42.2 BLAS-like Primitives: AXPY, DOT, GEMV

Core building blocks for linear algebra mirror BLAS patterns: AXPY (y ← a·x + y), DOT (dot product), and GEMV (matrix–vector).

42.2.1 AXPY

void axpy(float a, float[] x, float[] y) {
  for (int i = 0; i < x.length; i++) y[i] += a * x[i];
}

42.2.2 DOT (Dot Product)

float dot(float[] x, float[] y) {
  float acc = 0f;
  for (int i = 0; i < x.length; i++) acc += x[i] * y[i];
  return acc;
}

Use Kahan summation to reduce floating‑point error when needed:

float dotKahan(float[] x, float[] y) {
  float sum = 0f, c = 0f; // compensation
  for (int i = 0; i < x.length; i++) {
    float prod = x[i] * y[i];
    float yk = prod - c;
    float t = sum + yk;
    c = (t - sum) - yk;
    sum = t;
  }
  return sum;
}

42.2.3 GEMV (Matrix–Vector)

Row‑major matrix times vector:

void gemv(float[] A, int M, int N, float[] x, float[] y) {
  for (int i = 0; i < M; i++) {
    float acc = 0f;
    int base = i * N;
    for (int j = 0; j < N; j++) acc += A[base + j] * x[j];
    y[i] = acc;
  }
}

Transpose GEMV (A^T · x) for column‑wise operations:

void gemvT(float[] A, int M, int N, float[] x, float[] y) {
  java.util.Arrays.fill(y, 0f);
  for (int i = 0; i < M; i++) {
    int base = i * N;
    float xi = x[i];
    for (int j = 0; j < N; j++) y[j] += A[base + j] * xi;
  }
}