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計算物理学2 第2回演習
ver. 2018/4/19 以下の例題を Fortran もしくは C 言語を用いてプログラム化せよ。
四則演算
(1) 標準入力から実数 a と b を読み込み、その和、差、積、商を標準出力に出力。
・b にゼロを代入するとどうなりますか。
DO ループ
(2) および の N=1 から 30 までの値。
(3) N! (階乗)の値(N が 1 から 30 まで)
・N がいくつまで正確に計算できますか。
(4) 整数 N=1,...30 に対して単精度と倍精度で N の値を出力(sqrt(x)関数を使う)。
配列
(5) 行列 A(1:3,1:3)と B(1:3,1:3)の積の計算(Fortran の場合は MATMUL を使う場合と 使わない場合二通り)。
ただし A の行列要素 a
ijは b
ijは とする。
(6) n が 5 までのルジャンドル多項式の直交性を以下の方法で評価しなさい。
まず区間[-1,1]を N-1 等分し、それぞれの座標を要素数 N の配列 x(1:N)にセットする。端 点を x(1)=-1, x(N)=1 として残りの点 x(2), x(3), ...., x(N)も求めて値を配列にセット。
配 列 P(1:N,0:5) を 用 意 し 、 同 じ [-1,1] 区 間 で ル ジ ャ ン ド ル 多 項 式 P
0(x), P
1(x), P
2(x),P
3(x),P
4(x),P
5(x)の値をセット。(例えば x=x(i)でのルジャンドル多項式 P
2(x(i))の値 を配列 P(i,2)に代入) ただし n=0,1,2,3,4,5 のルジャンドル多項式は以下で与えられる。
X
N k=1k
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X
Nk=1
k
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a
ij= e
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b
ij= e
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2
ルジャンドル多項式の配列を用いて、区分求積法で次の積分を計算する。
Δx は N から計算できる。I
nmを 6x6 の行列として出力。
・N の値を 100, 1000,10000 と変えるとどうなるか。
・区分求積の和を N-1 までにした時と N までにした時で何が変わるか。
(7) 標準入力から2つの角度 a と b を実数で読み取り(単位:度)、cos(a+b)と sin(a+b) を 3 通りの方法で計算。
1. cos(a+b), sin(a+b)をそのまま出力(ラジアンに変換)。
2. 角度 a 回転させる 2 次元の行列、角度 b 回転させる行列を配列を使って作 り、それらの行列の積を計算して cos(a+b)と sin(a+b)に対応する成分を出力。ただし角
度θ回転させる回転行列は で与えられる。
3. 複素数 e
iaと e
ibの積を計算し、その実部と虚部から出力。
P0(x) =1 P1(x) =x P2(x) =1
2(3x2 1) P3(x) =1
2(5x3 3x) P4(x) =1
8(35x4 30x2+ 3) P5(x) =1
8(63x5 70x3+ 15x)
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Inm = Z 1
1
Pn(x)Pm(x)dx⇠
NX1 i=1
Pn(x(i))Pm(x(i)) x
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✓cos✓ sin✓ sin✓ cos✓
◆
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