列赋值
m(::, 2) := 5.0
m
res40: breeze.linalg.DenseVector[Double] = DenseVector(5.0, 5.0, 5.0)
res41: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 5.0 4.0
5.0 0.0 5.0 8.0
9.0 0.0 5.0 12.0
val a1 = DenseMatrix((1.0, 2.0, 3.0), (4.0, 5.0, 6.0))
val a2 = DenseMatrix((7.0, 8.0, 9.0), (10.0, 11.0, 12.0))
a1: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0
4.0 5.0 6.0
a2: breeze.linalg.DenseMatrix[Double] =
7.0 8.0 9.0
10.0 11.0 12.0
//垂直连接矩阵
DenseMatrix.vertcat(a1, a2)
res42: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
10.0 11.0 12.0
//横向连接矩阵
DenseMatrix.horzcat(a1, a2)
res43: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0 7.0 8.0 9.0
4.0 5.0 6.0 10.0 11.0 12.0
//向量连接
DenseVector.vertcat(DenseVector(20, 21, 22), DenseVector(23, 24, 25))
DenseVector.horzcat(DenseVector(20, 21, 22), DenseVector(23, 24, 25))
res44: breeze.linalg.DenseVector[Int] = DenseVector(20, 21, 22, 23, 24, 25)
res45: breeze.linalg.DenseMatrix[Int] =
20 23
21 24
22 25
Breeze数值计算函数
//元素加法
a1 + a2
res46: breeze.linalg.DenseMatrix[Double] =
8.0 10.0 12.0
14.0 16.0 18.0
//元素乘法
a1 :* a2
res47: breeze.linalg.DenseMatrix[Double] =
7.0 16.0 27.0
40.0 55.0 72.0
//元素除法
a1 :/ a2
res48: breeze.linalg.DenseMatrix[Double] =
0.14285714285714285 0.25 0.3333333333333333
0.4 0.45454545454545453 0.5
//元素比较
a1 :< a2
res49: breeze.linalg.DenseMatrix[Boolean] =
true true true
true true true
//元素相等
a1 :== a2
res50: breeze.linalg.DenseMatrix[Boolean] =
false false false
false false false
//元素追加
a1 :+=2.0
res51: breeze.linalg.DenseMatrix[Double] =
3.0 4.0 5.0
6.0 7.0 8.0
//元素追乘
a1 :*=2.0
res52: breeze.linalg.DenseMatrix[Double] =
6.0 8.0 10.0
12.0 14.0 16.0
//向量点积
DenseVector(1, 2, 3, 4) dot DenseVector(1, 1, 1, 1)
res53: Int = 10
//元素最大值
max(a1)
res54: Double = 16.0
//元素最小值
min(a1)
res55: Double = 6.0
//元素最大值的位置
argmax(a1)
res56: (Int, Int) = (1,2)
//元素最小值的位置
argmin(a1)
res57: (Int, Int) = (0,0)
Breeze求和函数
val m1 = DenseMatrix((1.0, 2.0, 3.0, 4.0), (5.0, 6.0, 7.0, 8.0), (9.0, 10.0, 11.0, 12.0))
m1: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0 4.0
5.0 6.0 7.0 8.0
9.0 10.0 11.0 12.0
//元素求和
sum(m1)
res58: Double = 78.0
//每一列求和
sum(m1, Axis._0)
res59: breeze.linalg.DenseMatrix[Double] = 15.0 18.0 21.0 24.0
//每一行求和
sum(m1, Axis._1)
res60: breeze.linalg.DenseVector[Double] = DenseVector(10.0, 26.0, 42.0)
//对角线元素和
trace(lowerTriangular(m1))
res61: Double = 18.0
//累积和
val a3 = new DenseVector[Int](Array(10 to 20: _*))
accumulate(a3)
a3: breeze.linalg.DenseVector[Int] = DenseVector(10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20)
res62: breeze.linalg.DenseVector[Int] = DenseVector(10, 21, 33, 46, 60, 75, 91, 108, 126, 145, 165)
Breeze布尔函数
val c = DenseVector(true, false, true)
val d = DenseVector(false, true, true)
//元素与操作
c :& d
res63: breeze.linalg.DenseVector[Boolean] = DenseVector(false, false, true)
//元素或操作
c :| d
res64: breeze.linalg.DenseVector[Boolean] = DenseVector(true, true, true)
//元素非操作
!c
res65: breeze.linalg.DenseVector[Boolean] = DenseVector(false, true, false)
val e = DenseVector[Int](-3, 0, 2)
e: breeze.linalg.DenseVector[Int] = DenseVector(-3, 0, 2)
//存在非零元素
any(e)
res66: Boolean = true
//所有元素非零
all(e)
res67: Boolean = false
Breeze线性代数函数
val f = DenseMatrix((1.0, 2.0, 3.0), (4.0, 5.0, 6.0), (7.0, 8.0, 9.0))
val g = DenseMatrix((1.0, 1.0, 1.0), (1.0, 1.0, 1.0), (1.0, 1.0, 1.0))
f: breeze.linalg.DenseMatrix[Double] =
1.0 2.0 3.0
4.0 5.0 6.0
7.0 8.0 9.0
g: breeze.linalg.DenseMatrix[Double] =
1.0 1.0 1.0
1.0 1.0 1.0
1.0 1.0 1.0
//线性求解，AX = B，求解X
f \ g
res68: breeze.linalg.DenseMatrix[Double] =
-2.5 -2.5 -2.5
4.0 4.0 4.0
-1.5 -1.5 -1.5
//转置
f.t
res69: breeze.linalg.DenseMatrix[Double] =
1.0 4.0 7.0
2.0 5.0 8.0
3.0 6.0 9.0
//求特征值
det(f)
res70: Double = 6.661338147750939E-16
//求逆
inv(f)
res71: breeze.linalg.DenseMatrix[Double] =
-4.503599627370499E15 9.007199254740992E15 -4.503599627370495E15
9.007199254740998E15 -1.8014398509481984E16 9.007199254740991E15
-4.503599627370498E15 9.007199254740992E15 -4.5035996273704955E15
//求伪逆
pinv(f)
res72: breeze.linalg.DenseMatrix[Double] =
-3.7720834019330525E14 7.544166803866101E14 -3.77208340193305E14
7.544166803866094E14 -1.5088333607732208E15 7.544166803866108E14
-3.772083401933041E14 7.544166803866104E14 -3.772083401933055E14
//特征值和特征向量
eig(f)
res73: breeze.linalg.eig.DenseEig =
Eig(DenseVector(16.116843969807043, -1.1168439698070427, -1.3036777264747022E-15),
DenseVector(0.0, 0.0, 0.0),
-0.23197068724628617 -0.7858302387420671 0.40824829046386363
-0.5253220933012336 -0.08675133925662833 -0.816496580927726
-0.8186734993561815 0.61232756022881 0.4082482904638625 )
//奇异值分解
val svd.SVD(u,s,v) = svd(g)
u: breeze.linalg.DenseMatrix[Double] =
-0.5773502691896255 -0.5773502691896257 -0.5773502691896256
-0.5773502691896256 -0.2113248654051871 0.7886751345948126
-0.5773502691896256 0.7886751345948129 -0.21132486540518708
s: breeze.linalg.DenseVector[Double] = DenseVector(3.0000000000000004, 0.0, 0.0)
v: breeze.linalg.DenseMatrix[Double] =
-0.5773502691896256 -0.5773502691896257 -0.5773502691896256
0.0 -0.7071067811865474 0.7071067811865477
0.816496580927726 -0.4082482904638629 -0.4082482904638628
//求矩阵的秩
rank(f)
res74: Int = 2
//矩阵长度
f.size
res75: Int = 9
//矩阵行数
f.rows
res76: Int = 3
//矩阵列数
f.cols
res77: Int = 3
Breeze取整函数
val h = DenseVector(-1.2, 0.7, 2.3)
h: breeze.linalg.DenseVector[Double] = DenseVector(-1.2, 0.7, 2.3)
//四舍五入
round(h)
res78: breeze.linalg.DenseVector[Long] = DenseVector(-1, 1, 2)
//大于它的最小整数
ceil(h)
res79: breeze.linalg.DenseVector[Double] = DenseVector(-1.0, 1.0, 3.0)
//小于它的最大整数
floor(h)
res80: breeze.linalg.DenseVector[Double] = DenseVector(-2.0, 0.0, 2.0)
//符号函数
signum(h)
res81: breeze.linalg.DenseVector[Double] = DenseVector(-1.0, 1.0, 1.0)
//取正数
abs(h)
res82: breeze.linalg.DenseVector[Double] = DenseVector(1.2, 0.7, 2.3)