! ============================================================================== ! 通过 gfortran ./test.f90 -o ./run && ./run 运行 ! 程序名： ! 第9章习题，及书上例题 ! 目的: ! ! 修订记录: ! 日期 编程者 改动描述 ! =================== ============= ===================================== ! 2021-05-06 18:56:24 Sola 例9-1 模仿正常的上三角解法 ! 2021-05-07 00:20:30 Sola 例9-1 高斯-亚当消元法(书上解法确实好) ! 2021-05-07 17:48:03 Sola 习题9-4 判断程序正误 ! 2021-05-07 18:00:54 Sola 习题9-5 判断输出结果，跳过 ! 2021-05-07 19:17:23 Sola 习题9-6 矩阵乘法子程序, 显式结构 ! 2021-05-07 19:17:53 Sola 习题9-7 矩阵乘法子程序, 显式接口不定结构 ! 2021-05-07 19:20:11 Sola 习题9-8 用不定结构形参数组修改例9-1 ! 2021-05-07 20:22:22 Sola 习题9-9 测试图9-6的子程序 ! 2021-05-07 21:42:49 Sola 习题9-11 测试程序运行 ! 2021-05-07 21:59:54 Sola 习题9-12 修改测试程序 ! 2021-05-07 22:16:18 Sola 习题9-13 模拟掷色子 ! 2021-05-08 13:02:55 Sola 习题9-14 创建逐元过程运算 ! 2021-05-08 14:59:48 Sola 习题9-15 将9-14换成pure纯函数 ! 2021-05-09 00:52:48 Sola 习题9-16 高阶最小二乘回归 ! 2021-05-09 13:49:16 Sola 习题9-17 噪声值对拟合结果的影响 ! 2021-05-09 14:36:52 Sola 习题9-18 任意高阶最小二乘回归（9-16直接写了，跳过） ! 2021-05-09 14:38:55 Sola 习题9-19 4阶最小二乘回归噪声影响 ! 2021-05-09 15:12:56 Sola 习题9-20 利用高阶最小二乘回归进行插值处理 ! 2021-05-09 15:24:01 Sola 习题9-21 推理，推理范围外的值 ! 程序结构: ! ! ============================================================================== ! 模块： module Chapter9 implicit none ! 数据字典 ! 声明常数 REAL, PARAMETER :: PI=3.14159265 ! PI值 REAL, PARAMETER :: e=2.718281828459 ! 自然对数 INTEGER, PARAMETER :: arrayLength=20 ! 数组基准长度 ! Example 9-1 ! real, dimension(:), allocatable :: arrayX ! read, dimension(:, :), allocatable :: N integer :: m, n ! 声明变量 ! 创建显式接口 contains ! subroutine SubName(varName1,varName2) ! implicit none ! ! 数据字典 ! end subroutine SubName ! 上三角消元法 ! subroutine UpperTriangleElimination(matrixInput, arrayX, m, n) ! implicit none ! integer, intent(in) :: m, n ! 矩阵大小与X解向量长度 ! real, dimension(m, m+1), intent(inout) :: matrixInput ! 输入矩阵(输入) ! real, dimension(n), intent(inout) :: arrayX ! 解向量(输入输出) ! real, dimension(m+1) :: matrixTemp ! 临时向量, 用于矩阵行交换 ! integer :: i ! 局部变量: 循环参数 ! if ( m /= 1 ) then ! 如果矩阵行数不为1 ! if ( abs(matrixInput(1, 1)) < 1.0E-30 ) then ! 如果矩阵首位为0 ! do i = 2, m ! 从其余行选择不为0的交换 ! if ( abs(matrixInput(i, 1)) >= 1.0E-30 ) then ! matrixTemp = matrixInput(i, :) ! matrixInput(i, :) = matrixInput(1, :) ! matrixInput(1, :) = matrixTemp ! exit ! end if ! end do ! end if ! matrixInput(1, :) = matrixInput(1, :)/matrixInput(1, 1) ! 单位化矩阵第一行 ! do i = 2, m ! 消去其他行所有第一列的系数 ! matrixInput(i, :) = matrixInput(i, :) - matrixInput(1, :)*matrixInput(i, 1) ! end do ! if ( all(matrixInput(2:m, 2) == 0) ) then ! 检查第二列第二行到最后一行不为0 ! stop "输入矩阵不满秩，无法求解" ! 如果为0, 则输入矩阵不满秩, 无解 ! else ! call GaussAdamElimination(matrixInput(2:m, 2:m+1), arrayX, m-1, n) ! 递归调用自身 ! end if ! arrayX(n-m+1) = matrixInput(1, m+1) - matrixInput(1, 2:m)*arrayX(n-m+2:n) ! 计算当前位X值 ! else ! arrayX(n) = matrixInput(1, m+1)/matrixInput(1, 1) ! 如果矩阵行数为1, 直接计算当前位X值 ! end if ! end subroutine UpperTriangleElimination ! 高斯亚当消元法 会破坏输入数组 subroutine GaussAdamElimination(matrixInput, m, arrayX, maxLimit, error) implicit none integer, intent(in) :: m ! 输入的方程组数量 integer, intent(in) :: maxLimit ! 矩阵大小与X解向量长度 real, dimension(maxLimit, maxLimit), intent(inout) :: matrixInput ! 输入矩阵(输入) real, dimension(maxLimit), intent(inout) :: arrayX ! 输入常数向量, 输出X解向量 integer, intent(out) :: error ! 错误值 real :: temp ! 临时值, 用于数值交换 integer :: maxLocate ! 最大值所在行 integer :: i, n ! 局部变量: 循环参数 do n = 1, m maxLocate = n ! 初始化绝对值最大的位置 do i = n+1, m ! 执行最大支点技术，减少舍入误差 if ( abs(matrixInput(i, n)) > abs(matrixInput(maxLocate, n)) ) then maxLocate = i end if end do if ( abs(matrixInput(maxLocate, n)) < 1.0E-30 ) then error = 1 ! 如果绝对值最大为0, 则矩阵不满秩, 无解 return ! 退出运算 end if if ( maxLocate /= n ) then ! 如果绝对值最大值位置发生了变化, 则交换 do i = 1, m temp = matrixInput(maxLocate, i) matrixInput(maxLocate, i) = matrixInput(n, i) matrixInput(n, i) = temp end do temp = arrayX(maxLocate) arrayX(maxLocate) = arrayX(n) arrayX(n) = temp end if do i = 1, m ! 消去其他行所有第n列的系数 if ( i /= n ) then temp = -matrixInput(i, n)/matrixInput(n, n) ! 这里务必要设置一个临时值保存数值, 不然会被修改 matrixInput(i, :) = matrixInput(i, :) + matrixInput(n, :)*temp arrayX(i) = arrayX(i) + arrayX(n)*temp end if end do end do do i = 1, m ! 产生X的解向量 arrayX(i) = arrayX(i)/matrixInput(i, i) matrixInput(i, i) = 1. end do error = 0 end subroutine GaussAdamElimination end module Chapter9 ! ============================================================================== ! 主程序： program ProName ! use ModName implicit none ! 数据字典 ! 声明常量 ! 声明变量 ! 变量初始化 ! 数据输入 ! 运算过程 ! 结果输出 ! call Exercises9_4 ! call Exercises9_6 ! call Exercises9_8 ! call Exercises9_9 ! call Exercises9_11 ! call Exercises9_12 ! call Exercises9_13 ! call Exercises9_14 ! call Exercises9_15 ! call Exercises9_16 end program ProName ! ============================================================================== ! 子程序 ! subroutine SubName(varName1,varName2) ! use MouName ! implicit none ! Type, intent(inout) :: varName ! end subroutine SubName ! ============================================================================== ! 函数 ! function FunName(varName1,varName2) ! use MouName ! implicit none ! end function FunName ! ============================================================================== ! 习题9-4 判断程序正误 subroutine Exercises9_4 implicit none real :: a=3., b=4., output integer :: i=0 call sub1(a, I, output) write(*,*) 'The output is', output contains subroutine sub1(x, j, junk) real, intent(in) :: x integer, intent(in) :: j real, intent(out) :: junk junk = (x-j)/b end subroutine sub1 end subroutine Exercises9_4 subroutine Exercises9_6 implicit none real, dimension(:, :), allocatable :: arrayInput1, arrayInput2, arrayResult ! 输入输出矩阵 integer :: errorLevel ! 错误码 integer :: i ! 循环参数 ! 9-6 a allocate(arrayInput1(3, 3), stat=errorLevel) ! 分配数组 if (errorLevel /= 0) print *, "arrayInput1: Allocation request denied" allocate(arrayInput2(3, 3), stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput2: Allocation request denied" allocate(arrayResult(3, 3), stat=errorLevel) if (errorLevel /= 0) print *, "arrayResult: Allocation request denied" arrayInput1(1, :) = (/ 2., -1., 2./) ! 定义数组 arrayInput1(2, :) = (/-1., -3., 4./) arrayInput1(3, :) = (/ 2., 4., 2./) arrayInput2(1, :) = (/ 1., 2., 3./) arrayInput2(2, :) = (/ 2., 1., 2./) arrayInput2(3, :) = (/ 3., 2., 1./) call MatrixMultplication(arrayInput1, 3, 3, arrayInput2, 3, 3, arrayResult, errorLevel) ! 调用子程序计算矩阵乘积 if (errorLevel /= 0) print *, "无法进行矩阵乘法" write(*,'(3F8.2)') (arrayResult(i, :), i = 1, 3) ! 输出计算结果 arrayResult(:,:) = matmul(arrayInput1, arrayInput2) ! 调用内部函数计算矩阵乘积 write(*,'(3F8.2)') (arrayResult(i, :), i = 1, 3) ! 输出计算结果 call MatrixMul(arrayInput1, arrayInput2, arrayResult, errorLevel) ! 调用子程序计算矩阵乘积 if (errorLevel /= 0) print *, "无法进行矩阵乘法" write(*,'(3F8.2)') (arrayResult(i, :), i = 1, 3) ! 输出计算结果 if (allocated(arrayResult)) deallocate(arrayResult, stat=errorLevel) ! 释放数组 if (errorLevel /= 0) print *, "arrayResult: Deallocation request denied" if (allocated(arrayInput2)) deallocate(arrayInput2, stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput2: Deallocation request denied" if (allocated(arrayInput1)) deallocate(arrayInput1, stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput1: Deallocation request denied" ! 9-6 b allocate(arrayInput1(4, 3), stat=errorLevel) ! 分配数组 if (errorLevel /= 0) print *, "arrayInput1: Allocation request denied" allocate(arrayInput2(3, 1), stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput2: Allocation request denied" allocate(arrayResult(4, 1), stat=errorLevel) if (errorLevel /= 0) print *, "arrayResult: Allocation request denied" arrayInput1(1, :) = (/ 1., -1., -2./) ! 定义数组 arrayInput1(2, :) = (/ 2., 2., 0./) arrayInput1(3, :) = (/ 3., 3., 3./) arrayInput1(4, :) = (/ 5., 4., 4./) arrayInput2(:, 1) = (/-2., 5., 2./) call MatrixMultplication(arrayInput1, 4, 3, arrayInput2, 3, 1, arrayResult, errorLevel) ! 调用子程序计算矩阵乘积 if (errorLevel /= 0) print *, "无法进行矩阵乘法" write(*,'(F8.2)') (arrayResult(i, :), i = 1, 4) ! 输出计算结果 arrayResult(:,:) = matmul(arrayInput1, arrayInput2) ! 调用内部函数计算矩阵乘积 write(*,'(F8.2)') (arrayResult(i, :), i = 1, 4) ! 输出计算结果 call MatrixMul(arrayInput1, arrayInput2, arrayResult, errorLevel) ! 调用子程序计算矩阵乘积 if (errorLevel /= 0) print *, "无法进行矩阵乘法" write(*,'(F8.2)') (arrayResult(i, :), i = 1, 4) ! 输出计算结果 if (allocated(arrayResult)) deallocate(arrayResult, stat=errorLevel) ! 释放数组 if (errorLevel /= 0) print *, "arrayResult: Deallocation request denied" if (allocated(arrayInput2)) deallocate(arrayInput2, stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput2: Deallocation request denied" if (allocated(arrayInput1)) deallocate(arrayInput1, stat=errorLevel) if (errorLevel /= 0) print *, "arrayInput1: Deallocation request denied" contains ! 矩阵乘法子程序, 显式结构 subroutine MatrixMultplication(array1, x1, y1, array2, x2, y2, array3, errorLevel) integer, intent(in) :: x1, y1, x2, y2 ! 矩阵大小 real, dimension(x1, y1), intent(in) :: array1 ! 矩阵1 real, dimension(x2, y2), intent(in) :: array2 ! 矩阵2 real, dimension(x1, y2), intent(out) :: array3 ! 输出矩阵 integer, intent(out) :: errorLevel ! 错误代码 integer :: i, j, k ! 循环参数 if ( y1 /= x2 ) then ! 判断乘法是否可行 errorLevel = 1 ! 不可行, 错误码=1 return ! 退出 end if array3 = 0. ! 初始化输出矩阵 ! forall(i = 1:x1, j = 1:y1, k = 1:y2) ! forall结构计算 ! array3(i, k) = array1(i, j)*array2(j, k) + array3(i, k) ! end forall do i = 1, x1 ! do嵌套计算 do j = 1, y1 do k = 1, y2 array3(i, k) = array1(i, j)*array2(j, k) + array3(i, k) end do end do end do errorLevel = 0 ! 运算完成, 错误码=0 end subroutine MatrixMultplication ! 矩阵乘法子程序, 显式接口不定结构 subroutine MatrixMul(array1, array2, array3, errorLevel) real, dimension(:, :), intent(in) :: array1 ! 矩阵1 real, dimension(:, :), intent(in) :: array2 ! 矩阵2 real, dimension(:, :), intent(out) :: array3 ! 输出矩阵 integer, intent(out) :: errorLevel ! 错误代码 integer :: i, j, k ! 循环参数 if ( size(array1, 2) /= size(array2, 1) ) then ! 判断乘法是否可行 errorLevel = 1 ! 不可行, 错误码=1 return ! 退出 end if array3 = 0. ! 初始化输出矩阵 ! forall(i = 1:x1, j = 1:y1, k = 1:y2) ! forall结构计算 ! array3(i, k) = array1(i, j)*array2(j, k) + array3(i, k) ! end forall do i = 1, size(array1, 1) ! do嵌套计算 do j = 1, size(array1, 2) do k = 1, size(array2, 2) array3(i, k) = array1(i, j)*array2(j, k) + array3(i, k) end do end do end do errorLevel = 0 ! 运算完成, 错误码=0 end subroutine MatrixMul end subroutine Exercises9_6 ! 习题9-8 subroutine Exercises9_8 implicit none real, dimension(3, 3) :: arrayInput1, arrayInput2 ! 输入矩阵 real, dimension(3) :: arrayX1, arrayX2 ! 常数向量 integer :: errorLevel ! 错误码 integer :: i ! 循环参数 arrayInput1(1, :) = (/ 1.0, 1.0, 1.0/) ! 输入数据1 arrayInput1(2, :) = (/ 2.0, 1.0, 1.0/) arrayInput1(3, :) = (/ 1.0, 3.0, 2.0/) arrayX1(:) = (/ 1.0, 2.0, 4.0/) arrayInput2(1, :) = (/ 1.0, 1.0, 1.0/) ! 输入数据2 arrayInput2(2, :) = (/ 2.0, 6.0, 4.0/) arrayInput2(3, :) = (/ 1.0, 3.0, 2.0/) arrayX2(:) = (/ 1.0, 8.0, 4.0/) call GAEli(arrayInput1, arrayX1, errorLevel) ! 调用子程序解方程组 if (errorLevel /= 0) then write(*,*) 'Error!' ! 如果无解, 输出错误 else write(*,'(4F8.2)') (arrayInput1(i, :), arrayX1(i), i = 1, 3) ! 如果有解, 输出计算后的矩阵 end if call GAEli(arrayInput2, arrayX2, errorLevel) ! 调用子程序解方程组 if (errorLevel /= 0) then write(*,*) 'Error!' ! 如果无解, 输出错误 else write(*,'(4F8.2)') (arrayInput2(i, :), arrayX2(i), i = 1, 3) ! 如果有解, 输出计算后的矩阵 end if contains ! 高斯亚当消元法, 不定结构, 破坏输入矩阵 subroutine GAEli(matrixInput, arrayX, errorLevel) implicit none real, dimension(:, :), intent(inout) :: matrixInput ! 输入矩阵(输入) real, dimension(:), intent(inout) :: arrayX ! 输入常数向量, 输出X解向量 integer, intent(out) :: errorLevel ! 错误值 real :: temp ! 临时值, 用于数值交换 integer :: maxLocate ! 最大值所在行 integer :: i, n, m ! 局部变量: 循环参数 m = size(matrixInput, 1) do n = 1, m maxLocate = n ! 初始化绝对值最大的位置 do i = n+1, m ! 执行最大支点技术，减少舍入误差 if ( abs(matrixInput(i, n)) > abs(matrixInput(maxLocate, n)) ) then maxLocate = i end if end do if ( abs(matrixInput(maxLocate, n)) < 1.0E-30 ) then errorLevel = 1 ! 如果绝对值最大为0, 则矩阵不满秩, 无解 return ! 退出运算 end if if ( maxLocate /= n ) then ! 如果绝对值最大值位置发生了变化, 则交换 do i = 1, m temp = matrixInput(maxLocate, i) matrixInput(maxLocate, i) = matrixInput(n, i) matrixInput(n, i) = temp end do temp = arrayX(maxLocate) arrayX(maxLocate) = arrayX(n) arrayX(n) = temp end if do i = 1, m ! 消去其他行所有第n列的系数 if ( i /= n ) then temp = -matrixInput(i, n)/matrixInput(n, n) ! 这里务必要设置一个临时值保存数值, 不然会被修改 matrixInput(i, :) = matrixInput(i, :) + matrixInput(n, :)*temp arrayX(i) = arrayX(i) + arrayX(n)*temp end if end do end do do i = 1, m ! 产生X的解向量 arrayX(i) = arrayX(i)/matrixInput(i, i) matrixInput(i, i) = 1. end do errorLevel = 0 end subroutine GAEli end subroutine Exercises9_8 ! 习题9-8 subroutine Exercises9_9 implicit none real, dimension(3, 3) :: arrayInput1, arrayInput2 ! 输入矩阵 real, dimension(3) :: arrayX1, arrayX2 ! 常数向量 integer :: errorLevel ! 错误码 integer :: i ! 循环参数 arrayInput1(1, :) = (/ 1.0, 1.0, 1.0/) ! 输入数据1 arrayInput1(2, :) = (/ 2.0, 1.0, 1.0/) arrayInput1(3, :) = (/ 1.0, 3.0, 2.0/) arrayX1(:) = (/ 1.0, 2.0, 4.0/) arrayInput2(1, :) = (/ 1.0, 1.0, 1.0/) ! 输入数据2 arrayInput2(2, :) = (/ 2.0, 6.0, 4.0/) arrayInput2(3, :) = (/ 1.0, 3.0, 2.0/) arrayX2(:) = (/ 1.0, 8.0, 4.0/) call GAEli1(arrayInput1, arrayX1, errorLevel) ! 调用子程序解方程组 if (errorLevel /= 0) then write(*,*) 'Error!' ! 如果无解, 输出错误 else write(*,'(A, I1, A, F6.2)') ('x', i, ' = ', arrayX1(i), i = 1, 3) ! 如果有解, 输出各解 end if call GAEli1(arrayInput2, arrayX2, errorLevel) ! 调用子程序解方程组 if (errorLevel /= 0) then write(*,*) 'Error!' ! 如果无解, 输出错误 else write(*,'(A, I1, F6.2)') ('x', i, ' = ', arrayX2(i), i = 1, 3) ! 如果有解, 输出各解 end if contains ! 高斯-亚当消元法，不破坏输入矩阵 subroutine GAEli1(matrixInput1, arrayX, errorLevel) implicit none real, dimension(:, :), intent(inout) :: matrixInput1 ! 输入矩阵(输入) real, dimension(size(matrixInput1, 1), size(matrixInput1, 2)) :: matrixInput ! 运算矩阵 real, dimension(:), intent(inout) :: arrayX ! 输入常数向量, 输出X解向量 integer, intent(out) :: errorLevel ! 错误值 real :: temp ! 临时值, 用于数值交换 integer :: maxLocate ! 最大值所在行 integer :: i, n, m ! 局部变量: 循环参数 matrixInput = matrixInput1 m = size(matrixInput, 1) do n = 1, m maxLocate = n ! 初始化绝对值最大的位置 do i = n+1, m ! 执行最大支点技术，减少舍入误差 if ( abs(matrixInput(i, n)) > abs(matrixInput(maxLocate, n)) ) then maxLocate = i end if end do if ( abs(matrixInput(maxLocate, n)) < 1.0E-30 ) then errorLevel = 1 ! 如果绝对值最大为0, 则矩阵不满秩, 无解 return ! 退出运算 end if if ( maxLocate /= n ) then ! 如果绝对值最大值位置发生了变化, 则交换 do i = 1, m temp = matrixInput(maxLocate, i) matrixInput(maxLocate, i) = matrixInput(n, i) matrixInput(n, i) = temp end do temp = arrayX(maxLocate) arrayX(maxLocate) = arrayX(n) arrayX(n) = temp end if do i = 1, m ! 消去其他行所有第n列的系数 if ( i /= n ) then temp = -matrixInput(i, n)/matrixInput(n, n) ! 这里务必要设置一个临时值保存数值, 不然会被修改 matrixInput(i, :) = matrixInput(i, :) + matrixInput(n, :)*temp arrayX(i) = arrayX(i) + arrayX(n)*temp end if end do end do do i = 1, m ! 产生X的解向量 arrayX(i) = arrayX(i)/matrixInput(i, i) matrixInput(i, i) = 1. end do errorLevel = 0 end subroutine GAEli1 end subroutine Exercises9_9 ! 习题9-11 9-12 subroutine Exercises9_11 implicit none real, allocatable, dimension(:) :: a integer :: istat allocate(a(6), stat=istat) a = (/ 1., 2., 3., 4., 5., 6./) write(*,'(A, 6F4.1)') 'Main: Array a before call = ', a call test_alloc(a) write(*,'(A, 6F4.1)') 'Main: Array a after call = ', a contains subroutine test_alloc(array) real, dimension(:), allocatable, intent(inout) :: array integer :: i integer :: istat if (allocated(array)) then write(*,'(A)') 'Sub: the array is allocated' write(*,'(A, 6F4.1)') 'Sub: Array on entry = ', array else write(*,*) 'In sub: the array is not allocated' end if if (allocated(array)) then deallocate(array, stat=istat) end if allocate(array(5), stat=istat) do i = 1, 5 array(i) = 6 - i end do write(*,'(A, 6F4.1)') 'Sub: Array on exit = ', array end subroutine test_alloc end subroutine Exercises9_11 ! 习题9-13 掷色子 subroutine Exercises9_13 implicit none integer :: x, y ! 定义俩变量 call throw(x, y) ! 获取俩随机值 write(*,*) x, y ! 打印 contains subroutine throw(random1, random2) ! out属性形参可以作为参数传递给过程 integer, intent(out) :: random1, random2 ! 定义俩变量储存随机数 call die(random1) ! 获取第一个随机数 call die(random2) ! 获取第二个随机数 end subroutine throw subroutine die(random) integer, intent(out) :: random ! 定义随机整数 real :: temp ! 随机数临时值 call random_seed() ! 根据时间日期获取随机数种子 call random_number(temp) ! 获取一个[0,1)的随机数给临时值 random = int(temp*6) + 1 ! 变化其区间为[1,7), 并向下取整 end subroutine die end subroutine Exercises9_13 ! 创建逐元过程运算, 这种过程内函数, 直接在函数体写定义, 调用部分可以不考虑 subroutine Exercises9_14 implicit none real, parameter :: PI=3.14159265 ! PI值 real, dimension(2, 3) :: array1, array2, array3 ! 计算用数组 integer :: i ! 循环参数 array1(1, :) = (/ 10.0, 20.0, 30.0/) ! 定义输入数组 array1(2, :) = (/ 40.0, 50.0, 60.0/) array2 = FunSin(array1) ! 计算值 write(*,1) '计算得到正弦值为:', (array2(i, :), i = 1, 2) ! 输出计算值, 以下相同 array3 = FunAsin(array2) write(*,1) '计算得到反正弦值为:', (array3(i, :), i = 1, 2) array2 = FunCos(array1) write(*,1) '计算得到余弦值为:', (array2(i, :), i = 1, 2) array3 = FunAcos(array2) write(*,1) '计算得到反余弦值为:', (array3(i, :), i = 1, 2) array2 = FunTan(array1) write(*,1) '计算得到正切为:', (array2(i, :), i = 1, 2) array3 = FunAtan(array2) write(*,1) '计算得到反正切为:', (array3(i, :), i = 1, 2) 1 format(1X, A,/& ! 输出格式 &1X, 3F8.2,/& &1X, 3F8.2) contains elemental function FunSin(degree) ! 定义函数 real, intent(in) :: degree ! 定义输入变量 real :: FunSin ! 定义函数输出类型 FunSin = sin(degree/360.*2*PI) ! 计算输出值, 以下相同 end function FunSin elemental function FunCos(degree) real, intent(in) :: degree real :: FunCos FunCos = cos(degree/360.*2*PI) end function FunCos elemental function FunTan(degree) real, intent(in) :: degree real :: FunTan FunTan = tan(degree/360.*2*PI) end function FunTan elemental function FunAsin(value) real, intent(in) :: value real :: FunAsin FunAsin = asin(value)*360./(2.*PI) end function FunAsin elemental function FunAcos(value) real, intent(in) :: value real :: FunAcos FunAcos = acos(value)*360./(2.*PI) end function FunAcos elemental function FunAtan(value) real, intent(in) :: value real :: FunAtan FunAtan = atan(value)*360./(2.*PI) end function FunAtan end subroutine Exercises9_14 ! 习题9-15 将9-14的内容修改为纯函数 subroutine Exercises9_15 implicit none real, parameter :: PI=3.14159265 ! PI值 real, dimension(2, 3) :: array1, array2, array3 ! 计算用数组 integer :: i ! 循环参数 array1(1, :) = (/ 10.0, 20.0, 30.0/) ! 定义输入数组 array1(2, :) = (/ 40.0, 50.0, 60.0/) array2 = FunSin(array1) ! 计算值 write(*,1) '计算得到正弦值为:', (array2(i, :), i = 1, 2) ! 输出计算值, 以下相同 array3 = FunAsin(array2) write(*,1) '计算得到反正弦值为:', (array3(i, :), i = 1, 2) array2 = FunCos(array1) write(*,1) '计算得到余弦值为:', (array2(i, :), i = 1, 2) array3 = FunAcos(array2) write(*,1) '计算得到反余弦值为:', (array3(i, :), i = 1, 2) array2 = FunTan(array1) write(*,1) '计算得到正切为:', (array2(i, :), i = 1, 2) array3 = FunAtan(array2) write(*,1) '计算得到反正切为:', (array3(i, :), i = 1, 2) 1 format(1X, A,/& ! 输出格式 &1X, 3F8.2,/& &1X, 3F8.2) contains pure function FunSin(degree) real, intent(in), dimension(:, :) :: degree real, dimension(size(degree, 1), size(degree, 2)) :: FunSin FunSin = sin(degree/360.*2*PI) end function FunSin pure function FunCos(degree) real, intent(in), dimension(:, :) :: degree real, dimension(size(degree, 1), size(degree, 2)) :: FunCos FunCos = cos(degree/360.*2*PI) end function FunCos pure function FunTan(degree) real, intent(in), dimension(:, :) :: degree real, dimension(size(degree, 1), size(degree, 2)) :: FunTan FunTan = tan(degree/360.*2*PI) end function FunTan pure function FunAsin(value) real, intent(in), dimension(:, :) :: value real, dimension(size(value, 1), size(value, 2)) :: FunAsin FunAsin = asin(value)*360./(2.*PI) end function FunAsin pure function FunAcos(value) real, intent(in), dimension(:, :) :: value real, dimension(size(value, 1), size(value, 2)) :: FunAcos FunAcos = acos(value)*360./(2.*PI) end function FunAcos pure function FunAtan(value) real, intent(in), dimension(:, :) :: value real, dimension(size(value, 1), size(value, 2)) :: FunAtan FunAtan = atan(value)*360./(2.*PI) end function FunAtan end subroutine Exercises9_15 ! 习题9-16 高阶最小二乘回归 首先第一个，发现了解多元一次方程组程序的错误，第二，发现了总之，很多很多错误，很有意义的一道题 subroutine Exercises9_16 implicit none real, dimension(:), allocatable :: pointX, pointY ! 坐标(x, y) real, dimension(:), allocatable :: arrayCoefficient ! 系数向量 integer :: errorLevel ! 错误码 integer :: i ! 循环参数 real :: temp open(unit=1, status='scratch', iostat=errorLevel) ! 打开临时文件 if ( errorLevel /= 0 ) stop "打开临时文件失败" ! 判断打开状态 write(1, '(A)') & ! 输入数据 &' 0.167, 0.333, 0.500, 0.667, 0.833, 1.000, 1.167, 1.333, 1.500& &, 1.667, 1.833, 2.000, 2.167, 2.333, 2.500, 2.667, 2.833, 3.000',& &' 49.9, 52.2, 50.6, 47.0, 47.7, 42.3, 37.9, 38.2, 38.0, 33.8, 26.7, 24.8, 22.0, 16.5, 14.0, 5.6, 2.9, 0.8',& &' -5.1,-12.9,-15.1, -6.8,-12.3,-18.0, -5.7, -6.3,-12.7,-13.7,-26.7,-31.3,-22.9,-25.6,-25.7,-25.2,-35.0,-27.9' rewind(unit=1, iostat=errorLevel) ! 回到初始行 if ( errorLevel /= 0 ) stop "读取临时文件失败" ! 判断执行状态 allocate(pointX(18), stat=errorLevel) ! 分配数组 if (errorLevel /= 0) print *, "pointX: Allocation request denied" ! 判断分配结果, 下同 allocate(pointY(18), stat=errorLevel) if (errorLevel /= 0) print *, "pointY: Allocation request denied" allocate(arrayCoefficient(3), stat=errorLevel) if (errorLevel /= 0) print *, "arrayCoefficient: Allocation request denied" read(1, *) pointX, pointY ! 读取数据 close(unit=1, iostat=errorLevel) ! 关闭临时文件 if ( errorLevel /= 0 ) stop "关闭临时文件失败" ! 判断关闭状态 ! write(*,'(18(F8.2, 1X))') pointX, pointY ! 输出读入值(供检验, 可注释) call MultipleRegression(pointX, pointY, arrayCoefficient, 2, errorLevel) ! 计算参数值 write(*,*) '习题9-16' write(*,1) arrayCoefficient ! 输出结果 1 format(1X, '(?) 计算得到的方程为：h = ', F8.2, ' + ', F8.2, ' t + ', F8.2, ' t^2') ! 习题9-17 write(*,*) '习题9-17' call ShowNoise2( 0.0, 0.0, '(a)') call ShowNoise2(-0.1, 0.1, '(b)') call ShowNoise2(-0.5, 0.5, '(c)') call ShowNoise2(-1.0, 1.0, '(d)') ! 习题9-19 write(*,*) '习题9-19' arrayCoefficient = (/0.0, 0.0, 0.0, 0.0, 0.0/) ! 重定义系数向量长度 call ShowNoise4( 0.0, 0.0, '(a)') call ShowNoise4(-0.1, 0.1, '(b)') call ShowNoise4(-0.5, 0.5, '(c)') call ShowNoise4(-1.0, 1.0, '(d)') ! 习题9-20 write(*,*) '习题9-20' open(unit=1, status='scratch', iostat=errorLevel) ! 打开临时文件 if ( errorLevel /= 0 ) stop "打开临时文件失败" ! 判断打开状态 write(1, '(A)') & ! 输入数据 &' 0.00, 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00, 10.00',& &'-23.22,-13.54, -4.14, -0.04, 3.92, 4.97, 3.96, -0.07, -5.67,-12.28,-20.25' rewind(unit=1, iostat=errorLevel) ! 回到初始行 if ( errorLevel /= 0 ) stop "关闭临时文件失败" ! 判断关闭状态 pointX = (/(i, i = 1, 10)/) ! 重定义坐标x向量长度 pointY = pointX ! 重定义坐标y向量长度 read(1,*) pointX, pointY ! 赋值坐标(x, y) close(unit=1, iostat=errorLevel) ! 关闭临时文件 arrayCoefficient = (/0.0, 0.0, 0.0/) ! 重定义系数向量长度 call MultipleRegression(pointX, pointY, arrayCoefficient, 2, errorLevel) ! 计算参数值 write(*,4) arrayCoefficient ! 输出结果 4 format(1X, '(?) 计算得到的方程为：y = ', F8.2, ' + ', F8.2, ' x + ', F8.2, ' x^2') write(*,*) ' 当x0取3.5时，得到y0 = ', arrayCoefficient(1) + arrayCoefficient(2)*3.5 + arrayCoefficient(3)*3.5**2 ! 习题9-21 write(*,*) '习题9-21' open(unit=1, status='scratch', iostat=errorLevel) ! 打开临时文件 if ( errorLevel /= 0 ) stop "打开临时文件失败" ! 判断打开状态 write(1, '(A)') & ! 输入数据 &' 0.00, 1.00, 2.00, 3.00, 4.00, 5.00, 6.00, 7.00, 8.00, 9.00, 10.00',& &'-14.22,-10.54, -5.09, -3.12, 0.92, 3.79, 6.99, 8.95, 11.33, 14.71, 18.75' rewind(unit=1, iostat=errorLevel) ! 回到初始行 if ( errorLevel /= 0 ) stop "关闭临时文件失败" ! 判断关闭状态 pointX = (/(i, i = 1, 10)/) ! 重定义坐标x向量长度 pointY = pointX ! 重定义坐标y向量长度 read(1,*) pointX, pointY ! 赋值坐标(x, y) close(unit=1, iostat=errorLevel) ! 关闭临时文件 arrayCoefficient = (/0.0, 0.0, 0.0/) ! 重定义系数向量长度 call MultipleRegression(pointX, pointY, arrayCoefficient, 2, errorLevel) ! 计算参数值 write(*,4) arrayCoefficient ! 输出结果 write(*,*) ' 当x0取14.0时，得到y0 = ', arrayCoefficient(1) + arrayCoefficient(2)*14.0 + arrayCoefficient(3)*14.0**2 if ( errorLevel /= 0 ) stop "读取临时文件失败" ! 判断执行状态 if (allocated(arrayCoefficient)) deallocate(arrayCoefficient, stat=errorLevel) ! 释放数组 if (errorLevel /= 0) print *, "arrayCoefficient: Deallocation request denied" ! 判断释放结果, 下同 if (allocated(pointY)) deallocate(pointY, stat=errorLevel) if (errorLevel /= 0) print *, "pointY: Deallocation request denied" if (allocated(pointX)) deallocate(pointX, stat=errorLevel) if (errorLevel /= 0) print *, "pointX: Deallocation request denied" contains ! 构建输入方程和解向量 subroutine MultipleRegression(arrayX, arrayY, arrayResult, order, errorLevel) real, dimension(:), intent(in) :: arrayX, arrayY ! 输入坐标(x, y) integer, intent(in) :: order ! 拟合阶数 real, dimension(order+1), intent(out) :: arrayResult ! 系数向量(同时也充当了常数向量) real, dimension(order+1, order+1) :: matrixCoefficient ! 系数矩阵 integer, intent(out) :: errorLevel ! 错误码(虽然主程序中没有做错误校验就是) integer :: i, j ! 循环参数 if ( order < 1 ) then ! 判断阶数是否符合要求 errorLevel = 1 return else do i = 1, order+1 ! 各参数的偏导方程 do j = 1, order+1 ! 某一参数偏导数取0时, 各系数 matrixCoefficient(i, j) = sum(arrayX**(i+j-2)) ! 给各系数赋值 end do arrayResult(i) = sum(arrayX**(i-1)*arrayY) ! 给常数向量赋值 end do ! write(*,'(3(F8.2, 1X))') matrixCoefficient, arrayResult ! 打印系数矩阵和常数向量(共检验, 可注释) call GAEli1(matrixCoefficient, arrayResult, errorLevel) ! 计算系数值 end if end subroutine MultipleRegression ! 调用方程组求解子程序，高斯-亚当消元法（不破坏输入数组） subroutine GAEli1(matrixInput1, arrayX, errorLevel) implicit none real, dimension(:, :), intent(inout) :: matrixInput1 ! 输入矩阵(输入) real, dimension(size(matrixInput1, 1), size(matrixInput1, 2)) :: matrixInput ! 运算矩阵 real, dimension(:), intent(inout) :: arrayX ! 输入常数向量, 输出X解向量 integer, intent(out) :: errorLevel ! 错误值 real :: temp ! 临时值, 用于数值交换 integer :: maxLocate ! 最大值所在行 integer :: i, n, m ! 局部变量: 循环参数 matrixInput = matrixInput1 m = size(matrixInput, 1) do n = 1, m maxLocate = n ! 初始化绝对值最大的位置 do i = n+1, m ! 执行最大支点技术，减少舍入误差 if ( abs(matrixInput(i, n)) > abs(matrixInput(maxLocate, n)) ) then maxLocate = i end if end do if ( abs(matrixInput(maxLocate, n)) < 1.0E-30 ) then errorLevel = 1 ! 如果绝对值最大为0, 则矩阵不满秩, 无解 return ! 退出运算 end if if ( maxLocate /= n ) then ! 如果绝对值最大值位置发生了变化, 则交换 do i = 1, m temp = matrixInput(maxLocate, i) matrixInput(maxLocate, i) = matrixInput(n, i) matrixInput(n, i) = temp end do temp = arrayX(maxLocate) arrayX(maxLocate) = arrayX(n) arrayX(n) = temp end if do i = 1, m ! 消去其他行所有第n列的系数 if ( i /= n ) then temp = -matrixInput(i, n)/matrixInput(n, n) ! 这里务必要设置一个临时值保存数值, 不然会被修改 matrixInput(i, :) = matrixInput(i, :) + matrixInput(n, :)*temp arrayX(i) = arrayX(i) + arrayX(n)*temp end if end do end do do i = 1, m ! 产生X的解向量 arrayX(i) = arrayX(i)/matrixInput(i, i) matrixInput(i, i) = 1. end do errorLevel = 0 end subroutine GAEli1 ! 用于产生噪声值 subroutine Noise(value, lowerBound, upperBound) real, intent(out) :: value ! 输出的随机数值 real, intent(in) :: lowerBound, upperBound ! 输出随机数值的上下界, 范围是[lowerBound, upperBound) call random_seed() ! 根据当前时间日期计算随机数种子 call random_number(value) ! 计算随机数 value = lowerBound + value*( upperBound - lowerBound ) ! 重分布以适应范围 end subroutine Noise ! 用于计算习题9-17, 这里采用的是二阶回归 subroutine ShowNoise2(lowerBound, upperBound, tag) real, intent(in) :: lowerBound, upperBound ! 噪声值上下界 character(len=3), intent(in) :: tag ! 当前题目标识符 pointX = (/0.0, 1.0, 2.0, 3.0, 4.0, 5.0/) ! 赋值坐标x do i = 1, 6 call Noise(temp, lowerBound, upperBound) ! 产生噪声值 pointY(i) = pointX(i)**2 - 4*pointX(i) + 3 + temp ! 赋值坐标y end do call MultipleRegression(pointX, pointY, arrayCoefficient, 2, errorLevel)! 计算参数值 write(*,2) trim(tag), arrayCoefficient ! 输出结果 2 format(1X, A,' 计算得到的方程为：h = ', F8.2, ' + ', F8.2, ' t + ', F8.2, ' t^2') end subroutine ShowNoise2 ! 用于计算习题9-19, 这里采用的是四阶回归 subroutine ShowNoise4(lowerBound, upperBound, tag) real, intent(in) :: lowerBound, upperBound ! 噪声值上下界 character(len=3), intent(in) :: tag ! 当前题目标识符 pointX = (/0.0, 1.0, 2.0, 3.0, 4.0, 5.0/) ! 赋值坐标x do i = 1, 6 call Noise(temp, lowerBound, upperBound) ! 产生噪声值 pointY(i) = pointX(i)**4 - 3*pointX(i)**3 - 4*pointX(i)**2 + 2*pointX(i) + 3 + temp end do ! 赋值坐标y call MultipleRegression(pointX, pointY, arrayCoefficient, 4, errorLevel)! 计算参数值 write(*,3) trim(tag), arrayCoefficient ! 输出结果 3 format(1X, A,' 计算得到的方程为：h = ', F8.2, ' + ', F8.2, ' t + ', F8.2,& & ' t^2 + ', F8.2, ' t^3 + ', F8.2, ' t^4') end subroutine ShowNoise4 end subroutine Exercises9_16